Average Error: 6.1 → 0.5
Time: 3.2s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -6.59417047227896231 \cdot 10^{254}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 5.35794824374080815 \cdot 10^{166}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -6.59417047227896231 \cdot 10^{254}:\\
\;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 5.35794824374080815 \cdot 10^{166}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{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 r271168 = x;
        double r271169 = y;
        double r271170 = z;
        double r271171 = t;
        double r271172 = r271170 - r271171;
        double r271173 = r271169 * r271172;
        double r271174 = a;
        double r271175 = r271173 / r271174;
        double r271176 = r271168 - r271175;
        return r271176;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r271177 = y;
        double r271178 = z;
        double r271179 = t;
        double r271180 = r271178 - r271179;
        double r271181 = r271177 * r271180;
        double r271182 = -6.594170472278962e+254;
        bool r271183 = r271181 <= r271182;
        double r271184 = x;
        double r271185 = a;
        double r271186 = r271185 / r271180;
        double r271187 = r271177 / r271186;
        double r271188 = r271184 - r271187;
        double r271189 = 5.357948243740808e+166;
        bool r271190 = r271181 <= r271189;
        double r271191 = r271181 / r271185;
        double r271192 = r271184 - r271191;
        double r271193 = r271177 / r271185;
        double r271194 = r271179 - r271178;
        double r271195 = fma(r271193, r271194, r271184);
        double r271196 = r271190 ? r271192 : r271195;
        double r271197 = r271183 ? r271188 : r271196;
        return r271197;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.7
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -6.594170472278962e+254

    1. Initial program 41.8

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

    3. Applied associate-/l*0.4

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

    if -6.594170472278962e+254 < (* y (- z t)) < 5.357948243740808e+166

    1. Initial program 0.3

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

    if 5.357948243740808e+166 < (* y (- z t))

    1. Initial program 24.1

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

    2. Simplified1.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -6.59417047227896231 \cdot 10^{254}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 5.35794824374080815 \cdot 10^{166}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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)))