Kahan p9 Example

Percentage Accurate: 66.0% → 99.9%

Time: 14.9s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (/ (+ x y) (* (hypot x y) (/ (hypot x y) (- 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) * (hypot(x, y) / (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) * (Math.hypot(x, y) / (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) * (math.hypot(x, y) / (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(x + y) / Float64(hypot(x, y) * Float64(hypot(x, y) / Float64(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) * (hypot(x, y) / (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[(x + y), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] * N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	66.0%
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 69.5%

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

2. Applied egg-rr 100.0%

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

   [Start] 69.5%	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   add-sqr-sqrt [=>] 69.5%	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}} \]
   times-frac [=>] 69.5%	\[ \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]
   hypot-def [=>] 69.5%	\[ \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}} \]
   hypot-def [=>] 100.0%	\[ \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \]

3. Applied egg-rr 100.0%

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

   [Start] 100.0%	\[ \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \]
   clear-num [=>] 100.0%	\[ \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x - y}}} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \]
   frac-times [=>] 100.0%	\[ \color{blue}{\frac{1 \cdot \left(x + y\right)}{\frac{\mathsf{hypot}\left(x, y\right)}{x - y} \cdot \mathsf{hypot}\left(x, y\right)}} \]
   *-un-lft-identity [<=] 100.0%	\[ \frac{\color{blue}{x + y}}{\frac{\mathsf{hypot}\left(x, y\right)}{x - y} \cdot \mathsf{hypot}\left(x, y\right)} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13632

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

Alternative 2
Accuracy	99.9%
Cost	13632

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

Alternative 3
Accuracy	91.8%
Cost	8004

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

Alternative 4
Accuracy	91.4%
Cost	1988

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

Alternative 5
Accuracy	84.1%
Cost	1232

\[\begin{array}{l} t_0 := -1 + \frac{x \cdot x}{y \cdot y}\\ t_1 := 1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-196}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	84.1%
Cost	1232

\[\begin{array}{l} t_0 := 1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{-159}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-195}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{y \cdot y}\\ \end{array} \]

Alternative 7
Accuracy	83.5%
Cost	1104

\[\begin{array}{l} t_0 := -1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;y \leq -2.16 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-219}:\\ \;\;\;\;\frac{x + y}{x}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-196}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-152}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	82.6%
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-165}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-219}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-196}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-128}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9
Accuracy	82.5%
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{-158}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-220}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-196}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-128}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10
Accuracy	82.5%
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{-159}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-219}:\\ \;\;\;\;\frac{x + y}{x}\\ \mathbf{elif}\;y \leq 10^{-193}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-128}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 11
Accuracy	69.3%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023263 
(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))))