Average Error: 5.5 → 1.2
Time: 21.2s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -3.6583546044435786 \cdot 10^{-85}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \le 2.2026776871235658 \cdot 10^{+77}:\\ \;\;\;\;x - \left(\frac{y \cdot z}{a} - \frac{1}{\frac{a}{t \cdot y}}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - t}{\sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -3.6583546044435786 \cdot 10^{-85}:\\
\;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\

\mathbf{elif}\;a \le 2.2026776871235658 \cdot 10^{+77}:\\
\;\;\;\;x - \left(\frac{y \cdot z}{a} - \frac{1}{\frac{a}{t \cdot y}}\right)\\

\mathbf{else}:\\
\;\;\;\;x - \frac{z - t}{\sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r12629984 = x;
        double r12629985 = y;
        double r12629986 = z;
        double r12629987 = t;
        double r12629988 = r12629986 - r12629987;
        double r12629989 = r12629985 * r12629988;
        double r12629990 = a;
        double r12629991 = r12629989 / r12629990;
        double r12629992 = r12629984 - r12629991;
        return r12629992;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r12629993 = a;
        double r12629994 = -3.6583546044435786e-85;
        bool r12629995 = r12629993 <= r12629994;
        double r12629996 = x;
        double r12629997 = y;
        double r12629998 = z;
        double r12629999 = t;
        double r12630000 = r12629998 - r12629999;
        double r12630001 = r12629993 / r12630000;
        double r12630002 = r12629997 / r12630001;
        double r12630003 = r12629996 - r12630002;
        double r12630004 = 2.2026776871235658e+77;
        bool r12630005 = r12629993 <= r12630004;
        double r12630006 = r12629997 * r12629998;
        double r12630007 = r12630006 / r12629993;
        double r12630008 = 1.0;
        double r12630009 = r12629999 * r12629997;
        double r12630010 = r12629993 / r12630009;
        double r12630011 = r12630008 / r12630010;
        double r12630012 = r12630007 - r12630011;
        double r12630013 = r12629996 - r12630012;
        double r12630014 = cbrt(r12629993);
        double r12630015 = r12630000 / r12630014;
        double r12630016 = r12630014 * r12630014;
        double r12630017 = r12629997 / r12630016;
        double r12630018 = r12630015 * r12630017;
        double r12630019 = r12629996 - r12630018;
        double r12630020 = r12630005 ? r12630013 : r12630019;
        double r12630021 = r12629995 ? r12630003 : r12630020;
        return r12630021;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original5.5
Target0.7
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -3.6583546044435786e-85

    1. Initial program 7.4

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied associate-/l*0.9

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

    if -3.6583546044435786e-85 < a < 2.2026776871235658e+77

    1. Initial program 1.5

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt2.3

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

    4. Applied times-frac5.6

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

    5. Taylor expanded around 0 1.5

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

    7. Applied clear-num1.6

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

    if 2.2026776871235658e+77 < a

    1. Initial program 9.4

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt9.6

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

    4. Applied times-frac1.1

      \[\leadsto x - \color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{a}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.6583546044435786 \cdot 10^{-85}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \le 2.2026776871235658 \cdot 10^{+77}:\\ \;\;\;\;x - \left(\frac{y \cdot z}{a} - \frac{1}{\frac{a}{t \cdot y}}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - t}{\sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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)))