Average Error: 6.0 → 0.4
Time: 14.4s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -4.51210365626401900297645952887463999678 \cdot 10^{201}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 9.346814402829446941667508921526282193068 \cdot 10^{281}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - t}{a} \cdot y\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -4.51210365626401900297645952887463999678 \cdot 10^{201}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 9.346814402829446941667508921526282193068 \cdot 10^{281}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r212932 = x;
        double r212933 = y;
        double r212934 = z;
        double r212935 = t;
        double r212936 = r212934 - r212935;
        double r212937 = r212933 * r212936;
        double r212938 = a;
        double r212939 = r212937 / r212938;
        double r212940 = r212932 - r212939;
        return r212940;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r212941 = y;
        double r212942 = z;
        double r212943 = t;
        double r212944 = r212942 - r212943;
        double r212945 = r212941 * r212944;
        double r212946 = -4.512103656264019e+201;
        bool r212947 = r212945 <= r212946;
        double r212948 = a;
        double r212949 = r212941 / r212948;
        double r212950 = r212943 - r212942;
        double r212951 = x;
        double r212952 = fma(r212949, r212950, r212951);
        double r212953 = 9.346814402829447e+281;
        bool r212954 = r212945 <= r212953;
        double r212955 = r212945 / r212948;
        double r212956 = r212951 - r212955;
        double r212957 = r212944 / r212948;
        double r212958 = r212957 * r212941;
        double r212959 = r212951 - r212958;
        double r212960 = r212954 ? r212956 : r212959;
        double r212961 = r212947 ? r212952 : r212960;
        return r212961;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.0
Target0.8
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -4.512103656264019e+201

    1. Initial program 27.3

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

    2. Simplified1.1

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

    if -4.512103656264019e+201 < (* y (- z t)) < 9.346814402829447e+281

    1. Initial program 0.3

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

    if 9.346814402829447e+281 < (* y (- z t))

    1. Initial program 50.1

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

    2. Taylor expanded around 0 50.1

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

    3. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -4.51210365626401900297645952887463999678 \cdot 10^{201}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 9.346814402829446941667508921526282193068 \cdot 10^{281}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - t}{a} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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 -1.07612662163899753e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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