Average Error: 6.0 → 0.4
Time: 3.0s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.58610512582075527 \cdot 10^{286} \lor \neg \left(y \cdot \left(z - t\right) \le 3.55029444822809126 \cdot 10^{190}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \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}, 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 r360958 = x;
        double r360959 = y;
        double r360960 = z;
        double r360961 = t;
        double r360962 = r360960 - r360961;
        double r360963 = r360959 * r360962;
        double r360964 = a;
        double r360965 = r360963 / r360964;
        double r360966 = r360958 - r360965;
        return r360966;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r360967 = y;
        double r360968 = z;
        double r360969 = t;
        double r360970 = r360968 - r360969;
        double r360971 = r360967 * r360970;
        double r360972 = -2.5861051258207553e+286;
        bool r360973 = r360971 <= r360972;
        double r360974 = 3.5502944482280913e+190;
        bool r360975 = r360971 <= r360974;
        double r360976 = !r360975;
        bool r360977 = r360973 || r360976;
        double r360978 = a;
        double r360979 = r360967 / r360978;
        double r360980 = r360969 - r360968;
        double r360981 = x;
        double r360982 = fma(r360979, r360980, r360981);
        double r360983 = r360971 / r360978;
        double r360984 = r360981 - r360983;
        double r360985 = r360977 ? r360982 : r360984;
        return r360985;
}

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

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

    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

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

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, 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)))