Average Error: 6.1 → 0.4
Time: 10.8s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 2.358413248954256348213035766917854469977 \cdot 10^{175}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 2.358413248954256348213035766917854469977 \cdot 10^{175}\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 r136950 = x;
        double r136951 = y;
        double r136952 = z;
        double r136953 = t;
        double r136954 = r136952 - r136953;
        double r136955 = r136951 * r136954;
        double r136956 = a;
        double r136957 = r136955 / r136956;
        double r136958 = r136950 - r136957;
        return r136958;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r136959 = y;
        double r136960 = z;
        double r136961 = t;
        double r136962 = r136960 - r136961;
        double r136963 = r136959 * r136962;
        double r136964 = -inf.0;
        bool r136965 = r136963 <= r136964;
        double r136966 = 2.3584132489542563e+175;
        bool r136967 = r136963 <= r136966;
        double r136968 = !r136967;
        bool r136969 = r136965 || r136968;
        double r136970 = a;
        double r136971 = r136959 / r136970;
        double r136972 = r136961 - r136960;
        double r136973 = x;
        double r136974 = fma(r136971, r136972, r136973);
        double r136975 = r136963 / r136970;
        double r136976 = r136973 - r136975;
        double r136977 = r136969 ? r136974 : r136976;
        return r136977;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.6
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)) < -inf.0 or 2.3584132489542563e+175 < (* y (- z t))

    1. Initial program 35.2

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

    2. Simplified0.6

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

    if -inf.0 < (* y (- z t)) < 2.3584132489542563e+175

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 2.358413248954256348213035766917854469977 \cdot 10^{175}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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)))