Average Error: 6.4 → 1.3
Time: 48.5s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.292867508096600443770196014802005652851 \cdot 10^{72}:\\ \;\;\;\;x + y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)\\ \mathbf{elif}\;y \le 6.304490465562008580777481844716509272673 \cdot 10^{-114}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4.292867508096600443770196014802005652851 \cdot 10^{72}:\\
\;\;\;\;x + y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)\\

\mathbf{elif}\;y \le 6.304490465562008580777481844716509272673 \cdot 10^{-114}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r15143461 = x;
        double r15143462 = y;
        double r15143463 = z;
        double r15143464 = t;
        double r15143465 = r15143463 - r15143464;
        double r15143466 = r15143462 * r15143465;
        double r15143467 = a;
        double r15143468 = r15143466 / r15143467;
        double r15143469 = r15143461 - r15143468;
        return r15143469;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r15143470 = y;
        double r15143471 = -4.2928675080966004e+72;
        bool r15143472 = r15143470 <= r15143471;
        double r15143473 = x;
        double r15143474 = t;
        double r15143475 = a;
        double r15143476 = r15143474 / r15143475;
        double r15143477 = z;
        double r15143478 = r15143477 / r15143475;
        double r15143479 = r15143476 - r15143478;
        double r15143480 = r15143470 * r15143479;
        double r15143481 = r15143473 + r15143480;
        double r15143482 = 6.3044904655620086e-114;
        bool r15143483 = r15143470 <= r15143482;
        double r15143484 = r15143477 - r15143474;
        double r15143485 = r15143470 * r15143484;
        double r15143486 = r15143485 / r15143475;
        double r15143487 = r15143473 - r15143486;
        double r15143488 = r15143483 ? r15143487 : r15143481;
        double r15143489 = r15143472 ? r15143481 : r15143488;
        return r15143489;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.4
Target0.7
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.2928675080966004e+72 or 6.3044904655620086e-114 < y

    1. Initial program 13.3

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

    2. Simplified3.2

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

    4. Applied fma-udef3.2

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

    6. Applied *-un-lft-identity3.2

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

    7. Applied add-cube-cbrt3.8

      \[\leadsto \left(t - z\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot a} + x\]

    8. Applied times-frac3.8

      \[\leadsto \left(t - z\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{a}\right)} + x\]

    9. Applied associate-*r*9.2

      \[\leadsto \color{blue}{\left(\left(t - z\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}\right) \cdot \frac{\sqrt[3]{y}}{a}} + x\]

    10. Simplified9.2

      \[\leadsto \color{blue}{\left(\sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \left(t - z\right)\right)\right)} \cdot \frac{\sqrt[3]{y}}{a} + x\]

    11. Taylor expanded around 0 13.3

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

    12. Simplified1.7

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

    if -4.2928675080966004e+72 < y < 6.3044904655620086e-114

    1. Initial program 1.0

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.292867508096600443770196014802005652851 \cdot 10^{72}:\\ \;\;\;\;x + y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)\\ \mathbf{elif}\;y \le 6.304490465562008580777481844716509272673 \cdot 10^{-114}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))