Average Error: 22.3 → 0.1
Time: 8.2s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -489641527.8911863:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\ \mathbf{elif}\;y \leq 12600.971116972236:\\ \;\;\;\;\left(1 + \frac{y \cdot x}{y + 1}\right) - \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{1}{y \cdot y} + \frac{x + -1}{y}\right)\right)\\ \end{array} \]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -489641527.8911863:\\
\;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\

\mathbf{elif}\;y \leq 12600.971116972236:\\
\;\;\;\;\left(1 + \frac{y \cdot x}{y + 1}\right) - \frac{y}{y + 1}\\

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


\end{array}

(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= y -489641527.8911863)
   (- (+ x (/ 1.0 y)) (/ x y))
   (if (<= y 12600.971116972236)
     (- (+ 1.0 (/ (* y x) (+ y 1.0))) (/ y (+ y 1.0)))
     (-
      (+ (+ x (/ x (* y y))) (/ 1.0 (pow y 3.0)))
      (+ (/ x (pow y 3.0)) (+ (/ 1.0 (* y y)) (/ (+ x -1.0) 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 (y <= -489641527.8911863) {
		tmp = (x + (1.0 / y)) - (x / y);
	} else if (y <= 12600.971116972236) {
		tmp = (1.0 + ((y * x) / (y + 1.0))) - (y / (y + 1.0));
	} else {
		tmp = ((x + (x / (y * y))) + (1.0 / pow(y, 3.0))) - ((x / pow(y, 3.0)) + ((1.0 / (y * y)) + ((x + -1.0) / 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

Original22.3
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 3 regimes

  2. if y < -489641527.891186297

    1. Initial program 45.5

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

    2. Simplified30.0

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

    3. Taylor expanded in x around 0 45.5

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

    4. Applied associate-/l*_binary6430.0

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

    5. Applied clear-num_binary6430.1

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

    6. Taylor expanded in y around inf 0.1

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

    7. Simplified0.1

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

    if -489641527.891186297 < y < 12600.971116972236

    1. Initial program 0.1

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

    2. Simplified0.1

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

    3. Taylor expanded in x around 0 0.1

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

    if 12600.971116972236 < y

    1. Initial program 44.4

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

    2. Simplified27.8

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

    3. Taylor expanded in y around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \left(\frac{x}{{y}^{2}} + x\right)\right)\right) - \left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)} \]

    4. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -489641527.8911863:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\ \mathbf{elif}\;y \leq 12600.971116972236:\\ \;\;\;\;\left(1 + \frac{y \cdot x}{y + 1}\right) - \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{1}{y \cdot y} + \frac{x + -1}{y}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021291 
(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))))