Average Error: 24.8 → 6.3
Time: 4.2s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.3433926726132482 \cdot 10^{154}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 1.34066223352094801 \cdot 10^{50}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.3433926726132482 \cdot 10^{154}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le 1.34066223352094801 \cdot 10^{50}:\\
\;\;\;\;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 r220663 = x;
        double r220664 = y;
        double r220665 = r220663 * r220664;
        double r220666 = z;
        double r220667 = r220665 * r220666;
        double r220668 = r220666 * r220666;
        double r220669 = t;
        double r220670 = a;
        double r220671 = r220669 * r220670;
        double r220672 = r220668 - r220671;
        double r220673 = sqrt(r220672);
        double r220674 = r220667 / r220673;
        return r220674;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r220675 = z;
        double r220676 = -1.3433926726132482e+154;
        bool r220677 = r220675 <= r220676;
        double r220678 = -1.0;
        double r220679 = x;
        double r220680 = y;
        double r220681 = r220679 * r220680;
        double r220682 = r220678 * r220681;
        double r220683 = 1.340662233520948e+50;
        bool r220684 = r220675 <= r220683;
        double r220685 = r220675 * r220675;
        double r220686 = t;
        double r220687 = a;
        double r220688 = r220686 * r220687;
        double r220689 = r220685 - r220688;
        double r220690 = sqrt(r220689);
        double r220691 = r220675 / r220690;
        double r220692 = r220680 * r220691;
        double r220693 = r220679 * r220692;
        double r220694 = r220684 ? r220693 : r220681;
        double r220695 = r220677 ? r220682 : r220694;
        return r220695;
}

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

Original24.8
Target7.5
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.3433926726132482e+154

    1. Initial program 54.0

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

    2. Taylor expanded around -inf 1.0

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

    if -1.3433926726132482e+154 < z < 1.340662233520948e+50

    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 *-un-lft-identity11.4

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

    4. Applied sqrt-prod11.4

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

    5. Applied times-frac9.5

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

    6. Simplified9.5

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

    8. Applied associate-*l*8.7

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

    if 1.340662233520948e+50 < z

    1. Initial program 37.2

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

    3. Applied *-un-lft-identity37.2

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

    4. Applied sqrt-prod37.2

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

    5. Applied times-frac34.1

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

    6. Simplified34.1

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

    8. Applied associate-*l*34.2

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

    9. Taylor expanded around inf 4.0

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.3433926726132482 \cdot 10^{154}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 1.34066223352094801 \cdot 10^{50}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(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)))))