Average Error: 5.8 → 0.4
Time: 2.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -9.995484289270434089051779709495019769676 \cdot 10^{266}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 8.372301841351568798694844482890536564686 \cdot 10^{181}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -9.995484289270434089051779709495019769676 \cdot 10^{266}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 8.372301841351568798694844482890536564686 \cdot 10^{181}:\\
\;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r395944 = x;
        double r395945 = y;
        double r395946 = z;
        double r395947 = t;
        double r395948 = r395946 - r395947;
        double r395949 = r395945 * r395948;
        double r395950 = a;
        double r395951 = r395949 / r395950;
        double r395952 = r395944 + r395951;
        return r395952;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r395953 = y;
        double r395954 = z;
        double r395955 = t;
        double r395956 = r395954 - r395955;
        double r395957 = r395953 * r395956;
        double r395958 = -9.995484289270434e+266;
        bool r395959 = r395957 <= r395958;
        double r395960 = a;
        double r395961 = r395953 / r395960;
        double r395962 = x;
        double r395963 = fma(r395961, r395956, r395962);
        double r395964 = 8.372301841351569e+181;
        bool r395965 = r395957 <= r395964;
        double r395966 = 1.0;
        double r395967 = r395966 / r395960;
        double r395968 = r395957 * r395967;
        double r395969 = r395962 + r395968;
        double r395970 = r395960 / r395956;
        double r395971 = r395953 / r395970;
        double r395972 = r395962 + r395971;
        double r395973 = r395965 ? r395969 : r395972;
        double r395974 = r395959 ? r395963 : r395973;
        return r395974;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original5.8
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 3 regimes

  2. if (* y (- z t)) < -9.995484289270434e+266

    1. Initial program 46.2

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

    2. Simplified0.2

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

    if -9.995484289270434e+266 < (* y (- z t)) < 8.372301841351569e+181

    1. Initial program 0.3

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

    3. Applied div-inv0.4

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

    if 8.372301841351569e+181 < (* y (- z t))

    1. Initial program 23.4

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

    3. Applied associate-/l*0.9

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -9.995484289270434089051779709495019769676 \cdot 10^{266}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 8.372301841351568798694844482890536564686 \cdot 10^{181}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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)))