Average Error: 22.6 → 7.6
Time: 5.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -534323128112500830000 \lor \neg \left(y \le 1225799455048050930000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{\frac{\sqrt[3]{y + 1} \cdot \sqrt[3]{y \cdot y - 1 \cdot 1}}{\sqrt[3]{y - 1}}}}{\sqrt[3]{y + 1}}, x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -534323128112500830000 \lor \neg \left(y \le 1225799455048050930000\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{\frac{\sqrt[3]{y + 1} \cdot \sqrt[3]{y \cdot y - 1 \cdot 1}}{\sqrt[3]{y - 1}}}}{\sqrt[3]{y + 1}}, x - 1, 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 <= -5.3432312811250083e+20) || !(y <= 1.225799455048051e+21))) {
		VAR = ((double) fma(((double) (x / y)), ((double) (((double) (1.0 / y)) - 1.0)), x));
	} else {
		VAR = ((double) fma(((double) (((double) (y / ((double) (((double) (((double) cbrt(((double) (y + 1.0)))) * ((double) cbrt(((double) (((double) (y * y)) - ((double) (1.0 * 1.0)))))))) / ((double) cbrt(((double) (y - 1.0)))))))) / ((double) cbrt(((double) (y + 1.0)))))), ((double) (x - 1.0)), 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

Original22.6
Target0.2
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -5.3432312811250083e+20 or 1.225799455048051e+21 < y

    1. Initial program 46.7

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

    2. Simplified30.0

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

    3. Taylor expanded around inf 14.9

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

    4. Simplified14.9

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

    if -5.3432312811250083e+20 < y < 1.225799455048051e+21

    1. Initial program 1.2

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

    2. Simplified1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt1.2

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

    5. Applied associate-/r*1.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{\sqrt[3]{y + 1} \cdot \sqrt[3]{y + 1}}}{\sqrt[3]{y + 1}}}, x - 1, 1\right)\]
    6. Using strategy rm

    7. Applied flip-+1.2

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

    8. Applied cbrt-div1.2

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

    9. Applied associate-*r/1.2

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -534323128112500830000 \lor \neg \left(y \le 1225799455048050930000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{\frac{\sqrt[3]{y + 1} \cdot \sqrt[3]{y \cdot y - 1 \cdot 1}}{\sqrt[3]{y - 1}}}}{\sqrt[3]{y + 1}}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))