Kahan p9 Example

Average Accuracy: 68.2% → 99.9%

Time: 14.8s

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	68.2%
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 68.2%

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

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] 68.2	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   add-sqr-sqrt [=>] 68.2	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}} \]
   times-frac [=>] 68.1	\[ \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]
   hypot-def [=>] 68.2	\[ \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}} \]
   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	92.6%
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 2
Accuracy	92.3%
Cost	1988

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

Alternative 3
Accuracy	92.4%
Cost	1988

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

Alternative 4
Accuracy	82.4%
Cost	1360

\[\begin{array}{l} t_0 := \frac{x - y}{\left(y - x\right) + \frac{x}{\frac{y}{x}}}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-232}:\\ \;\;\;\;1 + \frac{\frac{y}{\frac{x}{y}} \cdot -2}{x}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-159}:\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	82.4%
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-161}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-232}:\\ \;\;\;\;1 + \frac{\frac{y}{\frac{x}{y}} \cdot -2}{x}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-159}:\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{y \cdot y}\\ \end{array} \]

Alternative 6
Accuracy	81.8%
Cost	1104

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-161}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-232}:\\ \;\;\;\;\frac{y}{x} + 1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 7
Accuracy	82.3%
Cost	1104

\[\begin{array}{l} t_0 := 1 - \frac{y}{x} \cdot \frac{y}{x}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-163}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 8
Accuracy	82.4%
Cost	1104

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-161}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-232}:\\ \;\;\;\;1 + \frac{\frac{y}{\frac{x}{y}} \cdot -2}{x}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-154}:\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 9
Accuracy	81.9%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-162}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-232}:\\ \;\;\;\;\frac{y}{x} + 1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{-151}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 10
Accuracy	81.6%
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-163}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-232}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-133}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 11
Accuracy	81.6%
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-161}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-232}:\\ \;\;\;\;\frac{y}{x} + 1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 12
Accuracy	65.5%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023151 
(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))))