Kahan p9 Example

Average Error: 20.1 → 0.0

Time: 10.1s

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	20.1
Target	0.1
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 20.1

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

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	0.2
Cost	13632

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

Alternative 2
Error	4.6
Cost	8196

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

Alternative 3
Error	4.6
Cost	1988

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

Alternative 4
Error	6.5
Cost	1752

\[\begin{array}{l} t_0 := \left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ t_1 := 1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{if}\;y \leq -7.224230589813143 \cdot 10^{-24}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -6.912635693132585 \cdot 10^{-186}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.208182775869543 \cdot 10^{-210}:\\ \;\;\;\;\frac{x + y}{y \cdot \frac{y}{x - y}}\\ \mathbf{elif}\;y \leq -3.5773975902606055 \cdot 10^{-236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.1501473155494471 \cdot 10^{-244}:\\ \;\;\;\;-1 + \frac{\frac{2}{y}}{\frac{\frac{y}{x}}{x}}\\ \mathbf{elif}\;y \leq 5.0432457117699704 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	11.7
Cost	1232

\[\begin{array}{l} t_0 := -1 + \frac{\frac{2}{y}}{\frac{\frac{y}{x}}{x}}\\ \mathbf{if}\;y \leq -4.948842715265956 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.7046811496283202 \cdot 10^{-99}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.7514879357174125 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.0451869875353209 \cdot 10^{-128}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	11.7
Cost	1232

\[\begin{array}{l} t_0 := \frac{x + y}{y \cdot \frac{y}{x - y}}\\ \mathbf{if}\;y \leq -4.948842715265956 \cdot 10^{-80}:\\ \;\;\;\;-1 + \frac{\frac{2}{y}}{\frac{\frac{y}{x}}{x}}\\ \mathbf{elif}\;y \leq -1.7046811496283202 \cdot 10^{-99}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.7514879357174125 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.0451869875353209 \cdot 10^{-128}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	11.7
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y \leq -4.948842715265956 \cdot 10^{-80}:\\ \;\;\;\;-1 + \frac{\frac{2}{y}}{\frac{\frac{y}{x}}{x}}\\ \mathbf{elif}\;y \leq -1.7046811496283202 \cdot 10^{-99}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.7514879357174125 \cdot 10^{-128}:\\ \;\;\;\;\frac{x + y}{y \cdot \frac{y}{x - y}}\\ \mathbf{elif}\;y \leq 2.0451869875353209 \cdot 10^{-128}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{y \cdot y}\\ \end{array} \]

Alternative 8
Error	11.2
Cost	1232

\[\begin{array}{l} t_0 := 1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{if}\;y \leq -4.948842715265956 \cdot 10^{-80}:\\ \;\;\;\;-1 + \frac{\frac{2}{y}}{\frac{\frac{y}{x}}{x}}\\ \mathbf{elif}\;y \leq -1.7046811496283202 \cdot 10^{-99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.7514879357174125 \cdot 10^{-128}:\\ \;\;\;\;\frac{x + y}{y \cdot \frac{y}{x - y}}\\ \mathbf{elif}\;y \leq 2.0451869875353209 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{y \cdot y}\\ \end{array} \]

Alternative 9
Error	11.9
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -4.948842715265956 \cdot 10^{-80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.7046811496283202 \cdot 10^{-99}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.7514879357174125 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.0451869875353209 \cdot 10^{-128}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10
Error	42.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022311 
(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))))