Average Error: 6.1 → 2.4
Time: 15.5s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.149518351871894988729473359872954987335 \cdot 10^{-296}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right) + x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\left(\sqrt[3]{y} \cdot \left(t - z\right)\right) \cdot \left|\frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right|}{\sqrt[3]{a}} + x\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.149518351871894988729473359872954987335 \cdot 10^{-296}:\\
\;\;\;\;\frac{y}{a} \cdot \left(t - z\right) + x\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\left(\sqrt[3]{y} \cdot \left(t - z\right)\right) \cdot \left|\frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right|}{\sqrt[3]{a}} + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r220495 = x;
        double r220496 = y;
        double r220497 = z;
        double r220498 = t;
        double r220499 = r220497 - r220498;
        double r220500 = r220496 * r220499;
        double r220501 = a;
        double r220502 = r220500 / r220501;
        double r220503 = r220495 - r220502;
        return r220503;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r220504 = y;
        double r220505 = -3.149518351871895e-296;
        bool r220506 = r220504 <= r220505;
        double r220507 = a;
        double r220508 = r220504 / r220507;
        double r220509 = t;
        double r220510 = z;
        double r220511 = r220509 - r220510;
        double r220512 = r220508 * r220511;
        double r220513 = x;
        double r220514 = r220512 + r220513;
        double r220515 = cbrt(r220504);
        double r220516 = r220515 * r220515;
        double r220517 = cbrt(r220507);
        double r220518 = r220517 * r220517;
        double r220519 = r220516 / r220518;
        double r220520 = sqrt(r220519);
        double r220521 = r220515 * r220511;
        double r220522 = r220515 / r220517;
        double r220523 = fabs(r220522);
        double r220524 = r220521 * r220523;
        double r220525 = r220524 / r220517;
        double r220526 = r220520 * r220525;
        double r220527 = r220526 + r220513;
        double r220528 = r220506 ? r220514 : r220527;
        return r220528;
}

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

Original6.1
Target0.7
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.149518351871895e-296

    1. Initial program 6.3

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

    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

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

    if -3.149518351871895e-296 < y

    1. Initial program 5.9

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

    2. Simplified2.3

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

    4. Applied fma-udef2.3

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

    6. Applied add-cube-cbrt2.8

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

    7. Applied add-cube-cbrt2.9

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

    8. Applied times-frac2.9

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

    9. Applied associate-*l*1.0

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

    11. Applied add-sqr-sqrt1.0

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

    12. Applied associate-*l*1.0

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

    13. Simplified2.7

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.149518351871894988729473359872954987335 \cdot 10^{-296}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right) + x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\left(\sqrt[3]{y} \cdot \left(t - z\right)\right) \cdot \left|\frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right|}{\sqrt[3]{a}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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 -1.07612662163899753e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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