Average Error: 6.2 → 1.4
Time: 3.0s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -9.123265245159162379441648817674026348617 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;a \le 160974762298423667326976:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\sqrt{a}} \cdot \frac{z - t}{\sqrt{a}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -9.123265245159162379441648817674026348617 \cdot 10^{-287}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\mathbf{elif}\;a \le 160974762298423667326976:\\
\;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r307921 = x;
        double r307922 = y;
        double r307923 = z;
        double r307924 = t;
        double r307925 = r307923 - r307924;
        double r307926 = r307922 * r307925;
        double r307927 = a;
        double r307928 = r307926 / r307927;
        double r307929 = r307921 - r307928;
        return r307929;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r307930 = a;
        double r307931 = -9.123265245159162e-287;
        bool r307932 = r307930 <= r307931;
        double r307933 = y;
        double r307934 = r307933 / r307930;
        double r307935 = t;
        double r307936 = z;
        double r307937 = r307935 - r307936;
        double r307938 = x;
        double r307939 = fma(r307934, r307937, r307938);
        double r307940 = 1.6097476229842367e+23;
        bool r307941 = r307930 <= r307940;
        double r307942 = 1.0;
        double r307943 = r307936 - r307935;
        double r307944 = r307933 * r307943;
        double r307945 = r307930 / r307944;
        double r307946 = r307942 / r307945;
        double r307947 = r307938 - r307946;
        double r307948 = sqrt(r307930);
        double r307949 = r307933 / r307948;
        double r307950 = r307943 / r307948;
        double r307951 = r307949 * r307950;
        double r307952 = r307938 - r307951;
        double r307953 = r307941 ? r307947 : r307952;
        double r307954 = r307932 ? r307939 : r307953;
        return r307954;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.7
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -9.123265245159162e-287

    1. Initial program 6.4

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

    2. Simplified2.0

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

    if -9.123265245159162e-287 < a < 1.6097476229842367e+23

    1. Initial program 1.0

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied clear-num1.0

      \[\leadsto x - \color{blue}{\frac{1}{\frac{a}{y \cdot \left(z - t\right)}}}\]

    if 1.6097476229842367e+23 < a

    1. Initial program 10.0

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt10.1

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

    4. Applied times-frac0.7

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -9.123265245159162379441648817674026348617 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;a \le 160974762298423667326976:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\sqrt{a}} \cdot \frac{z - t}{\sqrt{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))