Average Error: 24.7 → 6.4
Time: 4.8s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.1315032243066199 \cdot 10^{154}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{elif}\;z \le 1.2924444412571481 \cdot 10^{68}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.1315032243066199 \cdot 10^{154}:\\
\;\;\;\;x \cdot \left(-1 \cdot y\right)\\

\mathbf{elif}\;z \le 1.2924444412571481 \cdot 10^{68}:\\
\;\;\;\;\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 r366009 = x;
        double r366010 = y;
        double r366011 = r366009 * r366010;
        double r366012 = z;
        double r366013 = r366011 * r366012;
        double r366014 = r366012 * r366012;
        double r366015 = t;
        double r366016 = a;
        double r366017 = r366015 * r366016;
        double r366018 = r366014 - r366017;
        double r366019 = sqrt(r366018);
        double r366020 = r366013 / r366019;
        return r366020;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r366021 = z;
        double r366022 = -1.1315032243066199e+154;
        bool r366023 = r366021 <= r366022;
        double r366024 = x;
        double r366025 = -1.0;
        double r366026 = y;
        double r366027 = r366025 * r366026;
        double r366028 = r366024 * r366027;
        double r366029 = 1.292444441257148e+68;
        bool r366030 = r366021 <= r366029;
        double r366031 = r366024 * r366026;
        double r366032 = r366021 * r366021;
        double r366033 = t;
        double r366034 = a;
        double r366035 = r366033 * r366034;
        double r366036 = r366032 - r366035;
        double r366037 = sqrt(r366036);
        double r366038 = r366021 / r366037;
        double r366039 = r366031 * r366038;
        double r366040 = r366030 ? r366039 : r366031;
        double r366041 = r366023 ? r366028 : r366040;
        return r366041;
}

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
Target7.5
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.1315032243066199e+154

    1. Initial program 53.4

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

    3. Applied *-un-lft-identity53.4

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

    4. Applied sqrt-prod53.4

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

    5. Applied times-frac53.0

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

    6. Simplified53.0

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

    8. Applied associate-*l*53.0

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

    9. Taylor expanded around -inf 0.9

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

    if -1.1315032243066199e+154 < z < 1.292444441257148e+68

    1. Initial program 11.0

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

    3. Applied *-un-lft-identity11.0

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

    4. Applied sqrt-prod11.0

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

    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

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

    if 1.292444441257148e+68 < z

    1. Initial program 39.8

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

    2. Taylor expanded around inf 2.7

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.1315032243066199 \cdot 10^{154}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{elif}\;z \le 1.2924444412571481 \cdot 10^{68}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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)))))