Average Error: 24.6 → 7.2
Time: 4.7s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.767471237781379 \cdot 10^{82}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 6.47554042515165421 \cdot 10^{62}:\\ \;\;\;\;x \cdot \left(\frac{y}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.767471237781379 \cdot 10^{82}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le 6.47554042515165421 \cdot 10^{62}:\\
\;\;\;\;x \cdot \left(\frac{y}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\\

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

\end{array}
double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (((double) (x * y)) * z)) / ((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a))))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((z <= -1.7674712377813788e+82)) {
		VAR = ((double) (-1.0 * ((double) (x * y))));
	} else {
		double VAR_1;
		if ((z <= 6.475540425151654e+62)) {
			VAR_1 = ((double) (x * ((double) (((double) (y / ((double) (((double) cbrt(((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a)))))))) * ((double) cbrt(((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a)))))))))))) * ((double) (z / ((double) cbrt(((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a))))))))))))));
		} else {
			VAR_1 = ((double) (x * y));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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
Target7.5
Herbie7.2
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.7674712377813788e+82

    1. Initial program 41.6

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

    2. Taylor expanded around -inf 2.1

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

    if -1.7674712377813788e+82 < z < 6.475540425151654e+62

    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-frac10.0

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

    6. Simplified10.0

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

    8. Applied associate-*l*10.0

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

    10. Applied add-cube-cbrt10.7

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

    11. Applied *-un-lft-identity10.7

      \[\leadsto x \cdot \left(y \cdot \frac{\color{blue}{1 \cdot z}}{\left(\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}\right) \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\]

    12. Applied times-frac10.7

      \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)}\right)\]

    13. Applied associate-*r*11.1

      \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot \frac{1}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right) \cdot \frac{z}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)}\]

    14. Simplified11.1

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

    if 6.475540425151654e+62 < z

    1. Initial program 38.7

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

    2. Taylor expanded around inf 3.1

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.767471237781379 \cdot 10^{82}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 6.47554042515165421 \cdot 10^{62}:\\ \;\;\;\;x \cdot \left(\frac{y}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020140 
(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)))))