Average Error: 22.4 → 0.1
Time: 3.7s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
↓
\[\begin{array}{l} t_0 := \frac{1}{y \cdot y}\\ t_1 := \left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\\ \mathbf{if}\;y \leq -12257.264768591502:\\ \;\;\;\;t_1 - \left(\frac{x}{{y}^{3}} + \left(t_0 + \frac{x + -1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 180128.4190764583:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x + -1}{y + 1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - \left(t_0 - \frac{1}{y}\right)\\ \end{array} \]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
t_0 := \frac{1}{y \cdot y}\\
t_1 := \left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\\
\mathbf{if}\;y \leq -12257.264768591502:\\
\;\;\;\;t_1 - \left(\frac{x}{{y}^{3}} + \left(t_0 + \frac{x + -1}{y}\right)\right)\\

\mathbf{elif}\;y \leq 180128.4190764583:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x + -1}{y + 1}, 1\right)\\

\mathbf{else}:\\
\;\;\;\;t_1 - \left(t_0 - \frac{1}{y}\right)\\


\end{array}

(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* y y)))
        (t_1 (+ (+ x (/ x (* y y))) (/ 1.0 (pow y 3.0)))))
   (if (<= y -12257.264768591502)
     (- t_1 (+ (/ x (pow y 3.0)) (+ t_0 (/ (+ x -1.0) y))))
     (if (<= y 180128.4190764583)
       (fma y (/ (+ x -1.0) (+ y 1.0)) 1.0)
       (- t_1 (- t_0 (/ 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 t_0 = 1.0 / (y * y);
	double t_1 = (x + (x / (y * y))) + (1.0 / pow(y, 3.0));
	double tmp;
	if (y <= -12257.264768591502) {
		tmp = t_1 - ((x / pow(y, 3.0)) + (t_0 + ((x + -1.0) / y)));
	} else if (y <= 180128.4190764583) {
		tmp = fma(y, ((x + -1.0) / (y + 1.0)), 1.0);
	} else {
		tmp = t_1 - (t_0 - (1.0 / y));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

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 3 regimes

  2. if y < -12257.2647685915017

    1. Initial program 44.4

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

    2. Simplified28.6

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

    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

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

    if -12257.2647685915017 < y < 180128.419076458289

    1. Initial program 0.1

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

    2. Simplified0.0

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

    if 180128.419076458289 < y

    1. Initial program 44.5

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

    2. Simplified29.1

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

    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

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

    5. Taylor expanded in x around 0 0.3

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

    6. Simplified0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12257.264768591502:\\ \;\;\;\;\left(\left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{1}{y \cdot y} + \frac{x + -1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 180128.4190764583:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x + -1}{y + 1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\right) - \left(\frac{1}{y \cdot y} - \frac{1}{y}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))))