Average Error: 6.0 → 0.5
Time: 14.3s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -7.735106615385173103480731659147183712856 \cdot 10^{273}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 6.02197539231061011906492216449992580951 \cdot 10^{152}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\left(z - t\right) \cdot y \le -7.735106615385173103480731659147183712856 \cdot 10^{273}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\mathbf{elif}\;\left(z - t\right) \cdot y \le 6.02197539231061011906492216449992580951 \cdot 10^{152}:\\
\;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r13479393 = x;
        double r13479394 = y;
        double r13479395 = z;
        double r13479396 = t;
        double r13479397 = r13479395 - r13479396;
        double r13479398 = r13479394 * r13479397;
        double r13479399 = a;
        double r13479400 = r13479398 / r13479399;
        double r13479401 = r13479393 - r13479400;
        return r13479401;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r13479402 = z;
        double r13479403 = t;
        double r13479404 = r13479402 - r13479403;
        double r13479405 = y;
        double r13479406 = r13479404 * r13479405;
        double r13479407 = -7.735106615385173e+273;
        bool r13479408 = r13479406 <= r13479407;
        double r13479409 = a;
        double r13479410 = r13479405 / r13479409;
        double r13479411 = r13479403 - r13479402;
        double r13479412 = x;
        double r13479413 = fma(r13479410, r13479411, r13479412);
        double r13479414 = 6.02197539231061e+152;
        bool r13479415 = r13479406 <= r13479414;
        double r13479416 = r13479406 / r13479409;
        double r13479417 = r13479412 - r13479416;
        double r13479418 = r13479415 ? r13479417 : r13479413;
        double r13479419 = r13479408 ? r13479413 : r13479418;
        return r13479419;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.0
Target0.7
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -7.735106615385173e+273 or 6.02197539231061e+152 < (* y (- z t))

    1. Initial program 29.7

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

    2. Simplified0.7

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

    if -7.735106615385173e+273 < (* y (- z t)) < 6.02197539231061e+152

    1. Initial program 0.5

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -7.735106615385173103480731659147183712856 \cdot 10^{273}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 6.02197539231061011906492216449992580951 \cdot 10^{152}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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 (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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