Average Error: 5.9 → 0.9
Time: 18.9s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.129129962210888 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;y \le 1.2672317070069067 \cdot 10^{-167}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{y}}}, \frac{t - z}{\frac{\sqrt[3]{a}}{\sqrt{y}}}, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -7.129129962210888 \cdot 10^{-45}:\\
\;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\

\mathbf{elif}\;y \le 1.2672317070069067 \cdot 10^{-167}:\\
\;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{y}}}, \frac{t - z}{\frac{\sqrt[3]{a}}{\sqrt{y}}}, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r9838295 = x;
        double r9838296 = y;
        double r9838297 = z;
        double r9838298 = t;
        double r9838299 = r9838297 - r9838298;
        double r9838300 = r9838296 * r9838299;
        double r9838301 = a;
        double r9838302 = r9838300 / r9838301;
        double r9838303 = r9838295 - r9838302;
        return r9838303;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r9838304 = y;
        double r9838305 = -7.129129962210888e-45;
        bool r9838306 = r9838304 <= r9838305;
        double r9838307 = t;
        double r9838308 = z;
        double r9838309 = r9838307 - r9838308;
        double r9838310 = a;
        double r9838311 = r9838309 / r9838310;
        double r9838312 = x;
        double r9838313 = fma(r9838311, r9838304, r9838312);
        double r9838314 = 1.2672317070069067e-167;
        bool r9838315 = r9838304 <= r9838314;
        double r9838316 = r9838304 * r9838309;
        double r9838317 = r9838316 / r9838310;
        double r9838318 = r9838312 + r9838317;
        double r9838319 = 1.0;
        double r9838320 = cbrt(r9838310);
        double r9838321 = r9838320 * r9838320;
        double r9838322 = sqrt(r9838304);
        double r9838323 = r9838321 / r9838322;
        double r9838324 = r9838319 / r9838323;
        double r9838325 = r9838320 / r9838322;
        double r9838326 = r9838309 / r9838325;
        double r9838327 = fma(r9838324, r9838326, r9838312);
        double r9838328 = r9838315 ? r9838318 : r9838327;
        double r9838329 = r9838306 ? r9838313 : r9838328;
        return r9838329;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original5.9
Target0.7
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -7.129129962210888e-45

    1. Initial program 12.0

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]

    2. Simplified3.2

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

    4. Applied fma-udef3.2

      \[\leadsto \color{blue}{\left(t - z\right) \cdot \frac{y}{a} + x}\]
    5. Using strategy rm

    6. Applied associate-*r/12.0

      \[\leadsto \color{blue}{\frac{\left(t - z\right) \cdot y}{a}} + x\]
    7. Using strategy rm

    8. Applied associate-/l*3.0

      \[\leadsto \color{blue}{\frac{t - z}{\frac{a}{y}}} + x\]
    9. Using strategy rm

    10. Applied *-un-lft-identity3.0

      \[\leadsto \frac{t - z}{\frac{a}{y}} + \color{blue}{1 \cdot x}\]

    11. Applied *-un-lft-identity3.0

      \[\leadsto \color{blue}{1 \cdot \frac{t - z}{\frac{a}{y}}} + 1 \cdot x\]

    12. Applied distribute-lft-out3.0

      \[\leadsto \color{blue}{1 \cdot \left(\frac{t - z}{\frac{a}{y}} + x\right)}\]

    13. Simplified1.0

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

    if -7.129129962210888e-45 < y < 1.2672317070069067e-167

    1. Initial program 0.6

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]

    2. Simplified1.8

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

    4. Applied fma-udef1.8

      \[\leadsto \color{blue}{\left(t - z\right) \cdot \frac{y}{a} + x}\]
    5. Using strategy rm

    6. Applied associate-*r/0.6

      \[\leadsto \color{blue}{\frac{\left(t - z\right) \cdot y}{a}} + x\]

    if 1.2672317070069067e-167 < y

    1. Initial program 8.0

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]

    2. Simplified2.8

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

    4. Applied fma-udef2.8

      \[\leadsto \color{blue}{\left(t - z\right) \cdot \frac{y}{a} + x}\]
    5. Using strategy rm

    6. Applied associate-*r/8.0

      \[\leadsto \color{blue}{\frac{\left(t - z\right) \cdot y}{a}} + x\]
    7. Using strategy rm

    8. Applied associate-/l*2.7

      \[\leadsto \color{blue}{\frac{t - z}{\frac{a}{y}}} + x\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt2.8

      \[\leadsto \frac{t - z}{\frac{a}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}} + x\]

    11. Applied add-cube-cbrt3.3

      \[\leadsto \frac{t - z}{\frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\sqrt{y} \cdot \sqrt{y}}} + x\]

    12. Applied times-frac3.3

      \[\leadsto \frac{t - z}{\color{blue}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{y}} \cdot \frac{\sqrt[3]{a}}{\sqrt{y}}}} + x\]

    13. Applied *-un-lft-identity3.3

      \[\leadsto \frac{\color{blue}{1 \cdot \left(t - z\right)}}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{y}} \cdot \frac{\sqrt[3]{a}}{\sqrt{y}}} + x\]

    14. Applied times-frac1.3

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{y}}} \cdot \frac{t - z}{\frac{\sqrt[3]{a}}{\sqrt{y}}}} + x\]

    15. Applied fma-def1.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.129129962210888 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;y \le 1.2672317070069067 \cdot 10^{-167}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{y}}}, \frac{t - z}{\frac{\sqrt[3]{a}}{\sqrt{y}}}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

  (- x (/ (* y (- z t)) a)))