Average Error: 6.2 → 0.4
Time: 3.3s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.6079589991147918 \cdot 10^{296} \lor \neg \left(y \cdot \left(z - t\right) \le 6.345330527820856 \cdot 10^{275}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -3.6079589991147918 \cdot 10^{296} \lor \neg \left(y \cdot \left(z - t\right) \le 6.345330527820856 \cdot 10^{275}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r331128 = x;
        double r331129 = y;
        double r331130 = z;
        double r331131 = t;
        double r331132 = r331130 - r331131;
        double r331133 = r331129 * r331132;
        double r331134 = a;
        double r331135 = r331133 / r331134;
        double r331136 = r331128 + r331135;
        return r331136;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r331137 = y;
        double r331138 = z;
        double r331139 = t;
        double r331140 = r331138 - r331139;
        double r331141 = r331137 * r331140;
        double r331142 = -3.607958999114792e+296;
        bool r331143 = r331141 <= r331142;
        double r331144 = 6.345330527820856e+275;
        bool r331145 = r331141 <= r331144;
        double r331146 = !r331145;
        bool r331147 = r331143 || r331146;
        double r331148 = a;
        double r331149 = r331137 / r331148;
        double r331150 = x;
        double r331151 = fma(r331149, r331140, r331150);
        double r331152 = r331141 / r331148;
        double r331153 = r331150 + r331152;
        double r331154 = r331147 ? r331151 : r331153;
        return r331154;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -3.607958999114792e+296 or 6.345330527820856e+275 < (* y (- z t))

    1. Initial program 53.2

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

    2. Simplified0.3

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

    if -3.607958999114792e+296 < (* y (- z t)) < 6.345330527820856e+275

    1. Initial program 0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.6079589991147918 \cdot 10^{296} \lor \neg \left(y \cdot \left(z - t\right) \le 6.345330527820856 \cdot 10^{275}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :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)))