math.fractions: add approximation.v and tests

vlib/math/fractions/approximations.v

@@ -0,0 +1,119 @@
+// Copyright (c) 2019-2020 Alexander Medvednikov. All rights reserved.
+// Use of this source code is governed by an MIT license
+// that can be found in the LICENSE file.
+module fractions
+
+import math
+
+const (
+	default_eps    = 1.0e-4
+	max_iterations = 50
+	zero           = fraction(0, 1)
+)
+
+// ------------------------------------------------------------------------
+// Unwrapped evaluation methods for fast evaluation of continued fractions.
+// ------------------------------------------------------------------------
+// We need these functions because the evaluation of continued fractions
+// always has to be done from the end. Also, the numerator-denominator pairs
+// are generated from front to end. This means building a result from a
+// previous one isn't possible. So we need unrolled versions to ensure that
+// we don't take too much of a performance penalty by calling eval_cf
+// several times.
+// ------------------------------------------------------------------------
+// eval_1 returns the result of evaluating a continued fraction series of length 1
+fn eval_1(whole i64, d []i64) Fraction {
+	return fraction(whole * d[0] + 1, d[0])
+}
+
+// eval_2 returns the result of evaluating a continued fraction series of length 2
+fn eval_2(whole i64, d []i64) Fraction {
+	den := d[0] * d[1] + 1
+	return fraction(whole * den + d[1], den)
+}
+
+// eval_3 returns the result of evaluating a continued fraction series of length 3
+fn eval_3(whole i64, d []i64) Fraction {
+	d1d2_plus_n2 := d[1] * d[2] + 1
+	den := d[0] * d1d2_plus_n2 + d[2]
+	return fraction(whole * den + d1d2_plus_n2, den)
+}
+
+// eval_cf evaluates a continued fraction series and returns a Fraction.
+fn eval_cf(whole i64, den []i64) Fraction {
+	count := den.len
+	// Offload some small-scale calculations
+	// to dedicated functions
+	match count {
+		1 {
+			return eval_1(whole, den)
+		}
+		2 {
+			return eval_2(whole, den)
+		}
+		3 {
+			return eval_3(whole, den)
+		}
+		else {
+			last := count - 1
+			mut n := 1
+			mut d := den[last]
+			// The calculations are done from back to front
+			for index := count - 2; index >= 0; index-- {
+				t := d
+				d = den[index] * d + n
+				n = t
+			}
+			return fraction(d * whole + n, d)
+		}
+	}
+}
+
+// approximate returns a Fraction that approcimates the given value to
+// within the default epsilon value (1.0e-4). This means the result will
+// be accurate to 3 places after the decimal.
+pub fn approximate(val f64) Fraction {
+	return approximate_with_eps(val, default_eps)
+}
+
+// approximate_with_eps returns a Fraction
+pub fn approximate_with_eps(val, eps f64) Fraction {
+	if val == 0.0 {
+		return zero
+	}
+	if eps < 0.0 {
+		panic('Epsilon value cannot be negative.')
+	}
+	if math.fabs(val) > math.max_i64 {
+		panic('Value out of range.')
+	}
+	// The integer part is separated first. Then we process the fractional
+	// part to generate numerators and denominators in tandem.
+	whole := i64(val)
+	mut frac := val - whole
+	// Quick exit for integers
+	if frac == 0.0 {
+		return fraction(whole, 1)
+	}
+	mut d := []i64{}
+	mut partial := zero
+	// We must complete the approximation within the maximum number of
+	// itertations allowed. If we can't panic.
+	// Empirically tested: the hardest constant to approximate is the
+	// golden ratio (math.phi) and for f64s, it only needs 38 iterations.
+	for _ in 0 .. max_iterations {
+		// We calculate the reciprocal. That's why the numerator is
+		// always 1.
+		frac = 1.0 / frac
+		den := i64(frac)
+		d << den
+		// eval_cf is called often so it needs to be performant
+		partial = eval_cf(whole, d)
+		// Check if we're done
+		if math.fabs(val - partial.f64()) < eps {
+			return partial
+		}
+		frac -= den
+	}
+	panic("Couldn\'t converge. Please create an issue on https://github.com/vlang/v")
+}