Average Error: 21.8 → 0.5
Time: 6.7s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1.4541039429164083 \cdot 10^{+33} \lor \neg \left(y \leq 270704.0980663581\right):\\ \;\;\;\;\left(x + \left(\frac{1}{y} + \frac{x}{y \cdot y}\right)\right) - \left(\frac{x}{y} + {y}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y \cdot \left(1 - x\right)\right) \cdot \frac{1}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1.4541039429164083 \cdot 10^{+33} \lor \neg \left(y \leq 270704.0980663581\right):\\
\;\;\;\;\left(x + \left(\frac{1}{y} + \frac{x}{y \cdot y}\right)\right) - \left(\frac{x}{y} + {y}^{-2}\right)\\

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

\end{array}
(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.4541039429164083e+33) (not (<= y 270704.0980663581)))
   (- (+ x (+ (/ 1.0 y) (/ x (* y y)))) (+ (/ x y) (pow y -2.0)))
   (- 1.0 (* (* y (- 1.0 x)) (/ 1.0 (+ y 1.0))))))
double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}
↓
double code(double x, double y) {
	double tmp;
	if ((y <= -1.4541039429164083e+33) || !(y <= 270704.0980663581)) {
		tmp = (x + ((1.0 / y) + (x / (y * y)))) - ((x / y) + pow(y, -2.0));
	} else {
		tmp = 1.0 - ((y * (1.0 - x)) * (1.0 / (y + 1.0)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.8
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.45410394291640831e33 or 270704.09806635813 < y

    1. Initial program 46.1

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    5. Applied pow2_binary64_175500.0

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

    6. Applied pow-flip_binary64_175430.0

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

    if -1.45410394291640831e33 < y < 270704.09806635813

    1. Initial program 0.9

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
    2. Using strategy rm

    3. Applied div-inv_binary64_174660.9

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4541039429164083 \cdot 10^{+33} \lor \neg \left(y \leq 270704.0980663581\right):\\ \;\;\;\;\left(x + \left(\frac{1}{y} + \frac{x}{y \cdot y}\right)\right) - \left(\frac{x}{y} + {y}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y \cdot \left(1 - x\right)\right) \cdot \frac{1}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))