Average Error: 24.6 → 7.5
Time: 4.2s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.145387718787729 \cdot 10^{27}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{elif}\;z \le 1.68945971621676345 \cdot 10^{31}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.145387718787729 \cdot 10^{27}:\\
\;\;\;\;x \cdot \left(-1 \cdot y\right)\\

\mathbf{elif}\;z \le 1.68945971621676345 \cdot 10^{31}:\\
\;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\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 r374012 = x;
        double r374013 = y;
        double r374014 = r374012 * r374013;
        double r374015 = z;
        double r374016 = r374014 * r374015;
        double r374017 = r374015 * r374015;
        double r374018 = t;
        double r374019 = a;
        double r374020 = r374018 * r374019;
        double r374021 = r374017 - r374020;
        double r374022 = sqrt(r374021);
        double r374023 = r374016 / r374022;
        return r374023;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r374024 = z;
        double r374025 = -1.1453877187877288e+27;
        bool r374026 = r374024 <= r374025;
        double r374027 = x;
        double r374028 = -1.0;
        double r374029 = y;
        double r374030 = r374028 * r374029;
        double r374031 = r374027 * r374030;
        double r374032 = 1.6894597162167635e+31;
        bool r374033 = r374024 <= r374032;
        double r374034 = r374029 * r374024;
        double r374035 = r374027 * r374034;
        double r374036 = r374024 * r374024;
        double r374037 = t;
        double r374038 = a;
        double r374039 = r374037 * r374038;
        double r374040 = r374036 - r374039;
        double r374041 = sqrt(r374040);
        double r374042 = r374035 / r374041;
        double r374043 = r374027 * r374029;
        double r374044 = r374033 ? r374042 : r374043;
        double r374045 = r374026 ? r374031 : r374044;
        return r374045;
}

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

Original24.6
Target7.6
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.1453877187877288e+27

    1. Initial program 34.6

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

    3. Applied *-un-lft-identity34.6

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

    4. Applied sqrt-prod34.6

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

    5. Applied times-frac32.4

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

    6. Simplified32.4

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

    8. Applied associate-*l*32.4

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

    9. Taylor expanded around -inf 3.9

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

    if -1.1453877187877288e+27 < z < 1.6894597162167635e+31

    1. Initial program 11.8

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

    3. Applied associate-*l*11.6

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

    if 1.6894597162167635e+31 < z

    1. Initial program 35.6

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

    2. Taylor expanded around inf 4.3

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.145387718787729 \cdot 10^{27}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{elif}\;z \le 1.68945971621676345 \cdot 10^{31}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))))