Average Error: 19.6 → 5.1
Time: 2.5s
Precision: binary64
\[0 < x \land x < 1 \land y < 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1.3302442616326837 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5902636281828094 \cdot 10^{-162} \lor \neg \left(y \leq 1.5861413695832953 \cdot 10^{-162}\right):\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1.3302442616326837 \cdot 10^{+154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \leq -1.5902636281828094 \cdot 10^{-162} \lor \neg \left(y \leq 1.5861413695832953 \cdot 10^{-162}\right):\\
\;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;1\\

\end{array}
(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= y -1.3302442616326837e+154)
   -1.0
   (if (or (<= y -1.5902636281828094e-162)
           (not (<= y 1.5861413695832953e-162)))
     (cbrt (pow (/ (- (* x x) (* y y)) (+ (* x x) (* y y))) 3.0))
     1.0)))
double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}
↓
double code(double x, double y) {
	double tmp;
	if (y <= -1.3302442616326837e+154) {
		tmp = -1.0;
	} else if ((y <= -1.5902636281828094e-162) || !(y <= 1.5861413695832953e-162)) {
		tmp = cbrt(pow((((x * x) - (y * y)) / ((x * x) + (y * y))), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.1
Herbie5.1
\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.33024426163268367e154

    1. Initial program 64.0

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

    2. Taylor expanded around 0 0

      \[\leadsto \color{blue}{-1}\]

    if -1.33024426163268367e154 < y < -1.59026362818280943e-162 or 1.58614136958329525e-162 < y

    1. Initial program 0.0

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube_binary64_5620.0

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \cdot \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right) \cdot \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}}\]

    4. Simplified0.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\right)}^{3}}}\]

    if -1.59026362818280943e-162 < y < 1.58614136958329525e-162

    1. Initial program 29.8

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

    2. Taylor expanded around inf 15.9

      \[\leadsto \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3302442616326837 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5902636281828094 \cdot 10^{-162} \lor \neg \left(y \leq 1.5861413695832953 \cdot 10^{-162}\right):\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 x 1.0) (< y 1.0))

  :herbie-target
  (if (< 0.5 (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))))