Average Error: 6.3 → 0.3
Time: 8.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.365941278162673333391660352650110325009 \cdot 10^{244}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 3.247318579970977390129000643251415184875 \cdot 10^{235}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -2.365941278162673333391660352650110325009 \cdot 10^{244}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 3.247318579970977390129000643251415184875 \cdot 10^{235}:\\
\;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\

\mathbf{else}:\\
\;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r389917 = x;
        double r389918 = y;
        double r389919 = z;
        double r389920 = t;
        double r389921 = r389919 - r389920;
        double r389922 = r389918 * r389921;
        double r389923 = a;
        double r389924 = r389922 / r389923;
        double r389925 = r389917 + r389924;
        return r389925;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r389926 = y;
        double r389927 = z;
        double r389928 = t;
        double r389929 = r389927 - r389928;
        double r389930 = r389926 * r389929;
        double r389931 = -2.3659412781626733e+244;
        bool r389932 = r389930 <= r389931;
        double r389933 = a;
        double r389934 = r389926 / r389933;
        double r389935 = x;
        double r389936 = fma(r389934, r389929, r389935);
        double r389937 = 3.2473185799709774e+235;
        bool r389938 = r389930 <= r389937;
        double r389939 = r389930 / r389933;
        double r389940 = r389939 + r389935;
        double r389941 = r389933 / r389926;
        double r389942 = r389929 / r389941;
        double r389943 = r389942 + r389935;
        double r389944 = r389938 ? r389940 : r389943;
        double r389945 = r389932 ? r389936 : r389944;
        return r389945;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.3
Target0.7
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -2.3659412781626733e+244

    1. Initial program 38.5

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

    2. Simplified0.3

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

    if -2.3659412781626733e+244 < (* y (- z t)) < 3.2473185799709774e+235

    1. Initial program 0.3

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

    2. Simplified2.8

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

    4. Applied fma-udef2.8

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

    5. Simplified2.8

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

    6. Taylor expanded around 0 0.3

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

    7. Simplified7.2

      \[\leadsto \color{blue}{y \cdot \frac{z - t}{a}} + x\]
    8. Using strategy rm

    9. Applied associate-*r/0.3

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

    if 3.2473185799709774e+235 < (* y (- z t))

    1. Initial program 37.2

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

    2. Simplified0.2

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

    4. Applied fma-udef0.2

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

    5. Simplified0.3

      \[\leadsto \color{blue}{\frac{z - t}{\frac{a}{y}}} + x\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.365941278162673333391660352650110325009 \cdot 10^{244}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 3.247318579970977390129000643251415184875 \cdot 10^{235}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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)))