Average Error: 22.6 → 7.6
Time: 5.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -12106626069498310 \lor \neg \left(y \le 166180124166105681923912094449664\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -12106626069498310 \lor \neg \left(y \le 166180124166105681923912094449664\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r782089 = 1.0;
        double r782090 = x;
        double r782091 = r782089 - r782090;
        double r782092 = y;
        double r782093 = r782091 * r782092;
        double r782094 = r782092 + r782089;
        double r782095 = r782093 / r782094;
        double r782096 = r782089 - r782095;
        return r782096;
}
↓
double f(double x, double y) {
        double r782097 = y;
        double r782098 = -1.210662606949831e+16;
        bool r782099 = r782097 <= r782098;
        double r782100 = 1.6618012416610568e+32;
        bool r782101 = r782097 <= r782100;
        double r782102 = !r782101;
        bool r782103 = r782099 || r782102;
        double r782104 = 1.0;
        double r782105 = x;
        double r782106 = 2.0;
        double r782107 = pow(r782097, r782106);
        double r782108 = r782105 / r782107;
        double r782109 = r782105 / r782097;
        double r782110 = r782108 - r782109;
        double r782111 = fma(r782104, r782110, r782105);
        double r782112 = r782097 + r782104;
        double r782113 = r782097 / r782112;
        double r782114 = r782105 - r782104;
        double r782115 = fma(r782113, r782114, r782104);
        double r782116 = r782103 ? r782111 : r782115;
        return r782116;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.6
Target0.2
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.210662606949831e+16 or 1.6618012416610568e+32 < y

    1. Initial program 47.7

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

    2. Simplified29.6

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

    4. Applied add-cube-cbrt30.4

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{\left(\sqrt[3]{y + 1} \cdot \sqrt[3]{y + 1}\right) \cdot \sqrt[3]{y + 1}}}, x - 1, 1\right)\]

    5. Applied *-un-lft-identity30.4

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{y + 1} \cdot \sqrt[3]{y + 1}\right) \cdot \sqrt[3]{y + 1}}, x - 1, 1\right)\]

    6. Applied times-frac30.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt[3]{y + 1} \cdot \sqrt[3]{y + 1}} \cdot \frac{y}{\sqrt[3]{y + 1}}}, x - 1, 1\right)\]

    7. Taylor expanded around inf 15.0

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

    8. Simplified15.0

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

    if -1.210662606949831e+16 < y < 1.6618012416610568e+32

    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. Recombined 2 regimes into one program.

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -12106626069498310 \lor \neg \left(y \le 166180124166105681923912094449664\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))))