Average Error: 22.4 → 0.1
Time: 6.9s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -3427262.9176493334 \lor \neg \left(y \leq 394937.3185555284\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) + \left(\frac{x + -1}{y \cdot y} - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -3427262.9176493334 \lor \neg \left(y \leq 394937.3185555284\right):\\
\;\;\;\;\left(x + \frac{1}{y}\right) + \left(\frac{x + -1}{y \cdot y} - \frac{x}{y}\right)\\

\mathbf{else}:\\
\;\;\;\;1 + y \cdot \frac{x + -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 -3427262.9176493334) (not (<= y 394937.3185555284)))
   (+ (+ x (/ 1.0 y)) (- (/ (+ x -1.0) (* y y)) (/ x y)))
   (+ 1.0 (* y (/ (+ 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 <= -3427262.9176493334) || !(y <= 394937.3185555284)) {
		tmp = (x + (1.0 / y)) + (((x + -1.0) / (y * y)) - (x / y));
	} else {
		tmp = 1.0 + (y * ((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

Original22.4
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3427262.9176493334 or 394937.31855552841 < y

    1. Initial program 45.8

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

    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. Taylor expanded around 0 0.0

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

    5. Simplified0.0

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

    if -3427262.9176493334 < y < 394937.31855552841

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    5. Applied sub-neg_binary64_188260.1

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3427262.9176493334 \lor \neg \left(y \leq 394937.3185555284\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) + \left(\frac{x + -1}{y \cdot y} - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2021093 
(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))))