Average Error: 22.7 → 7.4
Time: 3.3s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -56814521536770.6953 \lor \neg \left(y \le 10457115104856364\right):\\ \;\;\;\;x + \frac{x}{y} \cdot \left(\frac{1}{y} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(\frac{1}{\sqrt[3]{y + 1} \cdot \sqrt[3]{y + 1}} \cdot \frac{x - 1}{\sqrt[3]{y + 1}}\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -56814521536770.6953 \lor \neg \left(y \le 10457115104856364\right):\\
\;\;\;\;x + \frac{x}{y} \cdot \left(\frac{1}{y} - 1\right)\\

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

\end{array}
double code(double x, double y) {
	return ((double) (1.0 - ((double) (((double) (((double) (1.0 - x)) * y)) / ((double) (y + 1.0))))));
}
↓
double code(double x, double y) {
	double VAR;
	if (((y <= -56814521536770.695) || !(y <= 10457115104856364.0))) {
		VAR = ((double) (x + ((double) (((double) (x / y)) * ((double) (((double) (1.0 / y)) - 1.0))))));
	} else {
		VAR = ((double) (1.0 + ((double) (y * ((double) (((double) (1.0 / ((double) (((double) cbrt(((double) (y + 1.0)))) * ((double) cbrt(((double) (y + 1.0)))))))) * ((double) (((double) (x - 1.0)) / ((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

Original22.7
Target0.2
Herbie7.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -56814521536770.6953 or 10457115104856364 < y

    1. Initial program 46.6

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

    2. Simplified29.6

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

    3. Taylor expanded around inf 14.8

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

    4. Simplified14.8

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

    if -56814521536770.6953 < y < 10457115104856364

    1. Initial program 0.5

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

    2. Simplified0.5

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

    4. Applied add-cube-cbrt0.6

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

    5. Applied *-un-lft-identity0.6

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

    6. Applied times-frac0.6

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -56814521536770.6953 \lor \neg \left(y \le 10457115104856364\right):\\ \;\;\;\;x + \frac{x}{y} \cdot \left(\frac{1}{y} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(\frac{1}{\sqrt[3]{y + 1} \cdot \sqrt[3]{y + 1}} \cdot \frac{x - 1}{\sqrt[3]{y + 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))))