Average Error: 6.2 → 0.9
Time: 23.3s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.294102167235718175854547572327975906299 \cdot 10^{56}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.218276811061773599006994335448883110359 \cdot 10^{300}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -3.294102167235718175854547572327975906299 \cdot 10^{56}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 2.218276811061773599006994335448883110359 \cdot 10^{300}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r48280166 = x;
        double r48280167 = y;
        double r48280168 = z;
        double r48280169 = t;
        double r48280170 = r48280168 - r48280169;
        double r48280171 = r48280167 * r48280170;
        double r48280172 = a;
        double r48280173 = r48280171 / r48280172;
        double r48280174 = r48280166 + r48280173;
        return r48280174;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r48280175 = y;
        double r48280176 = z;
        double r48280177 = t;
        double r48280178 = r48280176 - r48280177;
        double r48280179 = r48280175 * r48280178;
        double r48280180 = -3.294102167235718e+56;
        bool r48280181 = r48280179 <= r48280180;
        double r48280182 = a;
        double r48280183 = r48280175 / r48280182;
        double r48280184 = x;
        double r48280185 = fma(r48280183, r48280178, r48280184);
        double r48280186 = 2.2182768110617736e+300;
        bool r48280187 = r48280179 <= r48280186;
        double r48280188 = r48280179 / r48280182;
        double r48280189 = r48280184 + r48280188;
        double r48280190 = r48280187 ? r48280189 : r48280185;
        double r48280191 = r48280181 ? r48280185 : r48280190;
        return r48280191;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.7
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -3.294102167235718e+56 or 2.2182768110617736e+300 < (* y (- z t))

    1. Initial program 22.6

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

    2. Simplified2.0

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

    if -3.294102167235718e+56 < (* y (- z t)) < 2.2182768110617736e+300

    1. Initial program 0.5

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.294102167235718175854547572327975906299 \cdot 10^{56}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.218276811061773599006994335448883110359 \cdot 10^{300}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :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)))