Average Error: 24.4 → 6.6
Time: 16.8s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.751960138449552 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \le 1.1757746865835248 \cdot 10^{+102}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.751960138449552 \cdot 10^{+153}:\\
\;\;\;\;y \cdot \left(-x\right)\\

\mathbf{elif}\;z \le 1.1757746865835248 \cdot 10^{+102}:\\
\;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r18010024 = x;
        double r18010025 = y;
        double r18010026 = r18010024 * r18010025;
        double r18010027 = z;
        double r18010028 = r18010026 * r18010027;
        double r18010029 = r18010027 * r18010027;
        double r18010030 = t;
        double r18010031 = a;
        double r18010032 = r18010030 * r18010031;
        double r18010033 = r18010029 - r18010032;
        double r18010034 = sqrt(r18010033);
        double r18010035 = r18010028 / r18010034;
        return r18010035;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r18010036 = z;
        double r18010037 = -7.751960138449552e+153;
        bool r18010038 = r18010036 <= r18010037;
        double r18010039 = y;
        double r18010040 = x;
        double r18010041 = -r18010040;
        double r18010042 = r18010039 * r18010041;
        double r18010043 = 1.1757746865835248e+102;
        bool r18010044 = r18010036 <= r18010043;
        double r18010045 = r18010036 * r18010036;
        double r18010046 = t;
        double r18010047 = a;
        double r18010048 = r18010046 * r18010047;
        double r18010049 = r18010045 - r18010048;
        double r18010050 = sqrt(r18010049);
        double r18010051 = r18010036 / r18010050;
        double r18010052 = r18010040 * r18010051;
        double r18010053 = r18010039 * r18010052;
        double r18010054 = r18010039 * r18010040;
        double r18010055 = r18010044 ? r18010053 : r18010054;
        double r18010056 = r18010038 ? r18010042 : r18010055;
        return r18010056;
}

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

Original24.4
Target7.8
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -7.751960138449552e+153

    1. Initial program 53.1

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

    2. Taylor expanded around -inf 1.2

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

    3. Simplified1.2

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

    if -7.751960138449552e+153 < z < 1.1757746865835248e+102

    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

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

    8. Applied associate-*l*9.1

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

    if 1.1757746865835248e+102 < z

    1. Initial program 43.6

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

    2. Taylor expanded around inf 2.8

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.751960138449552 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \le 1.1757746865835248 \cdot 10^{+102}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"

  :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)))))