Kahan p9 Example

Average Error: 20.2 → 0.0

Time: 10.4s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

FPCore

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

↓

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

Target

Original	20.2
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 20.2

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

2. Applied egg-rr 0.0

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

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	13632

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

Alternative 2
Error	13.1
Cost	1496

\[\begin{array}{l} t_0 := -1 + 2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -2.4027481049523924 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.6487267342967365 \cdot 10^{-180}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.5816840772633373 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.94088455111634 \cdot 10^{-181}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.484279074228259 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.088704166225584 \cdot 10^{-71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.0
Cost	1496

\[\begin{array}{l} t_0 := \frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ t_1 := -1 + 2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -2.4027481049523924 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.6487267342967365 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.5816840772633373 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.94088455111634 \cdot 10^{-181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.484279074228259 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.088704166225584 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	12.7
Cost	1496

\[\begin{array}{l} t_0 := 1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -2\right)\\ t_1 := -1 + 2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -2.4027481049523924 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.6487267342967365 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.5816840772633373 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.94088455111634 \cdot 10^{-181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.484279074228259 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.088704166225584 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	4.4
Cost	1356

\[\begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;y \leq -8.918987654442815 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -9.259737990795264 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9.816293967735295 \cdot 10^{-165}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	13.5
Cost	856

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4027481049523924 \cdot 10^{-156}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -4.6487267342967365 \cdot 10^{-180}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.5816840772633373 \cdot 10^{-216}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 8.94088455111634 \cdot 10^{-181}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.484279074228259 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.088704166225584 \cdot 10^{-71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7
Error	42.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022308 
(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))))