Average Error: 6.5 → 0.4
Time: 3.1s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -8.981442880940557861413613942616636285951 \cdot 10^{292} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 8.973171892835292774846855805866219421622 \cdot 10^{305}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -8.981442880940557861413613942616636285951 \cdot 10^{292} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 8.973171892835292774846855805866219421622 \cdot 10^{305}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, 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 r309127 = x;
        double r309128 = y;
        double r309129 = z;
        double r309130 = t;
        double r309131 = r309129 - r309130;
        double r309132 = r309128 * r309131;
        double r309133 = a;
        double r309134 = r309132 / r309133;
        double r309135 = r309127 - r309134;
        return r309135;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r309136 = y;
        double r309137 = z;
        double r309138 = t;
        double r309139 = r309137 - r309138;
        double r309140 = r309136 * r309139;
        double r309141 = a;
        double r309142 = r309140 / r309141;
        double r309143 = -8.981442880940558e+292;
        bool r309144 = r309142 <= r309143;
        double r309145 = 8.973171892835293e+305;
        bool r309146 = r309142 <= r309145;
        double r309147 = !r309146;
        bool r309148 = r309144 || r309147;
        double r309149 = r309136 / r309141;
        double r309150 = r309138 - r309137;
        double r309151 = x;
        double r309152 = fma(r309149, r309150, r309151);
        double r309153 = r309151 - r309142;
        double r309154 = r309148 ? r309152 : r309153;
        return r309154;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.5
Target0.7
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 2 regimes

  2. if (/ (* y (- z t)) a) < -8.981442880940558e+292 or 8.973171892835293e+305 < (/ (* y (- z t)) a)

    1. Initial program 57.8

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

    2. Simplified1.0

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

    if -8.981442880940558e+292 < (/ (* y (- z t)) a) < 8.973171892835293e+305

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -8.981442880940557861413613942616636285951 \cdot 10^{292} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 8.973171892835292774846855805866219421622 \cdot 10^{305}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))