Average Error: 22.6 → 7.6
Time: 3.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.34661344067154047 \cdot 10^{25} \lor \neg \left(y \le 2.5663705514386888 \cdot 10^{91}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.34661344067154047 \cdot 10^{25} \lor \neg \left(y \le 2.5663705514386888 \cdot 10^{91}\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 r577356 = 1.0;
        double r577357 = x;
        double r577358 = r577356 - r577357;
        double r577359 = y;
        double r577360 = r577358 * r577359;
        double r577361 = r577359 + r577356;
        double r577362 = r577360 / r577361;
        double r577363 = r577356 - r577362;
        return r577363;
}
↓
double f(double x, double y) {
        double r577364 = y;
        double r577365 = -1.3466134406715405e+25;
        bool r577366 = r577364 <= r577365;
        double r577367 = 2.5663705514386888e+91;
        bool r577368 = r577364 <= r577367;
        double r577369 = !r577368;
        bool r577370 = r577366 || r577369;
        double r577371 = x;
        double r577372 = r577371 / r577364;
        double r577373 = 1.0;
        double r577374 = r577373 / r577364;
        double r577375 = r577374 - r577373;
        double r577376 = fma(r577372, r577375, r577371);
        double r577377 = 3.0;
        double r577378 = pow(r577364, r577377);
        double r577379 = pow(r577373, r577377);
        double r577380 = r577378 + r577379;
        double r577381 = r577364 / r577380;
        double r577382 = r577364 * r577364;
        double r577383 = r577373 * r577373;
        double r577384 = r577364 * r577373;
        double r577385 = r577383 - r577384;
        double r577386 = r577382 + r577385;
        double r577387 = r577381 * r577386;
        double r577388 = r577371 - r577373;
        double r577389 = fma(r577387, r577388, r577373);
        double r577390 = r577370 ? r577376 : r577389;
        return r577390;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.6
Target0.2
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.3466134406715405e+25 or 2.5663705514386888e+91 < y

    1. Initial program 48.6

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

    2. Simplified29.5

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

    3. Taylor expanded around inf 12.7

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

    4. Simplified12.7

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

    if -1.3466134406715405e+25 < y < 2.5663705514386888e+91

    1. Initial program 4.6

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

    2. Simplified4.0

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

    4. Applied flip3-+4.0

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

    5. Applied associate-/r/4.0

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.34661344067154047 \cdot 10^{25} \lor \neg \left(y \le 2.5663705514386888 \cdot 10^{91}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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))))