Average Error: 6.4 → 0.4
Time: 3.3s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.885127437985216706617229964801861933916 \cdot 10^{296} \lor \neg \left(y \cdot \left(z - t\right) \le 1.675666324185826084917217485128398230202 \cdot 10^{291}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -2.885127437985216706617229964801861933916 \cdot 10^{296} \lor \neg \left(y \cdot \left(z - t\right) \le 1.675666324185826084917217485128398230202 \cdot 10^{291}\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 r298013 = x;
        double r298014 = y;
        double r298015 = z;
        double r298016 = t;
        double r298017 = r298015 - r298016;
        double r298018 = r298014 * r298017;
        double r298019 = a;
        double r298020 = r298018 / r298019;
        double r298021 = r298013 + r298020;
        return r298021;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r298022 = y;
        double r298023 = z;
        double r298024 = t;
        double r298025 = r298023 - r298024;
        double r298026 = r298022 * r298025;
        double r298027 = -2.8851274379852167e+296;
        bool r298028 = r298026 <= r298027;
        double r298029 = 1.675666324185826e+291;
        bool r298030 = r298026 <= r298029;
        double r298031 = !r298030;
        bool r298032 = r298028 || r298031;
        double r298033 = a;
        double r298034 = r298022 / r298033;
        double r298035 = x;
        double r298036 = fma(r298034, r298025, r298035);
        double r298037 = r298026 / r298033;
        double r298038 = r298035 + r298037;
        double r298039 = r298032 ? r298036 : r298038;
        return r298039;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.4
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)) < -2.8851274379852167e+296 or 1.675666324185826e+291 < (* y (- z t))

    1. Initial program 57.2

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

    2. Simplified0.2

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

    if -2.8851274379852167e+296 < (* y (- z t)) < 1.675666324185826e+291

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.885127437985216706617229964801861933916 \cdot 10^{296} \lor \neg \left(y \cdot \left(z - t\right) \le 1.675666324185826084917217485128398230202 \cdot 10^{291}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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)))