Average Error: 24.7 → 6.2
Time: 15.1s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.4291886083756086 \cdot 10^{+152}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 1.0460513044759073 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{z}{\sqrt{z \cdot z - t \cdot a}} \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.4291886083756086 \cdot 10^{+152}:\\
\;\;\;\;x \cdot \left(-y\right)\\

\mathbf{elif}\;z \le 1.0460513044759073 \cdot 10^{+153}:\\
\;\;\;\;\left(\frac{z}{\sqrt{z \cdot z - t \cdot a}} \cdot x\right) \cdot y\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r13037826 = x;
        double r13037827 = y;
        double r13037828 = r13037826 * r13037827;
        double r13037829 = z;
        double r13037830 = r13037828 * r13037829;
        double r13037831 = r13037829 * r13037829;
        double r13037832 = t;
        double r13037833 = a;
        double r13037834 = r13037832 * r13037833;
        double r13037835 = r13037831 - r13037834;
        double r13037836 = sqrt(r13037835);
        double r13037837 = r13037830 / r13037836;
        return r13037837;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r13037838 = z;
        double r13037839 = -1.4291886083756086e+152;
        bool r13037840 = r13037838 <= r13037839;
        double r13037841 = x;
        double r13037842 = y;
        double r13037843 = -r13037842;
        double r13037844 = r13037841 * r13037843;
        double r13037845 = 1.0460513044759073e+153;
        bool r13037846 = r13037838 <= r13037845;
        double r13037847 = r13037838 * r13037838;
        double r13037848 = t;
        double r13037849 = a;
        double r13037850 = r13037848 * r13037849;
        double r13037851 = r13037847 - r13037850;
        double r13037852 = sqrt(r13037851);
        double r13037853 = r13037838 / r13037852;
        double r13037854 = r13037853 * r13037841;
        double r13037855 = r13037854 * r13037842;
        double r13037856 = r13037841 * r13037842;
        double r13037857 = r13037846 ? r13037855 : r13037856;
        double r13037858 = r13037840 ? r13037844 : r13037857;
        return r13037858;
}

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

Original24.7
Target7.8
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.4291886083756086e+152

    1. Initial program 53.9

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

    2. Taylor expanded around -inf 1.0

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

    3. Simplified1.0

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

    if -1.4291886083756086e+152 < z < 1.0460513044759073e+153

    1. Initial program 11.3

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

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

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

    4. Applied sqrt-prod11.3

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

    5. Applied times-frac9.0

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

    6. Simplified9.0

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

    8. Applied associate-*l*8.5

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

    if 1.0460513044759073e+153 < z

    1. Initial program 54.3

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

    3. Applied *-un-lft-identity54.3

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

    4. Applied sqrt-prod54.3

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

    5. Applied times-frac54.3

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

    6. Simplified54.3

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

    8. Applied associate-*l*54.3

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

    9. Taylor expanded around inf 1.5

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.4291886083756086 \cdot 10^{+152}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 1.0460513044759073 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{z}{\sqrt{z \cdot z - t \cdot a}} \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))))