Kahan p9 Example

Average Error: 20.4 → 0.0

Time: 9.2s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

FPCore

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

↓

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

Target

Original	20.4
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 20.4

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

2. Simplified 20.6

   \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   Proof
   (*.f64 (-.f64 x y) (/.f64 (+.f64 x y) (fma.f64 y y (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 x y) (/.f64 (+.f64 x y) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y y) (*.f64 x x))))): 4 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 x y) (/.f64 (+.f64 x y) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) (*.f64 y y))))): 0 points increase in error, 4 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))): 3 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	5.3
Cost	1620

\[\begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{x + y}{y}}{\frac{y}{x - y}}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-189}:\\ \;\;\;\;1 + \frac{-2}{\frac{x}{y} \cdot \frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-166}:\\ \;\;\;\;-1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.8
Cost	1232

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot \frac{x}{y}\\ t_1 := -1 + t_0\\ t_2 := 1 + \frac{-2}{t_0}\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-189}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y}\\ \end{array} \]

Alternative 3
Error	10.7
Cost	1232

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot \frac{x}{y}\\ t_1 := 1 + \frac{-2}{t_0}\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{-129}:\\ \;\;\;\;-1 + \frac{\frac{x \cdot 2}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-198}:\\ \;\;\;\;-1 + t_0\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y}\\ \end{array} \]

Alternative 4
Error	10.8
Cost	1232

\[\begin{array}{l} t_0 := 1 + \frac{-2}{\frac{x}{y} \cdot \frac{x}{y}}\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{-127}:\\ \;\;\;\;-1 + \frac{\frac{x \cdot 2}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{x + y}{y}}{\frac{y}{x - y}}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-189}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y}\\ \end{array} \]

Alternative 5
Error	11.1
Cost	972

\[\begin{array}{l} t_0 := -1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -1.26 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-196}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y}\\ \end{array} \]

Alternative 6
Error	11.2
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-129}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-161}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-196}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y}\\ \end{array} \]

Alternative 7
Error	11.3
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -1.38 \cdot 10^{-129}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-196}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8
Error	21.2
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022343 
(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))))