Kahan p9 Example

Average Error: 20.3 → 0.0

Time: 11.0s

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	20.3
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 20.3

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

2. Simplified 20.6

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

   [Start] 20.3	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   associate-*r/ [<=] 20.6	\[ \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   *-commutative [<=] 20.6	\[ \color{blue}{\frac{x + y}{x \cdot x + y \cdot y} \cdot \left(x - y\right)} \]
   fma-def [=>] 20.6	\[ \frac{x + y}{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y\right)}} \cdot \left(x - y\right) \]

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	4.5
Cost	8580

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

Alternative 2
Error	4.5
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 3
Error	4.6
Cost	1988

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

Alternative 4
Error	11.0
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-190} \lor \neg \left(y \leq 1.58 \cdot 10^{-112}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-2}{\frac{x}{y}}}{\frac{x}{y}}\\ \end{array} \]

Alternative 5
Error	11.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-190}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-110}:\\ \;\;\;\;1 + \frac{\frac{-2}{\frac{x}{y}}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{y \cdot y}\\ \end{array} \]

Alternative 6
Error	11.2
Cost	905

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-190} \lor \neg \left(y \leq 1.58 \cdot 10^{-112}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \frac{y}{x}}{-x}\\ \end{array} \]

Alternative 7
Error	11.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-190} \lor \neg \left(y \leq 1.7 \cdot 10^{-119}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	11.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-190}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9
Error	11.7
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-190}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10
Error	21.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023125 
(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))))