Average Error: 24.7 → 5.8
Time: 19.8s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.828914214900401987678134109339515859521 \cdot 10^{120}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \le 1.239440162778036983903868388902715558197 \cdot 10^{98}:\\ \;\;\;\;\frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{\sqrt{z \cdot z - a \cdot t}} \cdot \sqrt[3]{\sqrt{z \cdot z - a \cdot t}}}} \cdot \left(y \cdot \left(x \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - a \cdot t}\right|}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(\frac{-1}{2}, \frac{a}{\frac{z}{t}}, z\right)} \cdot \left(y \cdot x\right)\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.828914214900401987678134109339515859521 \cdot 10^{120}:\\
\;\;\;\;y \cdot \left(-x\right)\\

\mathbf{elif}\;z \le 1.239440162778036983903868388902715558197 \cdot 10^{98}:\\
\;\;\;\;\frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{\sqrt{z \cdot z - a \cdot t}} \cdot \sqrt[3]{\sqrt{z \cdot z - a \cdot t}}}} \cdot \left(y \cdot \left(x \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - a \cdot t}\right|}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{z}{\mathsf{fma}\left(\frac{-1}{2}, \frac{a}{\frac{z}{t}}, z\right)} \cdot \left(y \cdot x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r10939837 = x;
        double r10939838 = y;
        double r10939839 = r10939837 * r10939838;
        double r10939840 = z;
        double r10939841 = r10939839 * r10939840;
        double r10939842 = r10939840 * r10939840;
        double r10939843 = t;
        double r10939844 = a;
        double r10939845 = r10939843 * r10939844;
        double r10939846 = r10939842 - r10939845;
        double r10939847 = sqrt(r10939846);
        double r10939848 = r10939841 / r10939847;
        return r10939848;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r10939849 = z;
        double r10939850 = -4.828914214900402e+120;
        bool r10939851 = r10939849 <= r10939850;
        double r10939852 = y;
        double r10939853 = x;
        double r10939854 = -r10939853;
        double r10939855 = r10939852 * r10939854;
        double r10939856 = 1.239440162778037e+98;
        bool r10939857 = r10939849 <= r10939856;
        double r10939858 = cbrt(r10939849);
        double r10939859 = r10939849 * r10939849;
        double r10939860 = a;
        double r10939861 = t;
        double r10939862 = r10939860 * r10939861;
        double r10939863 = r10939859 - r10939862;
        double r10939864 = sqrt(r10939863);
        double r10939865 = cbrt(r10939864);
        double r10939866 = r10939865 * r10939865;
        double r10939867 = sqrt(r10939866);
        double r10939868 = r10939858 / r10939867;
        double r10939869 = r10939858 * r10939858;
        double r10939870 = cbrt(r10939863);
        double r10939871 = fabs(r10939870);
        double r10939872 = r10939869 / r10939871;
        double r10939873 = r10939853 * r10939872;
        double r10939874 = r10939852 * r10939873;
        double r10939875 = r10939868 * r10939874;
        double r10939876 = -0.5;
        double r10939877 = r10939849 / r10939861;
        double r10939878 = r10939860 / r10939877;
        double r10939879 = fma(r10939876, r10939878, r10939849);
        double r10939880 = r10939849 / r10939879;
        double r10939881 = r10939852 * r10939853;
        double r10939882 = r10939880 * r10939881;
        double r10939883 = r10939857 ? r10939875 : r10939882;
        double r10939884 = r10939851 ? r10939855 : r10939883;
        return r10939884;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.7
Target7.6
Herbie5.8
\[\begin{array}{l} \mathbf{if}\;z \lt -3.192130590385276419686361646843883646209 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.976268120920894210257945708950453212935 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -4.828914214900402e+120

    1. Initial program 48.0

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

    2. Taylor expanded around -inf 1.7

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

    3. Simplified1.7

      \[\leadsto \color{blue}{-y \cdot x}\]

    if -4.828914214900402e+120 < z < 1.239440162778037e+98

    1. Initial program 11.0

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

    3. Applied *-un-lft-identity11.0

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

    4. Applied sqrt-prod11.0

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

    5. Applied times-frac9.6

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

    6. Simplified9.6

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

    8. Applied add-cube-cbrt10.0

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

    9. Applied sqrt-prod10.0

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

    10. Applied add-cube-cbrt10.3

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

    11. Applied times-frac10.3

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

    12. Applied associate-*r*9.5

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

    13. Simplified8.4

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

    15. Applied add-sqr-sqrt8.4

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

    16. Applied cbrt-prod8.3

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

    if 1.239440162778037e+98 < z

    1. Initial program 43.4

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

    3. Applied *-un-lft-identity43.4

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

    4. Applied sqrt-prod43.4

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

    5. Applied times-frac40.9

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

    6. Simplified40.9

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

    7. Taylor expanded around inf 6.0

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\color{blue}{z - \frac{1}{2} \cdot \frac{a \cdot t}{z}}}\]

    8. Simplified2.2

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{a}{\frac{z}{t}}, z\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.828914214900401987678134109339515859521 \cdot 10^{120}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \le 1.239440162778036983903868388902715558197 \cdot 10^{98}:\\ \;\;\;\;\frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{\sqrt{z \cdot z - a \cdot t}} \cdot \sqrt[3]{\sqrt{z \cdot z - a \cdot t}}}} \cdot \left(y \cdot \left(x \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - a \cdot t}\right|}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(\frac{-1}{2}, \frac{a}{\frac{z}{t}}, z\right)} \cdot \left(y \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

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