Kahan p9 Example

Average Error: 20.1 → 0.0

Time: 12.1s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

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	20.1
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 20.1

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

2. Simplified 20.3

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

   [Start] 20.1	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   associate-*r/ [<=] 20.3	\[ \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   +-commutative [=>] 20.3	\[ \left(x - y\right) \cdot \frac{x + y}{\color{blue}{y \cdot y + x \cdot x}} \]
   fma-def [=>] 20.3	\[ \left(x - y\right) \cdot \frac{x + y}{\color{blue}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]

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	5.0
Cost	1808

\[\begin{array}{l} t_0 := \frac{y}{x} \cdot \frac{y}{x}\\ t_1 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-195}:\\ \;\;\;\;\left(\frac{0}{x} - t_0\right) + \left(1 - t_0\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-169}:\\ \;\;\;\;\frac{x + y}{\left(\frac{x \cdot \left(-x\right)}{y} - \left(x + y\right)\right) - \frac{x}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	5.0
Cost	1740

\[\begin{array}{l} t_0 := \frac{y}{x} \cdot \frac{y}{x}\\ t_1 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-195}:\\ \;\;\;\;\left(\frac{0}{x} - t_0\right) + \left(1 - t_0\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-169}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	5.7
Cost	1488

\[\begin{array}{l} t_0 := \left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{if}\;y \leq -3.05 \cdot 10^{-56}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-196}:\\ \;\;\;\;1 + \frac{-2}{\frac{x}{\frac{y}{\frac{x}{y}}}}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-171}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	5.0
Cost	1488

\[\begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-195}:\\ \;\;\;\;1 + \frac{-2}{\frac{x}{\frac{y}{\frac{x}{y}}}}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	10.5
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-164} \lor \neg \left(y \leq 2.5 \cdot 10^{-195}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-2}{\frac{x}{\frac{y}{\frac{x}{y}}}}\\ \end{array} \]

Alternative 6
Error	10.5
Cost	968

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -7.7 \cdot 10^{-165}:\\ \;\;\;\;-1 + 2 \cdot t_0\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-195}:\\ \;\;\;\;1 + \frac{-2}{\frac{x}{\frac{y}{\frac{x}{y}}}}\\ \mathbf{else}:\\ \;\;\;\;-1 + t_0\\ \end{array} \]

Alternative 7
Error	10.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-177} \lor \neg \left(y \leq 8.4 \cdot 10^{-196}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	10.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{-176} \lor \neg \left(y \leq 2.7 \cdot 10^{-196}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \end{array} \]

Alternative 9
Error	11.3
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-180}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10
Error	21.1
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023057 
(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))))