Average Error: 24.5 → 5.4
Time: 18.3s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.506706062111452568496513260808126439647 \cdot 10^{153}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \le 3.243528684537981667725101402901133521761 \cdot 10^{132}:\\ \;\;\;\;\frac{x}{\frac{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\frac{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.506706062111452568496513260808126439647 \cdot 10^{153}:\\
\;\;\;\;-x \cdot y\\

\mathbf{elif}\;z \le 3.243528684537981667725101402901133521761 \cdot 10^{132}:\\
\;\;\;\;\frac{x}{\frac{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\frac{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot y\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r316429 = x;
        double r316430 = y;
        double r316431 = r316429 * r316430;
        double r316432 = z;
        double r316433 = r316431 * r316432;
        double r316434 = r316432 * r316432;
        double r316435 = t;
        double r316436 = a;
        double r316437 = r316435 * r316436;
        double r316438 = r316434 - r316437;
        double r316439 = sqrt(r316438);
        double r316440 = r316433 / r316439;
        return r316440;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r316441 = z;
        double r316442 = -6.5067060621114526e+153;
        bool r316443 = r316441 <= r316442;
        double r316444 = x;
        double r316445 = y;
        double r316446 = r316444 * r316445;
        double r316447 = -r316446;
        double r316448 = 3.2435286845379817e+132;
        bool r316449 = r316441 <= r316448;
        double r316450 = r316441 * r316441;
        double r316451 = t;
        double r316452 = a;
        double r316453 = r316451 * r316452;
        double r316454 = r316450 - r316453;
        double r316455 = sqrt(r316454);
        double r316456 = cbrt(r316455);
        double r316457 = r316456 * r316456;
        double r316458 = cbrt(r316441);
        double r316459 = r316458 * r316458;
        double r316460 = r316457 / r316459;
        double r316461 = r316444 / r316460;
        double r316462 = r316456 / r316458;
        double r316463 = r316445 / r316462;
        double r316464 = r316461 * r316463;
        double r316465 = r316449 ? r316464 : r316446;
        double r316466 = r316443 ? r316447 : r316465;
        return r316466;
}

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

Original24.5
Target7.8
Herbie5.4
\[\begin{array}{l} \mathbf{if}\;z \lt -3.192130590385276419686361646843883646209 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.976268120920894210257945708950453212935 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -6.5067060621114526e+153

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 0.9

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

    3. Simplified0.9

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

    if -6.5067060621114526e+153 < z < 3.2435286845379817e+132

    1. Initial program 11.4

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

    3. Applied associate-/l*9.2

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

    5. Applied add-cube-cbrt9.9

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

    6. Applied add-cube-cbrt9.4

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

    7. Applied times-frac9.4

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

    8. Applied times-frac7.6

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

    if 3.2435286845379817e+132 < z

    1. Initial program 48.8

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

    2. Taylor expanded around inf 1.4

      \[\leadsto \color{blue}{x \cdot y}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.506706062111452568496513260808126439647 \cdot 10^{153}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \le 3.243528684537981667725101402901133521761 \cdot 10^{132}:\\ \;\;\;\;\frac{x}{\frac{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\frac{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :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)))))