Kahan p9 Example

Average Error: 20.0 → 0.0

Time: 13.8s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

FPCore

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

↓

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

Target

Original	20.0
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 20.0

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

2. Simplified 20.3

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

   [Start] 20.0	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   associate-*r/ [<=] 20.3	\[ \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   *-commutative [<=] 20.3	\[ \color{blue}{\frac{x + y}{x \cdot x + y \cdot y} \cdot \left(x - y\right)} \]
   fma-def [=>] 20.3	\[ \frac{x + y}{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y\right)}} \cdot \left(x - y\right) \]

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	3.1
Cost	7812

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

Alternative 2
Error	5.0
Cost	1357

\[\begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-49}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-162} \lor \neg \left(y \leq 1.55 \cdot 10^{-162}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -2}{x} \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3
Error	4.6
Cost	1357

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+156}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-162} \lor \neg \left(y \leq 1.55 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -2}{x} \cdot \frac{y}{x}\\ \end{array} \]

Alternative 4
Error	10.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-143}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-175}:\\ \;\;\;\;1 + \frac{y \cdot -2}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y + \left(\frac{x}{\frac{y}{x}} - x\right)}\\ \end{array} \]

Alternative 5
Error	10.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-142} \lor \neg \left(y \leq 1.42 \cdot 10^{-174}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -2}{x} \cdot \frac{y}{x}\\ \end{array} \]

Alternative 6
Error	10.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-144} \lor \neg \left(y \leq 8 \cdot 10^{-174}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	10.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-143} \lor \neg \left(y \leq 1.5 \cdot 10^{-174}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \end{array} \]

Alternative 8
Error	11.1
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-143}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9
Error	21.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023066 
(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))))