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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -172408127.577771663665771484375 \lor \neg \left(y \le 167881476.735638082027435302734375\right):\\ \;\;\;\;\frac{1}{y} + \left(x - 1 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \le -172408127.577771663665771484375 \lor \neg \left(y \le 167881476.735638082027435302734375\right):\\
\;\;\;\;\frac{1}{y} + \left(x - 1 \cdot \frac{x}{y}\right)\\

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

\end{array}

double f(double x, double y) {
        double r680071 = 1.0;
        double r680072 = x;
        double r680073 = r680071 - r680072;
        double r680074 = y;
        double r680075 = r680073 * r680074;
        double r680076 = r680074 + r680071;
        double r680077 = r680075 / r680076;
        double r680078 = r680071 - r680077;
        return r680078;
}

↓

double f(double x, double y) {
        double r680079 = y;
        double r680080 = -172408127.57777166;
        bool r680081 = r680079 <= r680080;
        double r680082 = 167881476.73563808;
        bool r680083 = r680079 <= r680082;
        double r680084 = !r680083;
        bool r680085 = r680081 || r680084;
        double r680086 = 1.0;
        double r680087 = r680086 / r680079;
        double r680088 = x;
        double r680089 = r680088 / r680079;
        double r680090 = r680086 * r680089;
        double r680091 = r680088 - r680090;
        double r680092 = r680087 + r680091;
        double r680093 = r680079 + r680086;
        double r680094 = r680079 / r680093;
        double r680095 = r680088 - r680086;
        double r680096 = fma(r680094, r680095, r680086);
        double r680097 = r680085 ? r680092 : r680096;
        return r680097;
}

Original	22.9
Target	0.3
Herbie	0.2

Derivation

Split input into 2 regimes
if y < -172408127.57777166 or 167881476.73563808 < y
1. Initial program 46.2
  \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
2. Using strategy rm
3. Applied flip-+51.0
  \[\leadsto 1 - \frac{\left(1 - x\right) \cdot y}{\color{blue}{\frac{y \cdot y - 1 \cdot 1}{y - 1}}}\]
4. Applied associate-/r/51.0
  \[\leadsto 1 - \color{blue}{\frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)}\]
5. Taylor expanded around inf 0.1
  \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}}\]
6. Simplified0.1
  \[\leadsto \color{blue}{\frac{1}{y} + \left(x - 1 \cdot \frac{x}{y}\right)}\]
if -172408127.57777166 < y < 167881476.73563808
1. Initial program 0.2
  \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
2. Simplified0.2
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)}\]
Recombined 2 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -172408127.577771663665771484375 \lor \neg \left(y \le 167881476.735638082027435302734375\right):\\ \;\;\;\;\frac{1}{y} + \left(x - 1 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))

Error

Target

Derivation

`if y < -172408127.57777166 or 167881476.73563808 < y`

`if -172408127.57777166 < y < 167881476.73563808`

Reproduce