Average Error: 22.4 → 7.5
Time: 4.1s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2555995508154242 \lor \neg \left(y \le 41765098227347587000\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right), x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2555995508154242 \lor \neg \left(y \le 41765098227347587000\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\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 temp;
	if (((y <= -2555995508154242.0) || !(y <= 4.176509822734759e+19))) {
		temp = fma(1.0, ((x / pow(y, 2.0)) - (x / y)), x);
	} else {
		temp = fma(((y / ((y * y) - (1.0 * 1.0))) * (y - 1.0)), (x - 1.0), 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

Original22.4
Target0.2
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2555995508154242.0 or 4.176509822734759e+19 < y

    1. Initial program 47.2

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

    2. Simplified29.7

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

    4. Applied div-inv29.8

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

    5. Taylor expanded around inf 15.2

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

    6. Simplified15.2

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

    if -2555995508154242.0 < y < 4.176509822734759e+19

    1. Initial program 0.8

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

    2. Simplified0.8

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

    4. Applied flip-+0.8

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

    5. Applied associate-/r/0.8

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2555995508154242 \lor \neg \left(y \le 41765098227347587000\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right), x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +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))))