Average Error: 6.0 → 0.4
Time: 2.9s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.58610512582075527 \cdot 10^{286} \lor \neg \left(y \cdot \left(z - t\right) \le 3.55029444822809126 \cdot 10^{190}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -2.58610512582075527 \cdot 10^{286} \lor \neg \left(y \cdot \left(z - t\right) \le 3.55029444822809126 \cdot 10^{190}\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 r299704 = x;
        double r299705 = y;
        double r299706 = z;
        double r299707 = t;
        double r299708 = r299706 - r299707;
        double r299709 = r299705 * r299708;
        double r299710 = a;
        double r299711 = r299709 / r299710;
        double r299712 = r299704 + r299711;
        return r299712;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r299713 = y;
        double r299714 = z;
        double r299715 = t;
        double r299716 = r299714 - r299715;
        double r299717 = r299713 * r299716;
        double r299718 = -2.5861051258207553e+286;
        bool r299719 = r299717 <= r299718;
        double r299720 = 3.5502944482280913e+190;
        bool r299721 = r299717 <= r299720;
        double r299722 = !r299721;
        bool r299723 = r299719 || r299722;
        double r299724 = a;
        double r299725 = r299713 / r299724;
        double r299726 = x;
        double r299727 = fma(r299725, r299716, r299726);
        double r299728 = r299717 / r299724;
        double r299729 = r299726 + r299728;
        double r299730 = r299723 ? r299727 : r299729;
        return r299730;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.0
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)) < -2.5861051258207553e+286 or 3.5502944482280913e+190 < (* y (- z t))

    1. Initial program 34.5

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

    2. Simplified0.5

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

    if -2.5861051258207553e+286 < (* y (- z t)) < 3.5502944482280913e+190

    1. Initial program 0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.58610512582075527 \cdot 10^{286} \lor \neg \left(y \cdot \left(z - t\right) \le 3.55029444822809126 \cdot 10^{190}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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)))