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 r632294 = 1.0;
        double r632295 = x;
        double r632296 = r632294 - r632295;
        double r632297 = y;
        double r632298 = r632296 * r632297;
        double r632299 = r632297 + r632294;
        double r632300 = r632298 / r632299;
        double r632301 = r632294 - r632300;
        return r632301;
}
↓
double f(double x, double y) {
        double r632302 = y;
        double r632303 = -1.8933009259790781e+18;
        bool r632304 = r632302 <= r632303;
        double r632305 = 1.02337014478886e+20;
        bool r632306 = r632302 <= r632305;
        double r632307 = !r632306;
        bool r632308 = r632304 || r632307;
        double r632309 = x;
        double r632310 = r632309 / r632302;
        double r632311 = 1.0;
        double r632312 = r632311 / r632302;
        double r632313 = r632312 - r632311;
        double r632314 = fma(r632310, r632313, r632309);
        double r632315 = 3.0;
        double r632316 = pow(r632302, r632315);
        double r632317 = pow(r632311, r632315);
        double r632318 = r632316 + r632317;
        double r632319 = r632302 / r632318;
        double r632320 = r632302 * r632302;
        double r632321 = r632311 * r632311;
        double r632322 = r632302 * r632311;
        double r632323 = r632321 - r632322;
        double r632324 = r632320 + r632323;
        double r632325 = r632319 * r632324;
        double r632326 = r632309 - r632311;
        double r632327 = fma(r632325, r632326, r632311);
        double r632328 = r632308 ? r632314 : r632327;
        return r632328;
}

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))))