Average Error: 24.8 → 6.6
Time: 5.1s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.79622158684895449 \cdot 10^{111}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{elif}\;z \le 2.3472884937339154 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{\frac{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\frac{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.79622158684895449 \cdot 10^{111}:\\
\;\;\;\;x \cdot \left(-1 \cdot y\right)\\

\mathbf{elif}\;z \le 2.3472884937339154 \cdot 10^{-9}:\\
\;\;\;\;\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 r251630 = x;
        double r251631 = y;
        double r251632 = r251630 * r251631;
        double r251633 = z;
        double r251634 = r251632 * r251633;
        double r251635 = r251633 * r251633;
        double r251636 = t;
        double r251637 = a;
        double r251638 = r251636 * r251637;
        double r251639 = r251635 - r251638;
        double r251640 = sqrt(r251639);
        double r251641 = r251634 / r251640;
        return r251641;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r251642 = z;
        double r251643 = -2.7962215868489545e+111;
        bool r251644 = r251642 <= r251643;
        double r251645 = x;
        double r251646 = -1.0;
        double r251647 = y;
        double r251648 = r251646 * r251647;
        double r251649 = r251645 * r251648;
        double r251650 = 2.3472884937339154e-09;
        bool r251651 = r251642 <= r251650;
        double r251652 = r251642 * r251642;
        double r251653 = t;
        double r251654 = a;
        double r251655 = r251653 * r251654;
        double r251656 = r251652 - r251655;
        double r251657 = sqrt(r251656);
        double r251658 = cbrt(r251657);
        double r251659 = r251658 * r251658;
        double r251660 = cbrt(r251642);
        double r251661 = r251660 * r251660;
        double r251662 = r251659 / r251661;
        double r251663 = r251645 / r251662;
        double r251664 = r251658 / r251660;
        double r251665 = r251647 / r251664;
        double r251666 = r251663 * r251665;
        double r251667 = r251645 * r251647;
        double r251668 = r251651 ? r251666 : r251667;
        double r251669 = r251644 ? r251649 : r251668;
        return r251669;
}

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

Original24.8
Target7.7
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.7962215868489545e+111

    1. Initial program 45.7

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

    3. Applied associate-/l*43.6

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

    5. Applied *-un-lft-identity43.6

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

    6. Applied *-un-lft-identity43.6

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

    7. Applied sqrt-prod43.6

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

    8. Applied times-frac43.6

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

    9. Applied times-frac43.6

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

    10. Simplified43.6

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

    11. Taylor expanded around -inf 2.3

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

    if -2.7962215868489545e+111 < z < 2.3472884937339154e-09

    1. Initial program 12.0

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

    3. Applied associate-/l*10.9

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

    5. Applied add-cube-cbrt11.6

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

    6. Applied add-cube-cbrt11.3

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

    7. Applied times-frac11.3

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

    8. Applied times-frac8.8

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

    if 2.3472884937339154e-09 < z

    1. Initial program 31.8

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

    2. Taylor expanded around inf 5.8

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.79622158684895449 \cdot 10^{111}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{elif}\;z \le 2.3472884937339154 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{\frac{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\frac{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(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)))))