Average Error: 22.8 → 0.2
Time: 3.0s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -68835468.4205499142 \lor \neg \left(y \le 60021882.2951981202\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y \cdot \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1}\right) - \left(\frac{\left(1 - x\right) \cdot y}{{\left(y \cdot y\right)}^{3} - {\left(1 \cdot 1\right)}^{3}} \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(y \cdot y\right) \cdot \left(1 \cdot 1\right)\right)\right)\right) \cdot \left(-1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -68835468.4205499142 \lor \neg \left(y \le 60021882.2951981202\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

\mathbf{else}:\\
\;\;\;\;\left(1 - y \cdot \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1}\right) - \left(\frac{\left(1 - x\right) \cdot y}{{\left(y \cdot y\right)}^{3} - {\left(1 \cdot 1\right)}^{3}} \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(y \cdot y\right) \cdot \left(1 \cdot 1\right)\right)\right)\right) \cdot \left(-1\right)\\

\end{array}
double code(double x, double y) {
	return (1.0 - (((1.0 - x) * y) / (y + 1.0)));
}
↓
double code(double x, double y) {
	double VAR;
	if (((y <= -68835468.42054991) || !(y <= 60021882.29519812))) {
		VAR = ((1.0 * ((1.0 / y) - (x / y))) + x);
	} else {
		VAR = ((1.0 - (y * (((1.0 - x) * y) / ((y * y) - (1.0 * 1.0))))) - (((((1.0 - x) * y) / (pow((y * y), 3.0) - pow((1.0 * 1.0), 3.0))) * (((y * y) * (y * y)) + (((1.0 * 1.0) * (1.0 * 1.0)) + ((y * y) * (1.0 * 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

Original22.8
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -68835468.42054991 or 60021882.29519812 < y

    1. Initial program 46.2

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

    if -68835468.42054991 < y < 60021882.29519812

    1. Initial program 0.2

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

    3. Applied flip-+0.2

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

    4. Applied associate-/r/0.2

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

    6. Applied sub-neg0.2

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

    7. Applied distribute-lft-in0.2

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

    8. Applied associate--r+0.2

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

    9. Simplified0.2

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

    11. Applied flip3--0.2

      \[\leadsto \left(1 - y \cdot \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1}\right) - \frac{\left(1 - x\right) \cdot y}{\color{blue}{\frac{{\left(y \cdot y\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}{\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(y \cdot y\right) \cdot \left(1 \cdot 1\right)\right)}}} \cdot \left(-1\right)\]

    12. Applied associate-/r/0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -68835468.4205499142 \lor \neg \left(y \le 60021882.2951981202\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y \cdot \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1}\right) - \left(\frac{\left(1 - x\right) \cdot y}{{\left(y \cdot y\right)}^{3} - {\left(1 \cdot 1\right)}^{3}} \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(y \cdot y\right) \cdot \left(1 \cdot 1\right)\right)\right)\right) \cdot \left(-1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(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))))