Average Error: 22.4 → 7.3
Time: 3.5s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1448331083874109700 \lor \neg \left(y \le 1.83328374746711847 \cdot 10^{30}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\frac{1}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)\right), x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1448331083874109700 \lor \neg \left(y \le 1.83328374746711847 \cdot 10^{30}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot \left(\frac{1}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)\right), 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 <= -1.4483310838741097e+18) || !(y <= 1.8332837474671185e+30))) {
		VAR = ((double) fma(((double) (x / y)), ((double) (((double) (1.0 / y)) - 1.0)), x));
	} else {
		VAR = ((double) fma(((double) (y * ((double) (((double) (1.0 / ((double) (((double) (y * y)) - ((double) (1.0 * 1.0)))))) * ((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

Original22.4
Target0.2
Herbie7.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.4483310838741097e+18 or 1.8332837474671185e+30 < y

    1. Initial program 47.2

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

    2. Simplified30.0

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

    3. Taylor expanded around inf 14.5

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

    4. Simplified14.5

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

    if -1.4483310838741097e+18 < y < 1.8332837474671185e+30

    1. Initial program 1.4

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

    2. Simplified1.2

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

    4. Applied div-inv1.2

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

    6. Applied flip-+1.2

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

    7. Applied associate-/r/1.3

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1448331083874109700 \lor \neg \left(y \le 1.83328374746711847 \cdot 10^{30}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\frac{1}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)\right), x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +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))))