Kahan p9 Example

Average Error: 20.4 → 0.0

Time: 13.1s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

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(x + y) / Float64(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[(x + y), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] * N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	20.4
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 20.4

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

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

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	13632

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

Alternative 2
Error	4.6
Cost	8260

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

Alternative 3
Error	4.7
Cost	8196

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

Alternative 4
Error	4.7
Cost	2116

\[\begin{array}{l} t_0 := \frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ t_1 := \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1 + t_1\right)\\ \end{array} \]

Alternative 5
Error	5.0
Cost	1357

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-162} \lor \neg \left(y \leq 1.55 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -2}{x} \cdot \frac{y}{x}\\ \end{array} \]

Alternative 6
Error	10.8
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-171}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-147}:\\ \;\;\;\;1 + \frac{y \cdot -2}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y + \left(\frac{x}{\frac{y}{x}} - x\right)}\\ \end{array} \]

Alternative 7
Error	10.8
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-169} \lor \neg \left(y \leq 1.04 \cdot 10^{-147}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -2}{x} \cdot \frac{y}{x}\\ \end{array} \]

Alternative 8
Error	10.8
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-169}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-148}:\\ \;\;\;\;1 + \frac{y \cdot -2}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{y \cdot y}{x + y}}\\ \end{array} \]

Alternative 9
Error	11.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-166} \lor \neg \left(y \leq 1.35 \cdot 10^{-147}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	11.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-169} \lor \neg \left(y \leq 1.18 \cdot 10^{-147}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \end{array} \]

Alternative 11
Error	11.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-173} \lor \neg \left(y \leq 8 \cdot 10^{-148}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{y}{x}\\ \end{array} \]

Alternative 12
Error	11.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-166}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.24 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 13
Error	21.3
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023038 
(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))))