Kahan p9 Example

Average Error: 20.6 → 0.0

Time: 15.5s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

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(x - y) / Float64(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[(x - y), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] * N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	20.6
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 20.6

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

2. Simplified 20.7

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

   [Start] 20.6	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   associate-/l* [=>] 20.7	\[ \color{blue}{\frac{x - y}{\frac{x \cdot x + y \cdot y}{x + y}}} \]
   fma-def [=>] 20.7	\[ \frac{x - y}{\frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y\right)}}{x + y}} \]

3. Applied egg-rr 0.2

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

4. Simplified 0.0

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

   [Start] 0.2	\[ \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \left(\mathsf{hypot}\left(x, y\right) \cdot \frac{1}{x + y}\right)} \]
   associate-*r/ [=>] 0.0	\[ \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\frac{\mathsf{hypot}\left(x, y\right) \cdot 1}{x + y}}} \]
   *-rgt-identity [=>] 0.0	\[ \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\color{blue}{\mathsf{hypot}\left(x, y\right)}}{x + y}} \]
   +-commutative [=>] 0.0	\[ \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{\color{blue}{y + x}}} \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	5.0
Cost	2381

\[\begin{array}{l} t_0 := \frac{y}{x} \cdot \frac{y}{x}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-162} \lor \neg \left(y \leq 3.2 \cdot 10^{-164}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{y}{x} + \left(-1 - \left(t_0 + \left(\left(\frac{y}{x} + t_0\right) + -2\right)\right)\right)}}\\ \end{array} \]

Alternative 2
Error	5.0
Cost	1741

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-162} \lor \neg \left(y \leq 3.2 \cdot 10^{-164}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + \left(2 + \left(\left(-1 - \frac{y}{x}\right) + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -2\right)\right)\\ \end{array} \]

Alternative 3
Error	5.0
Cost	1485

\[\begin{array}{l} t_0 := \frac{y}{x} \cdot \frac{y}{x}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-162} \lor \neg \left(y \leq 3.2 \cdot 10^{-164}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t_0\right) - t_0\\ \end{array} \]

Alternative 4
Error	5.0
Cost	1357

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-162} \lor \neg \left(y \leq 3.2 \cdot 10^{-164}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-2}{\frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \]

Alternative 5
Error	11.0
Cost	969

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{-122} \lor \neg \left(y \leq 1.9 \cdot 10^{-146}\right):\\ \;\;\;\;-1 + t_0\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-2}{t_0}\\ \end{array} \]

Alternative 6
Error	11.4
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-127} \lor \neg \left(y \leq 1.05 \cdot 10^{-148}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	11.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-128} \lor \neg \left(y \leq 6.4 \cdot 10^{-152}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \end{array} \]

Alternative 8
Error	11.7
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-121}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-146}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9
Error	21.1
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023031 
(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))))