Average Error: 6.1 → 0.9
Time: 16.1s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -143671307292510012290142414877735627259900:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \le 1.510491293206764829542810454714574124313 \cdot 10^{-64}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x - \frac{z}{\frac{a}{y}}\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -143671307292510012290142414877735627259900:\\
\;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\

\mathbf{elif}\;a \le 1.510491293206764829542810454714574124313 \cdot 10^{-64}:\\
\;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r14155503 = x;
        double r14155504 = y;
        double r14155505 = z;
        double r14155506 = t;
        double r14155507 = r14155505 - r14155506;
        double r14155508 = r14155504 * r14155507;
        double r14155509 = a;
        double r14155510 = r14155508 / r14155509;
        double r14155511 = r14155503 - r14155510;
        return r14155511;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r14155512 = a;
        double r14155513 = -1.4367130729251001e+41;
        bool r14155514 = r14155512 <= r14155513;
        double r14155515 = x;
        double r14155516 = y;
        double r14155517 = z;
        double r14155518 = t;
        double r14155519 = r14155517 - r14155518;
        double r14155520 = r14155512 / r14155519;
        double r14155521 = r14155516 / r14155520;
        double r14155522 = r14155515 - r14155521;
        double r14155523 = 1.5104912932067648e-64;
        bool r14155524 = r14155512 <= r14155523;
        double r14155525 = r14155519 * r14155516;
        double r14155526 = r14155525 / r14155512;
        double r14155527 = r14155515 - r14155526;
        double r14155528 = r14155518 / r14155512;
        double r14155529 = r14155512 / r14155516;
        double r14155530 = r14155517 / r14155529;
        double r14155531 = r14155515 - r14155530;
        double r14155532 = fma(r14155528, r14155516, r14155531);
        double r14155533 = r14155524 ? r14155527 : r14155532;
        double r14155534 = r14155514 ? r14155522 : r14155533;
        return r14155534;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.6
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.4367130729251001e+41

    1. Initial program 10.7

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

    3. Applied associate-/l*0.3

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

    if -1.4367130729251001e+41 < a < 1.5104912932067648e-64

    1. Initial program 1.1

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

    if 1.5104912932067648e-64 < a

    1. Initial program 7.8

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

    2. Taylor expanded around 0 7.8

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

    3. Simplified1.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -143671307292510012290142414877735627259900:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \le 1.510491293206764829542810454714574124313 \cdot 10^{-64}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x - \frac{z}{\frac{a}{y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

  (- x (/ (* y (- z t)) a)))