Average Error: 25.2 → 7.4
Time: 5.9s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.1439133709880612 \cdot 10^{122}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le -4.59212318019564029 \cdot 10^{-25}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{elif}\;z \le 4.93564688867481586 \cdot 10^{38}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}{y} \cdot \sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.1439133709880612 \cdot 10^{122}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le -4.59212318019564029 \cdot 10^{-25}:\\
\;\;\;\;\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\

\mathbf{elif}\;z \le 4.93564688867481586 \cdot 10^{38}:\\
\;\;\;\;\frac{x \cdot z}{\frac{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}{y} \cdot \sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\\

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

\end{array}
double code(double x, double y, double z, double t, double a) {
	return (((x * y) * z) / sqrt(((z * z) - (t * a))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((z <= -5.143913370988061e+122)) {
		VAR = (-1.0 * (x * y));
	} else {
		double VAR_1;
		if ((z <= -4.59212318019564e-25)) {
			VAR_1 = (((x * y) / sqrt(sqrt(((z * z) - (t * a))))) * (z / sqrt(sqrt(((z * z) - (t * a))))));
		} else {
			double VAR_2;
			if ((z <= 4.935646888674816e+38)) {
				VAR_2 = ((x * z) / ((fabs(cbrt(((z * z) - (t * a)))) / y) * sqrt(cbrt(((z * z) - (t * a))))));
			} else {
				VAR_2 = (x * y);
			}
			VAR_1 = VAR_2;
		}
		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

Original25.2
Target7.6
Herbie7.4
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if z < -5.143913370988061e+122

    1. Initial program 47.1

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

    2. Taylor expanded around -inf 2.3

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

    if -5.143913370988061e+122 < z < -4.59212318019564e-25

    1. Initial program 8.5

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

    3. Applied add-sqr-sqrt8.5

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

    4. Applied sqrt-prod8.8

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

    5. Applied times-frac6.0

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

    if -4.59212318019564e-25 < z < 4.935646888674816e+38

    1. Initial program 12.8

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

    3. Applied add-cube-cbrt13.2

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

    4. Applied sqrt-prod13.2

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

    5. Applied times-frac12.8

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

    6. Simplified13.1

      \[\leadsto \color{blue}{\frac{x}{\frac{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}{y}}} \cdot \frac{z}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\]
    7. Using strategy rm

    8. Applied frac-times12.6

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

    if 4.935646888674816e+38 < z

    1. Initial program 36.6

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

    2. Taylor expanded around inf 3.8

      \[\leadsto \color{blue}{x \cdot y}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.1439133709880612 \cdot 10^{122}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le -4.59212318019564029 \cdot 10^{-25}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{elif}\;z \le 4.93564688867481586 \cdot 10^{38}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}{y} \cdot \sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 
(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)))))