Average Error: 22.6 → 7.6
Time: 2.5s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.56357964551490151 \cdot 10^{22} \lor \neg \left(y \le 11800200009954588700\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right)\right), x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.56357964551490151 \cdot 10^{22} \lor \neg \left(y \le 11800200009954588700\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right)\right), x - 1, 1\right)\\

\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 VAR;
	if (((y <= -2.5635796455149015e+22) || !(y <= 1.1800200009954589e+19))) {
		VAR = fma((x / y), ((1.0 / y) - 1.0), x);
	} else {
		VAR = fma(((1.0 / (pow(y, 3.0) + pow(1.0, 3.0))) * (y * ((y * y) + ((1.0 * 1.0) - (y * 1.0))))), (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 < -2.5635796455149015e+22 or 1.1800200009954589e+19 < y

    1. Initial program 46.7

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

    2. Simplified29.1

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

    3. Taylor expanded around inf 14.7

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

    4. Simplified14.7

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

    if -2.5635796455149015e+22 < y < 1.1800200009954589e+19

    1. Initial program 1.3

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

    2. Simplified1.2

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

    4. Applied clear-num1.3

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

    6. Applied flip3-+1.3

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

    7. Applied associate-/l/1.3

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

    8. Applied associate-/r/1.2

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.56357964551490151 \cdot 10^{22} \lor \neg \left(y \le 11800200009954588700\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right)\right), x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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))))