Kahan p9 Example

Average Accuracy: 68.5% → 99.9%

Time: 15.2s

Precision: binary64

Cost: 20224

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (/ (- (/ x (hypot x y)) (/ 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 / hypot(x, y)) - (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 / Math.hypot(x, y)) - (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 / math.hypot(x, y)) - (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(Float64(x / hypot(x, y)) - Float64(y / 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 / hypot(x, y)) - (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[(N[(x / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] - N[(y / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(x + y), $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 100.0%

   \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}} \]
   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 [=>] 68.4	\[ \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]
   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 [=>] 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%

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

   [Start] 100.0	\[ \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \]
   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 99.9%

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

   [Start] 100.0	\[ \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}} \]
   div-sub [=>] 99.9	\[ \frac{\color{blue}{\frac{x}{\mathsf{hypot}\left(x, y\right)} - \frac{y}{\mathsf{hypot}\left(x, y\right)}}}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}} \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	100.0%
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	92.9%
Cost	7565

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-160} \lor \neg \left(y \leq 7.6 \cdot 10^{-177}\right):\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \left(\frac{y}{x} + 1\right)\\ \end{array} \]

Alternative 4
Accuracy	92.6%
Cost	1357

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-160} \lor \neg \left(y \leq 7.6 \cdot 10^{-177}\right):\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \end{array} \]

Alternative 5
Accuracy	83.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{-144} \lor \neg \left(y \leq 8.6 \cdot 10^{-162}\right):\\ \;\;\;\;-1 + x \cdot \left(2 \cdot \frac{x}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \end{array} \]

Alternative 6
Accuracy	83.6%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-144}:\\ \;\;\;\;\left(1 - \frac{x}{y}\right) \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + x \cdot \left(2 \cdot \frac{x}{y \cdot y}\right)\\ \end{array} \]

Alternative 7
Accuracy	82.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-116}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-162}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8
Accuracy	83.0%
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -4.75 \cdot 10^{-144}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9
Accuracy	65.8%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023138 
(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))))