Kahan p9 Example

Average Error: 20.5 → 0.0

Time: 8.4s

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	20.5
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 20.5

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

2. Simplified 20.8

   \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(x, x, y \cdot y\right)}} \]
   Proof
   (*.f64 (-.f64 x y) (/.f64 (+.f64 x y) (fma.f64 x x (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 x y) (/.f64 (+.f64 x y) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x x) (*.f64 y y))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))): 43 points increase in error, 51 points decrease in error

3. Applied egg-rr 0.2

   \[\leadsto \left(x - y\right) \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\right)} \]

4. Applied egg-rr 0.0

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

5. 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	5.0
Cost	8260

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

Alternative 2
Error	5.0
Cost	1988

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

Alternative 3
Error	11.7
Cost	1232

\[\begin{array}{l} t_0 := 2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{if}\;y \leq -2.6753082947503586 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.692200988015536 \cdot 10^{-219}:\\ \;\;\;\;1 - \frac{y}{x \cdot \frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.2622630467922794 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.34189315169563 \cdot 10^{-97}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	11.6
Cost	1232

\[\begin{array}{l} t_0 := 2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{if}\;y \leq -2.6753082947503586 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.692200988015536 \cdot 10^{-219}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 1.2622630467922794 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.34189315169563 \cdot 10^{-97}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	12.6
Cost	1104

\[\begin{array}{l} t_0 := 1 - \frac{y}{x \cdot \frac{x}{y}}\\ \mathbf{if}\;y \leq -2.6753082947503586 \cdot 10^{-163}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.692200988015536 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.2058790097396788 \cdot 10^{-115}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.653774309196888 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6
Error	12.8
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6753082947503586 \cdot 10^{-163}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.692200988015536 \cdot 10^{-219}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.2058790097396788 \cdot 10^{-115}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.653774309196888 \cdot 10^{-83}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7
Error	42.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022300 
(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))))