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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -18067858855719976 \lor \neg \left(y \le 7.03747293035493089 \cdot 10^{31}\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{y + 1}, x - 1, 1\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \le -18067858855719976 \lor \neg \left(y \le 7.03747293035493089 \cdot 10^{31}\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\

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

\end{array}

double f(double x, double y) {
        double r676796 = 1.0;
        double r676797 = x;
        double r676798 = r676796 - r676797;
        double r676799 = y;
        double r676800 = r676798 * r676799;
        double r676801 = r676799 + r676796;
        double r676802 = r676800 / r676801;
        double r676803 = r676796 - r676802;
        return r676803;
}

↓

double f(double x, double y) {
        double r676804 = y;
        double r676805 = -18067858855719976.0;
        bool r676806 = r676804 <= r676805;
        double r676807 = 7.037472930354931e+31;
        bool r676808 = r676804 <= r676807;
        double r676809 = !r676808;
        bool r676810 = r676806 || r676809;
        double r676811 = 1.0;
        double r676812 = x;
        double r676813 = 2.0;
        double r676814 = pow(r676804, r676813);
        double r676815 = r676812 / r676814;
        double r676816 = r676812 / r676804;
        double r676817 = r676815 - r676816;
        double r676818 = fma(r676811, r676817, r676812);
        double r676819 = 1.0;
        double r676820 = r676804 + r676811;
        double r676821 = r676819 / r676820;
        double r676822 = r676804 * r676821;
        double r676823 = r676812 - r676811;
        double r676824 = fma(r676822, r676823, r676811);
        double r676825 = r676810 ? r676818 : r676824;
        return r676825;
}

Original	22.1
Target	0.2
Herbie	7.4

Derivation

Split input into 2 regimes
if y < -18067858855719976.0 or 7.037472930354931e+31 < y
1. Initial program 46.8
  \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
2. Simplified28.9
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)}\]
3. Using strategy rm
4. Applied div-inv29.0
  \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \frac{1}{y + 1}}, x - 1, 1\right)\]
5. Taylor expanded around inf 14.8
  \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{x}{y}}\]
6. Simplified14.8
  \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)}\]
if -18067858855719976.0 < y < 7.037472930354931e+31
1. Initial program 1.3
  \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
2. Simplified1.2
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)}\]
3. Using strategy rm
4. Applied div-inv1.2
  \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \frac{1}{y + 1}}, x - 1, 1\right)\]
Recombined 2 regimes into one program.
Final simplification7.4
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -18067858855719976 \lor \neg \left(y \le 7.03747293035493089 \cdot 10^{31}\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{y + 1}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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 < -18067858855719976.0 or 7.037472930354931e+31 < y`

`if -18067858855719976.0 < y < 7.037472930354931e+31`

Reproduce