Average Error: 6.1 → 0.6
Time: 3.5s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -\infty \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 3.04145977992159801 \cdot 10^{266}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -\infty \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 3.04145977992159801 \cdot 10^{266}\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 r309103 = x;
        double r309104 = y;
        double r309105 = z;
        double r309106 = t;
        double r309107 = r309105 - r309106;
        double r309108 = r309104 * r309107;
        double r309109 = a;
        double r309110 = r309108 / r309109;
        double r309111 = r309103 - r309110;
        return r309111;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r309112 = y;
        double r309113 = z;
        double r309114 = t;
        double r309115 = r309113 - r309114;
        double r309116 = r309112 * r309115;
        double r309117 = a;
        double r309118 = r309116 / r309117;
        double r309119 = -inf.0;
        bool r309120 = r309118 <= r309119;
        double r309121 = 3.041459779921598e+266;
        bool r309122 = r309118 <= r309121;
        double r309123 = !r309122;
        bool r309124 = r309120 || r309123;
        double r309125 = r309112 / r309117;
        double r309126 = r309114 - r309113;
        double r309127 = x;
        double r309128 = fma(r309125, r309126, r309127);
        double r309129 = r309127 - r309118;
        double r309130 = r309124 ? r309128 : r309129;
        return r309130;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (/ (* y (- z t)) a) < -inf.0 or 3.041459779921598e+266 < (/ (* y (- z t)) a)

    1. Initial program 52.1

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

    2. Simplified1.5

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

    if -inf.0 < (/ (* y (- z t)) a) < 3.041459779921598e+266

    1. Initial program 0.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -\infty \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 3.04145977992159801 \cdot 10^{266}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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)))