Average Error: 22.3 → 0.0
Time: 3.7s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -507940.76313948643 \lor \neg \left(y \leq 245860.08583667743\right):\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x + -1}{y + 1}, 1\right)\\ \end{array} \]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -507940.76313948643 \lor \neg \left(y \leq 245860.08583667743\right):\\
\;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) - \frac{x + -1}{y}\\

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


\end{array}

(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= y -507940.76313948643) (not (<= y 245860.08583667743)))
   (- (+ x (/ (+ x -1.0) (* y y))) (/ (+ x -1.0) y))
   (fma y (/ (+ x -1.0) (+ y 1.0)) 1.0)))
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 <= -507940.76313948643) || !(y <= 245860.08583667743)) {
		tmp = (x + ((x + -1.0) / (y * y))) - ((x + -1.0) / y);
	} else {
		tmp = fma(y, ((x + -1.0) / (y + 1.0)), 1.0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.3
Target0.2
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if y < -507940.76313948643 or 245860.085836677434 < y

    1. Initial program 45.5

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

    2. Simplified29.3

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

    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 associate--r+_binary640.0

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

    6. Simplified0.0

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

    if -507940.76313948643 < y < 245860.085836677434

    1. Initial program 0.0

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

    2. Simplified0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -507940.76313948643 \lor \neg \left(y \leq 245860.08583667743\right):\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x + -1}{y + 1}, 1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))))