Average Error: 24.5 → 6.1
Time: 4.5s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.48711103202709677 \cdot 10^{146}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le -1.56344281963474788 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{elif}\;z \le 1.8216614650284523 \cdot 10^{-230}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \le 2.966256665740778 \cdot 10^{111}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.48711103202709677 \cdot 10^{146}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le -1.56344281963474788 \cdot 10^{-286}:\\
\;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\

\mathbf{elif}\;z \le 1.8216614650284523 \cdot 10^{-230}:\\
\;\;\;\;\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\\

\mathbf{elif}\;z \le 2.966256665740778 \cdot 10^{111}:\\
\;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r277825 = x;
        double r277826 = y;
        double r277827 = r277825 * r277826;
        double r277828 = z;
        double r277829 = r277827 * r277828;
        double r277830 = r277828 * r277828;
        double r277831 = t;
        double r277832 = a;
        double r277833 = r277831 * r277832;
        double r277834 = r277830 - r277833;
        double r277835 = sqrt(r277834);
        double r277836 = r277829 / r277835;
        return r277836;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r277837 = z;
        double r277838 = -3.4871110320270968e+146;
        bool r277839 = r277837 <= r277838;
        double r277840 = -1.0;
        double r277841 = x;
        double r277842 = y;
        double r277843 = r277841 * r277842;
        double r277844 = r277840 * r277843;
        double r277845 = -1.5634428196347479e-286;
        bool r277846 = r277837 <= r277845;
        double r277847 = r277837 * r277837;
        double r277848 = t;
        double r277849 = a;
        double r277850 = r277848 * r277849;
        double r277851 = r277847 - r277850;
        double r277852 = sqrt(r277851);
        double r277853 = r277837 / r277852;
        double r277854 = r277842 * r277853;
        double r277855 = r277841 * r277854;
        double r277856 = 1.8216614650284523e-230;
        bool r277857 = r277837 <= r277856;
        double r277858 = r277843 * r277837;
        double r277859 = r277858 / r277852;
        double r277860 = 2.966256665740778e+111;
        bool r277861 = r277837 <= r277860;
        double r277862 = r277843 * r277853;
        double r277863 = r277861 ? r277862 : r277843;
        double r277864 = r277857 ? r277859 : r277863;
        double r277865 = r277846 ? r277855 : r277864;
        double r277866 = r277839 ? r277844 : r277865;
        return r277866;
}

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
Target7.6
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 5 regimes

  2. if z < -3.4871110320270968e+146

    1. Initial program 51.7

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

    2. Taylor expanded around -inf 1.4

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

    if -3.4871110320270968e+146 < z < -1.5634428196347479e-286

    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-frac8.6

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

    6. Simplified8.6

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

    8. Applied associate-*l*8.4

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

    if -1.5634428196347479e-286 < z < 1.8216614650284523e-230

    1. Initial program 16.5

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

    3. Applied *-un-lft-identity16.5

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

    4. Applied sqrt-prod16.5

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

    5. Applied times-frac19.3

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

    6. Simplified19.3

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

    8. Applied associate-*r/16.5

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

    if 1.8216614650284523e-230 < z < 2.966256665740778e+111

    1. Initial program 8.9

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

    3. Applied *-un-lft-identity8.9

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

    4. Applied sqrt-prod8.9

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

    5. Applied times-frac6.7

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

    6. Simplified6.7

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

    if 2.966256665740778e+111 < z

    1. Initial program 44.7

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

    2. Taylor expanded around inf 2.5

      \[\leadsto \color{blue}{x \cdot y}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.48711103202709677 \cdot 10^{146}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le -1.56344281963474788 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{elif}\;z \le 1.8216614650284523 \cdot 10^{-230}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \le 2.966256665740778 \cdot 10^{111}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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)))))