Average Error: 25.0 → 6.4
Time: 9.9s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.002837198668899603186184426685314231232 \cdot 10^{60}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \le 3.938152743281577115008790785650092236667 \cdot 10^{96}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.002837198668899603186184426685314231232 \cdot 10^{60}:\\
\;\;\;\;-x \cdot y\\

\mathbf{elif}\;z \le 3.938152743281577115008790785650092236667 \cdot 10^{96}:\\
\;\;\;\;\frac{x}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r365968 = x;
        double r365969 = y;
        double r365970 = r365968 * r365969;
        double r365971 = z;
        double r365972 = r365970 * r365971;
        double r365973 = r365971 * r365971;
        double r365974 = t;
        double r365975 = a;
        double r365976 = r365974 * r365975;
        double r365977 = r365973 - r365976;
        double r365978 = sqrt(r365977);
        double r365979 = r365972 / r365978;
        return r365979;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r365980 = z;
        double r365981 = -1.0028371986688996e+60;
        bool r365982 = r365980 <= r365981;
        double r365983 = x;
        double r365984 = y;
        double r365985 = r365983 * r365984;
        double r365986 = -r365985;
        double r365987 = 3.938152743281577e+96;
        bool r365988 = r365980 <= r365987;
        double r365989 = r365980 * r365980;
        double r365990 = t;
        double r365991 = a;
        double r365992 = r365990 * r365991;
        double r365993 = r365989 - r365992;
        double r365994 = sqrt(r365993);
        double r365995 = sqrt(r365994);
        double r365996 = cbrt(r365980);
        double r365997 = r365996 * r365996;
        double r365998 = r365995 / r365997;
        double r365999 = r365983 / r365998;
        double r366000 = r365995 / r365996;
        double r366001 = r365984 / r366000;
        double r366002 = r365999 * r366001;
        double r366003 = r365988 ? r366002 : r365985;
        double r366004 = r365982 ? r365986 : r366003;
        return r366004;
}

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
Target7.9
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;z \lt -3.192130590385276419686361646843883646209 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.976268120920894210257945708950453212935 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.0028371986688996e+60

    1. Initial program 39.0

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

    2. Taylor expanded around -inf 3.4

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

    3. Simplified3.4

      \[\leadsto \color{blue}{-x \cdot y}\]

    if -1.0028371986688996e+60 < z < 3.938152743281577e+96

    1. Initial program 11.4

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

    3. Applied associate-/l*10.1

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

    5. Applied add-cube-cbrt10.8

      \[\leadsto \frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}}\]

    6. Applied add-sqr-sqrt10.8

      \[\leadsto \frac{x \cdot y}{\frac{\sqrt{\color{blue}{\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

    7. Applied sqrt-prod10.8

      \[\leadsto \frac{x \cdot y}{\frac{\color{blue}{\sqrt{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt{\sqrt{z \cdot z - t \cdot a}}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

    8. Applied times-frac10.8

      \[\leadsto \frac{x \cdot y}{\color{blue}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}}\]

    9. Applied times-frac9.1

      \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}}\]

    if 3.938152743281577e+96 < z

    1. Initial program 43.9

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

    2. Taylor expanded around inf 2.7

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.002837198668899603186184426685314231232 \cdot 10^{60}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \le 3.938152743281577115008790785650092236667 \cdot 10^{96}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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)))))