Average Error: 25.1 → 6.1
Time: 9.1s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -9.34194979807773912 \cdot 10^{151}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 2.2349389384476415 \cdot 10^{125}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -9.34194979807773912 \cdot 10^{151}:\\
\;\;\;\;x \cdot \left(-y\right)\\

\mathbf{elif}\;z \le 2.2349389384476415 \cdot 10^{125}:\\
\;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r341953 = x;
        double r341954 = y;
        double r341955 = r341953 * r341954;
        double r341956 = z;
        double r341957 = r341955 * r341956;
        double r341958 = r341956 * r341956;
        double r341959 = t;
        double r341960 = a;
        double r341961 = r341959 * r341960;
        double r341962 = r341958 - r341961;
        double r341963 = sqrt(r341962);
        double r341964 = r341957 / r341963;
        return r341964;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r341965 = z;
        double r341966 = -9.341949798077739e+151;
        bool r341967 = r341965 <= r341966;
        double r341968 = x;
        double r341969 = y;
        double r341970 = -r341969;
        double r341971 = r341968 * r341970;
        double r341972 = 2.2349389384476415e+125;
        bool r341973 = r341965 <= r341972;
        double r341974 = r341965 * r341965;
        double r341975 = t;
        double r341976 = a;
        double r341977 = r341975 * r341976;
        double r341978 = r341974 - r341977;
        double r341979 = sqrt(r341978);
        double r341980 = r341965 / r341979;
        double r341981 = r341969 * r341980;
        double r341982 = r341968 * r341981;
        double r341983 = r341968 * r341969;
        double r341984 = r341973 ? r341982 : r341983;
        double r341985 = r341967 ? r341971 : r341984;
        return r341985;
}

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

Original25.1
Target7.5
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 < -9.341949798077739e+151

    1. Initial program 53.6

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

    3. Applied *-un-lft-identity53.6

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

    4. Applied sqrt-prod53.6

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

    5. Applied times-frac53.2

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

    6. Simplified53.2

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

    8. Applied associate-*l*53.2

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

    9. Taylor expanded around -inf 1.5

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

    10. Simplified1.5

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

    if -9.341949798077739e+151 < z < 2.2349389384476415e+125

    1. Initial program 11.5

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

    3. Applied *-un-lft-identity11.5

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

    4. Applied sqrt-prod11.5

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

    5. Applied times-frac9.3

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

    6. Simplified9.3

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

    8. Applied associate-*l*8.6

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

    if 2.2349389384476415e+125 < z

    1. Initial program 47.2

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

    2. Taylor expanded around inf 1.6

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.34194979807773912 \cdot 10^{151}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 2.2349389384476415 \cdot 10^{125}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))))