Average Error: 22.8 → 7.9
Time: 4.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.2287957848170708 \cdot 10^{34} \lor \neg \left(y \le 92001499739.104462\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.2287957848170708 \cdot 10^{34} \lor \neg \left(y \le 92001499739.104462\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\

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

\end{array}
double code(double x, double y) {
	return (1.0 - (((1.0 - x) * y) / (y + 1.0)));
}
↓
double code(double x, double y) {
	double temp;
	if (((y <= -2.228795784817071e+34) || !(y <= 92001499739.10446))) {
		temp = fma(1.0, ((x / pow(y, 2.0)) - (x / y)), x);
	} else {
		temp = (1.0 - (((1.0 - x) * y) / (y + 1.0)));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.8
Target0.2
Herbie7.9
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2.228795784817071e+34 or 92001499739.10446 < y

    1. Initial program 46.6

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

    2. Simplified29.5

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

    4. Applied add-cube-cbrt30.3

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

    5. Applied associate-/r*30.3

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

    7. Applied add-cube-cbrt30.3

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

    8. Applied cbrt-prod30.3

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

    9. Applied associate-*l*30.3

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

    10. Simplified30.6

      \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{\sqrt[3]{\sqrt[3]{y + 1} \cdot \sqrt[3]{y + 1}} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt[3]{y + 1}}\right)}^{4}}}}{\sqrt[3]{y + 1}}, x - 1, 1\right)\]

    11. Taylor expanded around inf 15.1

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

    12. Simplified15.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)}\]

    if -2.228795784817071e+34 < y < 92001499739.10446

    1. Initial program 1.5

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

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.2287957848170708 \cdot 10^{34} \lor \neg \left(y \le 92001499739.104462\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))))