Average Error: 22.3 → 0.1
Time: 8.7s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -373964.67688393424 \lor \neg \left(y \leq 382375.84537244635\right):\\ \;\;\;\;\left(x + \frac{x}{y \cdot y}\right) - \left(\frac{1}{y \cdot y} + \frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \frac{y}{y + 1}\\ \mathsf{fma}\left(t_0, x, 1\right) - t_0 \end{array}\\ \end{array} \]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -373964.67688393424 \lor \neg \left(y \leq 382375.84537244635\right):\\
\;\;\;\;\left(x + \frac{x}{y \cdot y}\right) - \left(\frac{1}{y \cdot y} + \frac{x + -1}{y}\right)\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \frac{y}{y + 1}\\
\mathsf{fma}\left(t_0, x, 1\right) - t_0
\end{array}\\


\end{array}

(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= y -373964.67688393424) (not (<= y 382375.84537244635)))
   (- (+ x (/ x (* y y))) (+ (/ 1.0 (* y y)) (/ (+ x -1.0) y)))
   (let* ((t_0 (/ y (+ y 1.0)))) (- (fma t_0 x 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 tmp;
	if ((y <= -373964.67688393424) || !(y <= 382375.84537244635)) {
		tmp = (x + (x / (y * y))) - ((1.0 / (y * y)) + ((x + -1.0) / y));
	} else {
		double t_0 = y / (y + 1.0);
		tmp = fma(t_0, x, 1.0) - t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.3
Target0.3
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 < -373964.67688393424 or 382375.84537244635 < y

    1. Initial program 45.5

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

    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

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

    if -373964.67688393424 < y < 382375.84537244635

    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

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -373964.67688393424 \lor \neg \left(y \leq 382375.84537244635\right):\\ \;\;\;\;\left(x + \frac{x}{y \cdot y}\right) - \left(\frac{1}{y \cdot y} + \frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x, 1\right) - \frac{y}{y + 1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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))))