Average Error: 6.1 → 0.5
Time: 3.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 2.995469952665014041540810221972499392166 \cdot 10^{202}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 2.995469952665014041540810221972499392166 \cdot 10^{202}\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 r242947 = x;
        double r242948 = y;
        double r242949 = z;
        double r242950 = t;
        double r242951 = r242949 - r242950;
        double r242952 = r242948 * r242951;
        double r242953 = a;
        double r242954 = r242952 / r242953;
        double r242955 = r242947 + r242954;
        return r242955;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r242956 = y;
        double r242957 = z;
        double r242958 = t;
        double r242959 = r242957 - r242958;
        double r242960 = r242956 * r242959;
        double r242961 = -inf.0;
        bool r242962 = r242960 <= r242961;
        double r242963 = 2.995469952665014e+202;
        bool r242964 = r242960 <= r242963;
        double r242965 = !r242964;
        bool r242966 = r242962 || r242965;
        double r242967 = a;
        double r242968 = r242956 / r242967;
        double r242969 = x;
        double r242970 = fma(r242968, r242959, r242969);
        double r242971 = r242960 / r242967;
        double r242972 = r242969 + r242971;
        double r242973 = r242966 ? r242970 : r242972;
        return r242973;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.8
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -inf.0 or 2.995469952665014e+202 < (* y (- z t))

    1. Initial program 39.5

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

    2. Simplified0.6

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

    if -inf.0 < (* y (- z t)) < 2.995469952665014e+202

    1. Initial program 0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 2.995469952665014041540810221972499392166 \cdot 10^{202}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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)))