Average Error: 25.0 → 6.1
Time: 4.9s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.0286624309368284 \cdot 10^{146}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 2.6367843017119099 \cdot 10^{123}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.0286624309368284 \cdot 10^{146}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le 2.6367843017119099 \cdot 10^{123}:\\
\;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r318751 = x;
        double r318752 = y;
        double r318753 = r318751 * r318752;
        double r318754 = z;
        double r318755 = r318753 * r318754;
        double r318756 = r318754 * r318754;
        double r318757 = t;
        double r318758 = a;
        double r318759 = r318757 * r318758;
        double r318760 = r318756 - r318759;
        double r318761 = sqrt(r318760);
        double r318762 = r318755 / r318761;
        return r318762;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r318763 = z;
        double r318764 = -3.0286624309368284e+146;
        bool r318765 = r318763 <= r318764;
        double r318766 = -1.0;
        double r318767 = x;
        double r318768 = y;
        double r318769 = r318767 * r318768;
        double r318770 = r318766 * r318769;
        double r318771 = 2.63678430171191e+123;
        bool r318772 = r318763 <= r318771;
        double r318773 = r318763 * r318763;
        double r318774 = t;
        double r318775 = a;
        double r318776 = r318774 * r318775;
        double r318777 = r318773 - r318776;
        double r318778 = sqrt(r318777);
        double r318779 = r318778 / r318763;
        double r318780 = r318768 / r318779;
        double r318781 = r318767 * r318780;
        double r318782 = 1.0;
        double r318783 = r318769 * r318782;
        double r318784 = r318772 ? r318781 : r318783;
        double r318785 = r318765 ? r318770 : r318784;
        return r318785;
}

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

Original25.0
Target8.0
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.0286624309368284e+146

    1. Initial program 52.7

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

    2. Taylor expanded around -inf 1.5

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

    if -3.0286624309368284e+146 < z < 2.63678430171191e+123

    1. Initial program 11.2

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

    3. Applied associate-/l*9.0

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

    5. Applied *-un-lft-identity9.0

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

    6. Applied *-un-lft-identity9.0

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

    7. Applied sqrt-prod9.0

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

    8. Applied times-frac9.0

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

    9. Applied times-frac8.5

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

    10. Simplified8.5

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

    if 2.63678430171191e+123 < z

    1. Initial program 48.7

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

    3. Applied *-un-lft-identity48.7

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

    4. Applied sqrt-prod48.7

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

    5. Applied times-frac46.9

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

    6. Simplified46.9

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

    7. Taylor expanded around inf 1.4

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.0286624309368284 \cdot 10^{146}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 2.6367843017119099 \cdot 10^{123}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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)))))