Average Error: 5.9 → 0.8
Time: 2.7s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.1473918177297942 \cdot 10^{219}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 4.65706236165514431 \cdot 10^{145}:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -1.1473918177297942 \cdot 10^{219}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 4.65706236165514431 \cdot 10^{145}:\\
\;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\

\mathbf{else}:\\
\;\;\;\;x - y \cdot \frac{z - t}{a}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r311855 = x;
        double r311856 = y;
        double r311857 = z;
        double r311858 = t;
        double r311859 = r311857 - r311858;
        double r311860 = r311856 * r311859;
        double r311861 = a;
        double r311862 = r311860 / r311861;
        double r311863 = r311855 - r311862;
        return r311863;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r311864 = y;
        double r311865 = z;
        double r311866 = t;
        double r311867 = r311865 - r311866;
        double r311868 = r311864 * r311867;
        double r311869 = -1.1473918177297942e+219;
        bool r311870 = r311868 <= r311869;
        double r311871 = a;
        double r311872 = r311864 / r311871;
        double r311873 = r311866 - r311865;
        double r311874 = x;
        double r311875 = fma(r311872, r311873, r311874);
        double r311876 = 4.6570623616551443e+145;
        bool r311877 = r311868 <= r311876;
        double r311878 = 1.0;
        double r311879 = r311871 / r311868;
        double r311880 = r311878 / r311879;
        double r311881 = r311874 - r311880;
        double r311882 = r311867 / r311871;
        double r311883 = r311864 * r311882;
        double r311884 = r311874 - r311883;
        double r311885 = r311877 ? r311881 : r311884;
        double r311886 = r311870 ? r311875 : r311885;
        return r311886;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -1.1473918177297942e+219

    1. Initial program 30.3

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

    2. Simplified0.6

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

    if -1.1473918177297942e+219 < (* y (- z t)) < 4.6570623616551443e+145

    1. Initial program 0.5

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

    3. Applied clear-num0.5

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

    if 4.6570623616551443e+145 < (* y (- z t))

    1. Initial program 19.7

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

    3. Applied *-un-lft-identity19.7

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

    4. Applied times-frac2.6

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

    5. Simplified2.6

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.1473918177297942 \cdot 10^{219}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 4.65706236165514431 \cdot 10^{145}:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :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)))