Average Error: 6.4 → 0.8
Time: 2.9s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -5.39575041880648583 \cdot 10^{136}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 1.0672653342869649 \cdot 10^{207}:\\ \;\;\;\;x - \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -5.39575041880648583 \cdot 10^{136}:\\
\;\;\;\;x - y \cdot \frac{z - t}{a}\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 1.0672653342869649 \cdot 10^{207}:\\
\;\;\;\;x - \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r362634 = x;
        double r362635 = y;
        double r362636 = z;
        double r362637 = t;
        double r362638 = r362636 - r362637;
        double r362639 = r362635 * r362638;
        double r362640 = a;
        double r362641 = r362639 / r362640;
        double r362642 = r362634 - r362641;
        return r362642;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r362643 = y;
        double r362644 = z;
        double r362645 = t;
        double r362646 = r362644 - r362645;
        double r362647 = r362643 * r362646;
        double r362648 = -5.395750418806486e+136;
        bool r362649 = r362647 <= r362648;
        double r362650 = x;
        double r362651 = a;
        double r362652 = r362646 / r362651;
        double r362653 = r362643 * r362652;
        double r362654 = r362650 - r362653;
        double r362655 = 1.0672653342869649e+207;
        bool r362656 = r362647 <= r362655;
        double r362657 = 1.0;
        double r362658 = r362657 / r362651;
        double r362659 = r362647 * r362658;
        double r362660 = r362650 - r362659;
        double r362661 = r362643 / r362651;
        double r362662 = r362645 - r362644;
        double r362663 = fma(r362661, r362662, r362650);
        double r362664 = r362656 ? r362660 : r362663;
        double r362665 = r362649 ? r362654 : r362664;
        return r362665;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.4
Target0.7
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -5.395750418806486e+136

    1. Initial program 20.6

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

    3. Applied *-un-lft-identity20.6

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

    4. Applied times-frac2.4

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

    5. Simplified2.4

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

    if -5.395750418806486e+136 < (* y (- z t)) < 1.0672653342869649e+207

    1. Initial program 0.4

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

    3. Applied div-inv0.5

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

    if 1.0672653342869649e+207 < (* y (- z t))

    1. Initial program 31.0

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

    2. Simplified0.4

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -5.39575041880648583 \cdot 10^{136}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 1.0672653342869649 \cdot 10^{207}:\\ \;\;\;\;x - \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))