Kahan p9 Example

Average Accuracy: 67.5% → 99.9%

Time: 16.4s

Precision: binary64

Cost: 14144

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x - y}{\mathsf{hypot}\left(x, y\right)}\\ \frac{t_0}{\frac{x - y}{t_0 \cdot \left(x + y\right)}} \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (hypot x y)))) (/ t_0 (/ (- x y) (* t_0 (+ x y))))))

C

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

↓

double code(double x, double y) {
	double t_0 = (x - y) / hypot(x, y);
	return t_0 / ((x - y) / (t_0 * (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) {
	double t_0 = (x - y) / Math.hypot(x, y);
	return t_0 / ((x - y) / (t_0 * (x + y)));
}

Python

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

↓

def code(x, y):
	t_0 = (x - y) / math.hypot(x, y)
	return t_0 / ((x - y) / (t_0 * (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)
	t_0 = Float64(Float64(x - y) / hypot(x, y))
	return Float64(t_0 / Float64(Float64(x - y) / Float64(t_0 * 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)
	t_0 = (x - y) / hypot(x, y);
	tmp = t_0 / ((x - y) / (t_0 * (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_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 / N[(N[(x - y), $MachinePrecision] / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	67.5%
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 67.5%

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

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] 67.5	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   add-sqr-sqrt [=>] 67.5	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}} \]
   times-frac [=>] 67.3	\[ \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]
   hypot-def [=>] 67.4	\[ \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}} \]
   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. Applied egg-rr 100.0%

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

   [Start] 99.9	\[ \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \]
   associate-*r/ [=>] 100.0	\[ \color{blue}{\frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \left(x + y\right)}{\mathsf{hypot}\left(x, y\right)}} \]
   associate-/l* [=>] 100.0	\[ \color{blue}{\frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

4. Applied egg-rr 66.7%

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

   [Start] 100.0	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}} \]
   flip-+ [=>] 67.2	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{\mathsf{hypot}\left(x, y\right)}{\color{blue}{\frac{x \cdot x - y \cdot y}{x - y}}}} \]
   associate-/r/ [=>] 67.0	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\color{blue}{\frac{\mathsf{hypot}\left(x, y\right)}{x \cdot x - y \cdot y} \cdot \left(x - y\right)}} \]
   sub-neg [=>] 67.0	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{\mathsf{hypot}\left(x, y\right)}{x \cdot x - y \cdot y} \cdot \color{blue}{\left(x + \left(-y\right)\right)}} \]
   distribute-lft-in [=>] 66.7	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\color{blue}{\frac{\mathsf{hypot}\left(x, y\right)}{x \cdot x - y \cdot y} \cdot x + \frac{\mathsf{hypot}\left(x, y\right)}{x \cdot x - y \cdot y} \cdot \left(-y\right)}} \]

5. Simplified 99.9%

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

   [Start] 66.7	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{\mathsf{hypot}\left(x, y\right)}{x \cdot x - y \cdot y} \cdot x + \frac{\mathsf{hypot}\left(x, y\right)}{x \cdot x - y \cdot y} \cdot \left(-y\right)} \]
   distribute-lft-out [=>] 67.0	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\color{blue}{\frac{\mathsf{hypot}\left(x, y\right)}{x \cdot x - y \cdot y} \cdot \left(x + \left(-y\right)\right)}} \]
   sub-neg [<=] 67.0	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{\mathsf{hypot}\left(x, y\right)}{x \cdot x - y \cdot y} \cdot \color{blue}{\left(x - y\right)}} \]
   *-commutative [<=] 67.0	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\color{blue}{\left(x - y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x \cdot x - y \cdot y}}} \]
   associate-*r/ [=>] 67.5	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\color{blue}{\frac{\left(x - y\right) \cdot \mathsf{hypot}\left(x, y\right)}{x \cdot x - y \cdot y}}} \]
   associate-/l* [=>] 67.2	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\color{blue}{\frac{x - y}{\frac{x \cdot x - y \cdot y}{\mathsf{hypot}\left(x, y\right)}}}} \]
   difference-of-squares [=>] 67.2	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{x - y}{\frac{\color{blue}{\left(x + y\right) \cdot \left(x - y\right)}}{\mathsf{hypot}\left(x, y\right)}}} \]
   associate-*r/ [<=] 99.9	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{x - y}{\color{blue}{\left(x + y\right) \cdot \frac{x - y}{\mathsf{hypot}\left(x, y\right)}}}} \]

6. Final simplification 99.9%

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

Alternatives

Alternative 1
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 2
Accuracy	100.0%
Cost	13632

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

Alternative 3
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 4
Accuracy	92.5%
Cost	1988

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

Alternative 5
Accuracy	82.5%
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-108}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{-170}:\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-196}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{-169}:\\ \;\;\;\;1 + \frac{\frac{y}{\frac{x}{y}}}{x} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6
Accuracy	82.3%
Cost	1105

\[\begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-106}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-169} \lor \neg \left(y \leq -3.1 \cdot 10^{-196}\right) \land y \leq 1.3 \cdot 10^{-169}:\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7
Accuracy	81.8%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-108}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-197}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-169}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8
Accuracy	81.8%
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-107}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-196}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9
Accuracy	66.4%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023140 
(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))))