Average Error: 5.9 → 1.5
Time: 2.5s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;z - t \le -5.8829093822384003 \cdot 10^{-14} \lor \neg \left(z - t \le 6.05878657433977089 \cdot 10^{-111}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;z - t \le -5.8829093822384003 \cdot 10^{-14} \lor \neg \left(z - t \le 6.05878657433977089 \cdot 10^{-111}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r311032 = x;
        double r311033 = y;
        double r311034 = z;
        double r311035 = t;
        double r311036 = r311034 - r311035;
        double r311037 = r311033 * r311036;
        double r311038 = a;
        double r311039 = r311037 / r311038;
        double r311040 = r311032 - r311039;
        return r311040;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r311041 = z;
        double r311042 = t;
        double r311043 = r311041 - r311042;
        double r311044 = -5.8829093822384e-14;
        bool r311045 = r311043 <= r311044;
        double r311046 = 6.058786574339771e-111;
        bool r311047 = r311043 <= r311046;
        double r311048 = !r311047;
        bool r311049 = r311045 || r311048;
        double r311050 = y;
        double r311051 = a;
        double r311052 = r311050 / r311051;
        double r311053 = r311042 - r311041;
        double r311054 = x;
        double r311055 = fma(r311052, r311053, r311054);
        double r311056 = r311043 / r311051;
        double r311057 = r311050 * r311056;
        double r311058 = r311054 - r311057;
        double r311059 = r311049 ? r311055 : r311058;
        return r311059;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original5.9
Target0.6
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- z t) < -5.8829093822384e-14 or 6.058786574339771e-111 < (- z t)

    1. Initial program 7.2

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

    2. Simplified1.6

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

    if -5.8829093822384e-14 < (- z t) < 6.058786574339771e-111

    1. Initial program 0.7

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

    3. Applied *-un-lft-identity0.7

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

    4. Applied times-frac0.9

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

    5. Simplified0.9

      \[\leadsto x - \color{blue}{y} \cdot \frac{z - t}{a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z - t \le -5.8829093822384003 \cdot 10^{-14} \lor \neg \left(z - t \le 6.05878657433977089 \cdot 10^{-111}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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)))