Average Error: 22.8 → 0.5
Time: 6.9s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 8.092986506409391 \cdot 10^{-05} \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.0000001086860406\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 8.092986506409391 \cdot 10^{-05} \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.0000001086860406\right):\\
\;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\

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

\end{array}
(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= (/ (* (- 1.0 x) y) (+ 1.0 y)) 8.092986506409391e-05)
         (not (<= (/ (* (- 1.0 x) y) (+ 1.0 y)) 1.0000001086860406)))
   (- 1.0 (* (- 1.0 x) (/ y (+ 1.0 y))))
   (- (+ x (/ 1.0 y)) (/ x y))))
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 (((((1.0 - x) * y) / (1.0 + y)) <= 8.092986506409391e-05) || !((((1.0 - x) * y) / (1.0 + y)) <= 1.0000001086860406)) {
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)));
	} else {
		tmp = (x + (1.0 / y)) - (x / y);
	}
	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

Original22.8
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 (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 8.0929865064093909e-5 or 1.0000001086860406 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

    1. Initial program 10.8

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

    3. Applied *-un-lft-identity_binary64_1474110.8

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

    4. Applied times-frac_binary64_147470.1

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

    5. Simplified0.1

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

    6. Simplified0.1

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

    if 8.0929865064093909e-5 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.0000001086860406

    1. Initial program 57.8

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

    2. Taylor expanded around inf 1.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 8.092986506409391 \cdot 10^{-05} \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.0000001086860406\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(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))))