Average Error: 24.2 → 6.5
Time: 5.6s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.1125528044552557 \cdot 10^{144}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 1.09631820124514439 \cdot 10^{69}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{\frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.1125528044552557 \cdot 10^{144}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le 1.09631820124514439 \cdot 10^{69}:\\
\;\;\;\;\left(x \cdot y\right) \cdot \frac{\frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot y\right) \cdot 1\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r422607 = x;
        double r422608 = y;
        double r422609 = r422607 * r422608;
        double r422610 = z;
        double r422611 = r422609 * r422610;
        double r422612 = r422610 * r422610;
        double r422613 = t;
        double r422614 = a;
        double r422615 = r422613 * r422614;
        double r422616 = r422612 - r422615;
        double r422617 = sqrt(r422616);
        double r422618 = r422611 / r422617;
        return r422618;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r422619 = z;
        double r422620 = -1.1125528044552557e+144;
        bool r422621 = r422619 <= r422620;
        double r422622 = -1.0;
        double r422623 = x;
        double r422624 = y;
        double r422625 = r422623 * r422624;
        double r422626 = r422622 * r422625;
        double r422627 = 1.0963182012451444e+69;
        bool r422628 = r422619 <= r422627;
        double r422629 = r422619 * r422619;
        double r422630 = t;
        double r422631 = a;
        double r422632 = r422630 * r422631;
        double r422633 = r422629 - r422632;
        double r422634 = sqrt(r422633);
        double r422635 = sqrt(r422634);
        double r422636 = r422619 / r422635;
        double r422637 = r422636 / r422635;
        double r422638 = r422625 * r422637;
        double r422639 = 1.0;
        double r422640 = r422625 * r422639;
        double r422641 = r422628 ? r422638 : r422640;
        double r422642 = r422621 ? r422626 : r422641;
        return r422642;
}

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

Original24.2
Target7.8
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.1125528044552557e+144

    1. Initial program 51.4

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

    3. Applied *-un-lft-identity51.4

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

    4. Applied sqrt-prod51.4

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

    5. Applied times-frac50.5

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

    6. Simplified50.5

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

    8. Applied add-sqr-sqrt50.5

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

    9. Applied sqrt-prod50.5

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

    10. Applied associate-/r*50.5

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

    11. Taylor expanded around -inf 1.4

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

    if -1.1125528044552557e+144 < z < 1.0963182012451444e+69

    1. Initial program 10.9

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

    3. Applied *-un-lft-identity10.9

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

    4. Applied sqrt-prod10.9

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

    5. Applied times-frac8.9

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

    6. Simplified8.9

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

    8. Applied add-sqr-sqrt8.9

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

    9. Applied sqrt-prod9.2

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

    10. Applied associate-/r*9.2

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

    if 1.0963182012451444e+69 < z

    1. Initial program 39.2

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

    3. Applied *-un-lft-identity39.2

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

    4. Applied sqrt-prod39.2

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

    5. Applied times-frac36.7

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

    6. Simplified36.7

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

    7. Taylor expanded around inf 2.9

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.1125528044552557 \cdot 10^{144}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 1.09631820124514439 \cdot 10^{69}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{\frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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)))))