Average Error: 6.3 → 0.9
Time: 9.1s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.131294303807309426502569446175008274808 \cdot 10^{-168}:\\ \;\;\;\;\left(y \cdot \frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{\sqrt[3]{t - z}}{\sqrt[3]{a}} + x\\ \mathbf{elif}\;a \le 8.709696976931573596806654735929958388855 \cdot 10^{77}:\\ \;\;\;\;\left(\frac{t \cdot y}{a} + \left(-\frac{z \cdot y}{a}\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a} + x\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.131294303807309426502569446175008274808 \cdot 10^{-168}:\\
\;\;\;\;\left(y \cdot \frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{\sqrt[3]{t - z}}{\sqrt[3]{a}} + x\\

\mathbf{elif}\;a \le 8.709696976931573596806654735929958388855 \cdot 10^{77}:\\
\;\;\;\;\left(\frac{t \cdot y}{a} + \left(-\frac{z \cdot y}{a}\right)\right) + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r342573 = x;
        double r342574 = y;
        double r342575 = z;
        double r342576 = t;
        double r342577 = r342575 - r342576;
        double r342578 = r342574 * r342577;
        double r342579 = a;
        double r342580 = r342578 / r342579;
        double r342581 = r342573 - r342580;
        return r342581;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r342582 = a;
        double r342583 = -1.1312943038073094e-168;
        bool r342584 = r342582 <= r342583;
        double r342585 = y;
        double r342586 = t;
        double r342587 = z;
        double r342588 = r342586 - r342587;
        double r342589 = cbrt(r342588);
        double r342590 = r342589 * r342589;
        double r342591 = cbrt(r342582);
        double r342592 = r342591 * r342591;
        double r342593 = r342590 / r342592;
        double r342594 = r342585 * r342593;
        double r342595 = r342589 / r342591;
        double r342596 = r342594 * r342595;
        double r342597 = x;
        double r342598 = r342596 + r342597;
        double r342599 = 8.709696976931574e+77;
        bool r342600 = r342582 <= r342599;
        double r342601 = r342586 * r342585;
        double r342602 = r342601 / r342582;
        double r342603 = r342587 * r342585;
        double r342604 = r342603 / r342582;
        double r342605 = -r342604;
        double r342606 = r342602 + r342605;
        double r342607 = r342606 + r342597;
        double r342608 = r342588 / r342582;
        double r342609 = r342585 * r342608;
        double r342610 = r342609 + r342597;
        double r342611 = r342600 ? r342607 : r342610;
        double r342612 = r342584 ? r342598 : r342611;
        return r342612;
}

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

Original6.3
Target0.7
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 < -1.1312943038073094e-168

    1. Initial program 7.1

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

    2. Simplified1.7

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

    4. Applied fma-udef1.7

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

    6. Applied div-inv1.7

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

    7. Applied associate-*l*3.0

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

    8. Simplified3.0

      \[\leadsto y \cdot \color{blue}{\frac{t - z}{a}} + x\]
    9. Using strategy rm

    10. Applied add-cube-cbrt3.5

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

    11. Applied add-cube-cbrt3.6

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

    12. Applied times-frac3.6

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

    13. Applied associate-*r*0.7

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

    if -1.1312943038073094e-168 < a < 8.709696976931574e+77

    1. Initial program 1.3

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

    2. Simplified3.4

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

    4. Applied fma-udef3.4

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

    6. Applied *-un-lft-identity3.4

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

    7. Applied add-cube-cbrt4.1

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

    8. Applied times-frac4.1

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

    9. Applied associate-*l*9.6

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

    11. Applied sub-neg9.6

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

    12. Applied distribute-lft-in9.6

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

    13. Applied distribute-lft-in9.6

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

    14. Simplified6.4

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

    15. Simplified1.3

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

    if 8.709696976931574e+77 < a

    1. Initial program 12.0

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

    2. Simplified2.2

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

    4. Applied fma-udef2.2

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

    6. Applied div-inv2.3

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

    7. Applied associate-*l*0.6

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

    8. Simplified0.6

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.131294303807309426502569446175008274808 \cdot 10^{-168}:\\ \;\;\;\;\left(y \cdot \frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{\sqrt[3]{t - z}}{\sqrt[3]{a}} + x\\ \mathbf{elif}\;a \le 8.709696976931573596806654735929958388855 \cdot 10^{77}:\\ \;\;\;\;\left(\frac{t \cdot y}{a} + \left(-\frac{z \cdot y}{a}\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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)))