Average Error: 6.7 → 0.4
Time: 19.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.035064490176370167637368676280506465886 \cdot 10^{200}:\\ \;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.143948490603520686127081681315653192418 \cdot 10^{204}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -2.035064490176370167637368676280506465886 \cdot 10^{200}:\\
\;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 2.143948490603520686127081681315653192418 \cdot 10^{204}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r209890 = x;
        double r209891 = y;
        double r209892 = z;
        double r209893 = t;
        double r209894 = r209892 - r209893;
        double r209895 = r209891 * r209894;
        double r209896 = a;
        double r209897 = r209895 / r209896;
        double r209898 = r209890 + r209897;
        return r209898;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r209899 = y;
        double r209900 = z;
        double r209901 = t;
        double r209902 = r209900 - r209901;
        double r209903 = r209899 * r209902;
        double r209904 = -2.03506449017637e+200;
        bool r209905 = r209903 <= r209904;
        double r209906 = a;
        double r209907 = r209906 / r209899;
        double r209908 = r209902 / r209907;
        double r209909 = x;
        double r209910 = r209908 + r209909;
        double r209911 = 2.1439484906035207e+204;
        bool r209912 = r209903 <= r209911;
        double r209913 = r209903 / r209906;
        double r209914 = r209909 + r209913;
        double r209915 = r209899 / r209906;
        double r209916 = fma(r209915, r209902, r209909);
        double r209917 = r209912 ? r209914 : r209916;
        double r209918 = r209905 ? r209910 : r209917;
        return r209918;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.7
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 3 regimes

  2. if (* y (- z t)) < -2.03506449017637e+200

    1. Initial program 30.1

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

    2. Simplified0.5

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

    4. Applied fma-udef0.5

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

    5. Simplified0.5

      \[\leadsto \color{blue}{\frac{z - t}{\frac{a}{y}}} + x\]

    if -2.03506449017637e+200 < (* y (- z t)) < 2.1439484906035207e+204

    1. Initial program 0.4

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

    if 2.1439484906035207e+204 < (* y (- z t))

    1. Initial program 31.6

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

    2. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.035064490176370167637368676280506465886 \cdot 10^{200}:\\ \;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.143948490603520686127081681315653192418 \cdot 10^{204}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +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 -1.07612662163899753e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))