Average Error: 22.9 → 0.1
Time: 3.5s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -130646298.08262084 \lor \neg \left(y \leq 217289467.81919855\right):\\ \;\;\;\;x + \frac{1}{y} \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -130646298.08262084 \lor \neg \left(y \leq 217289467.81919855\right):\\
\;\;\;\;x + \frac{1}{y} \cdot \left(1 - x\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{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 -130646298.08262084) (not (<= y 217289467.81919855)))
   (+ x (* (/ 1.0 y) (- 1.0 x)))
   (- 1.0 (* (- 1.0 x) (/ y (+ y 1.0))))))
double code(double x, double y) {
	return ((double) (1.0 - (((double) (((double) (1.0 - x)) * y)) / ((double) (y + 1.0)))));
}
↓
double code(double x, double y) {
	double tmp;
	if (((y <= -130646298.08262084) || !(y <= 217289467.81919855))) {
		tmp = ((double) (x + ((double) ((1.0 / y) * ((double) (1.0 - x))))));
	} else {
		tmp = ((double) (1.0 - ((double) (((double) (1.0 - x)) * (y / ((double) (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

Original22.9
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 < -130646298.082620844 or 217289467.81919855 < y

    1. Initial program 46.3

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

    3. Applied *-un-lft-identity_binary6446.3

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

    4. Applied times-frac_binary6430.1

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{1} \cdot \frac{y}{y + 1}}\]

    5. Simplified30.1

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

    6. Simplified30.1

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

    7. Taylor expanded around inf 0.2

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

    8. Simplified0.2

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

    if -130646298.082620844 < y < 217289467.81919855

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity_binary640.1

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

    4. Applied times-frac_binary640.1

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{1} \cdot \frac{y}{y + 1}}\]

    5. Simplified0.1

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -130646298.08262084 \lor \neg \left(y \leq 217289467.81919855\right):\\ \;\;\;\;x + \frac{1}{y} \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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))))