Average Error: 22.5 → 7.7
Time: 4.2s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1893300925979078144 \lor \neg \left(y \le 102337014478885994496\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1893300925979078144 \lor \neg \left(y \le 102337014478885994496\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), x - 1, 1\right)\\

\end{array}
double f(double x, double y) {
        double r634266 = 1.0;
        double r634267 = x;
        double r634268 = r634266 - r634267;
        double r634269 = y;
        double r634270 = r634268 * r634269;
        double r634271 = r634269 + r634266;
        double r634272 = r634270 / r634271;
        double r634273 = r634266 - r634272;
        return r634273;
}
↓
double f(double x, double y) {
        double r634274 = y;
        double r634275 = -1.8933009259790781e+18;
        bool r634276 = r634274 <= r634275;
        double r634277 = 1.02337014478886e+20;
        bool r634278 = r634274 <= r634277;
        double r634279 = !r634278;
        bool r634280 = r634276 || r634279;
        double r634281 = x;
        double r634282 = r634281 / r634274;
        double r634283 = 1.0;
        double r634284 = r634283 / r634274;
        double r634285 = r634284 - r634283;
        double r634286 = fma(r634282, r634285, r634281);
        double r634287 = 3.0;
        double r634288 = pow(r634274, r634287);
        double r634289 = pow(r634283, r634287);
        double r634290 = r634288 + r634289;
        double r634291 = r634274 / r634290;
        double r634292 = r634274 * r634274;
        double r634293 = r634283 * r634283;
        double r634294 = r634274 * r634283;
        double r634295 = r634293 - r634294;
        double r634296 = r634292 + r634295;
        double r634297 = r634291 * r634296;
        double r634298 = r634281 - r634283;
        double r634299 = fma(r634297, r634298, r634283);
        double r634300 = r634280 ? r634286 : r634299;
        return r634300;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.5
Target0.2
Herbie7.7
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.8933009259790781e+18 or 1.02337014478886e+20 < y

    1. Initial program 46.7

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

    2. Simplified29.5

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

    3. Taylor expanded around inf 15.2

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

    4. Simplified15.2

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

    if -1.8933009259790781e+18 < y < 1.02337014478886e+20

    1. Initial program 0.9

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

    2. Simplified0.9

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

    4. Applied flip3-+0.9

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

    5. Applied associate-/r/0.9

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

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1893300925979078144 \lor \neg \left(y \le 102337014478885994496\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))))