Average Error: 24.6 → 5.6
Time: 4.5s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.30694806017698615 \cdot 10^{154}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 2.34863444849625156 \cdot 10^{151}:\\ \;\;\;\;\frac{x}{\sqrt[3]{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot \sqrt[3]{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \cdot \frac{y}{\sqrt[3]{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.30694806017698615 \cdot 10^{154}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le 2.34863444849625156 \cdot 10^{151}:\\
\;\;\;\;\frac{x}{\sqrt[3]{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot \sqrt[3]{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \cdot \frac{y}{\sqrt[3]{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}}\\

\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 <= -1.3069480601769862e+154)) {
		VAR = (-1.0 * (x * y));
	} else {
		double VAR_1;
		if ((z <= 2.3486344484962516e+151)) {
			VAR_1 = ((x / (cbrt((sqrt(((z * z) - (t * a))) / z)) * cbrt((sqrt(((z * z) - (t * a))) / z)))) * (y / cbrt((sqrt(((z * z) - (t * a))) / z))));
		} else {
			VAR_1 = (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
Target7.3
Herbie5.6
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.3069480601769862e+154

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 1.0

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

    if -1.3069480601769862e+154 < z < 2.3486344484962516e+151

    1. Initial program 10.7

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

    3. Applied associate-/l*8.5

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt8.7

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

    6. Applied times-frac7.8

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

    if 2.3486344484962516e+151 < z

    1. Initial program 53.3

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

    2. Taylor expanded around inf 1.3

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.30694806017698615 \cdot 10^{154}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 2.34863444849625156 \cdot 10^{151}:\\ \;\;\;\;\frac{x}{\sqrt[3]{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot \sqrt[3]{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \cdot \frac{y}{\sqrt[3]{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +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)))))