Average Error: 22.8 → 0.1
Time: 4.3s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
↓
\[\begin{array}{l} t_0 := \left(\mathsf{fma}\left(x, {y}^{-2}, x\right) - {y}^{-2}\right) - \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -311785542274500.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 449605.74478200584:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
t_0 := \left(\mathsf{fma}\left(x, {y}^{-2}, x\right) - {y}^{-2}\right) - \frac{x + -1}{y}\\
\mathbf{if}\;y \leq -311785542274500.1:\\
\;\;\;\;t_0\\

\mathbf{elif}\;y \leq 449605.74478200584:\\
\;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (- (fma x (pow y -2.0) x) (pow y -2.0)) (/ (+ x -1.0) y))))
   (if (<= y -311785542274500.1)
     t_0
     (if (<= y 449605.74478200584)
       (- 1.0 (/ (* y (- 1.0 x)) (+ y 1.0)))
       t_0))))
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 = (fma(x, pow(y, -2.0), x) - pow(y, -2.0)) - ((x + -1.0) / y);
	double tmp;
	if (y <= -311785542274500.1) {
		tmp = t_0;
	} else if (y <= 449605.74478200584) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.8
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 < -311785542274500.12 or 449605.744782005844 < y

    1. Initial program 46.1

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

    2. Simplified29.5

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

    3. Taylor expanded in y around inf 0.0

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

    4. Simplified0.0

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

    5. Applied egg-rr0.0

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

    if -311785542274500.12 < y < 449605.744782005844

    1. Initial program 0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -311785542274500.1:\\ \;\;\;\;\left(\mathsf{fma}\left(x, {y}^{-2}, x\right) - {y}^{-2}\right) - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 449605.74478200584:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, {y}^{-2}, x\right) - {y}^{-2}\right) - \frac{x + -1}{y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))))