Average Error: 22.5 → 7.7
Time: 3.8s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.5109143016052069 \cdot 10^{33} \lor \neg \left(y \le 9.60502204570401 \cdot 10^{34}\right):\\ \;\;\;\;x + \frac{x}{y} \cdot \left(\frac{1}{y} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{\frac{x - 1}{\sqrt[3]{{\left(\sqrt[3]{y + 1}\right)}^{6}}}}{\sqrt[3]{y + 1}}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.5109143016052069 \cdot 10^{33} \lor \neg \left(y \le 9.60502204570401 \cdot 10^{34}\right):\\
\;\;\;\;x + \frac{x}{y} \cdot \left(\frac{1}{y} - 1\right)\\

\mathbf{else}:\\
\;\;\;\;1 + y \cdot \frac{\frac{x - 1}{\sqrt[3]{{\left(\sqrt[3]{y + 1}\right)}^{6}}}}{\sqrt[3]{y + 1}}\\

\end{array}
double code(double x, double y) {
	return ((double) (1.0 - (((double) (((double) (1.0 - x)) * y)) / ((double) (y + 1.0)))));
}
↓
double code(double x, double y) {
	double VAR;
	if (((y <= -2.510914301605207e+33) || !(y <= 9.605022045704014e+34))) {
		VAR = ((double) (x + ((double) ((x / y) * ((double) ((1.0 / y) - 1.0))))));
	} else {
		VAR = ((double) (1.0 + ((double) (y * ((((double) (x - 1.0)) / ((double) cbrt(((double) pow(((double) cbrt(((double) (y + 1.0)))), 6.0))))) / ((double) cbrt(((double) (y + 1.0)))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.5
Target0.3
Herbie7.7
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2.5109143016052069e33 or 9.60502204570401e34 < y

    1. Initial program 47.8

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

    2. Simplified29.7

      \[\leadsto \color{blue}{1 + y \cdot \frac{x - 1}{1 + y}}\]

    3. Taylor expanded around inf 14.3

      \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{x}{y}}\]

    4. Simplified14.3

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

    if -2.5109143016052069e33 < y < 9.60502204570401e34

    1. Initial program 2.4

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

    2. Simplified2.3

      \[\leadsto \color{blue}{1 + y \cdot \frac{x - 1}{1 + y}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt2.4

      \[\leadsto 1 + y \cdot \frac{x - 1}{\color{blue}{\left(\sqrt[3]{1 + y} \cdot \sqrt[3]{1 + y}\right) \cdot \sqrt[3]{1 + y}}}\]

    5. Applied associate-/r*2.4

      \[\leadsto 1 + y \cdot \color{blue}{\frac{\frac{x - 1}{\sqrt[3]{1 + y} \cdot \sqrt[3]{1 + y}}}{\sqrt[3]{1 + y}}}\]
    6. Using strategy rm

    7. Applied cbrt-unprod2.3

      \[\leadsto 1 + y \cdot \frac{\frac{x - 1}{\color{blue}{\sqrt[3]{\left(1 + y\right) \cdot \left(1 + y\right)}}}}{\sqrt[3]{1 + y}}\]

    8. Simplified2.4

      \[\leadsto 1 + y \cdot \frac{\frac{x - 1}{\sqrt[3]{\color{blue}{{\left(\sqrt[3]{1 + y}\right)}^{6}}}}}{\sqrt[3]{1 + y}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.5109143016052069 \cdot 10^{33} \lor \neg \left(y \le 9.60502204570401 \cdot 10^{34}\right):\\ \;\;\;\;x + \frac{x}{y} \cdot \left(\frac{1}{y} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{\frac{x - 1}{\sqrt[3]{{\left(\sqrt[3]{y + 1}\right)}^{6}}}}{\sqrt[3]{y + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))