Kahan p9 Example

Average Accuracy: 68.5% → 99.9%

Time: 12.5s

Precision: binary64

Cost: 13632

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]

↓

\[\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

↓

(FPCore (x y)
 :precision binary64
 (* (/ (- x y) (hypot x y)) (/ (+ x y) (hypot x y))))

C

double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

↓

double code(double x, double y) {
	return ((x - y) / hypot(x, y)) * ((x + y) / hypot(x, y));
}

Java

public static double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

↓

public static double code(double x, double y) {
	return ((x - y) / Math.hypot(x, y)) * ((x + y) / Math.hypot(x, y));
}

Python

def code(x, y):
	return ((x - y) * (x + y)) / ((x * x) + (y * y))

↓

def code(x, y):
	return ((x - y) / math.hypot(x, y)) * ((x + y) / math.hypot(x, y))

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
end

↓

function code(x, y)
	return Float64(Float64(Float64(x - y) / hypot(x, y)) * Float64(Float64(x + y) / hypot(x, y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
end

↓

function tmp = code(x, y)
	tmp = ((x - y) / hypot(x, y)) * ((x + y) / hypot(x, y));
end

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	68.5%
Target	99.9%
Herbie	99.9%

\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \]

Derivation

1. Initial program 68.5%

   \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]

2. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}} \]
   Proof

   [Start] 68.5	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   add-sqr-sqrt [=>] 68.5	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}} \]
   times-frac [=>] 68.5	\[ \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]
   hypot-def [=>] 68.5	\[ \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}} \]
   hypot-def [=>] 99.9	\[ \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \]

3. Final simplification 99.9%

   \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \]

Alternatives

Alternative 1
Accuracy	93.0%
Cost	2116

\[\begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ t_1 := \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(t_1 + -1\right)\\ \end{array} \]

Alternative 2
Accuracy	92.9%
Cost	1988

\[\begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \end{array} \]

Alternative 3
Accuracy	83.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{-149}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-190}:\\ \;\;\;\;1 + \frac{y \cdot -2}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \end{array} \]

Alternative 4
Accuracy	83.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-148} \lor \neg \left(y \leq 8.8 \cdot 10^{-191}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Accuracy	83.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-144} \lor \neg \left(y \leq 5.8 \cdot 10^{-190}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \end{array} \]

Alternative 6
Accuracy	83.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-141} \lor \neg \left(y \leq 9.6 \cdot 10^{-191}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{y}{x}\right) - \frac{y}{x}\\ \end{array} \]

Alternative 7
Accuracy	83.3%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{-148}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-190}:\\ \;\;\;\;\left(1 + \frac{y}{x}\right) - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \end{array} \]

Alternative 8
Accuracy	82.8%
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-149}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9
Accuracy	66.6%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :herbie-target
  (if (and (< 0.5 (fabs (/ x y))) (< (fabs (/ x y)) 2.0)) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))