Average Error: 25.3 → 6.4
Time: 4.7s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.76145830290999486 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le -8.5049186523994108 \cdot 10^{-253}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{elif}\;z \le 1.9983883551142221 \cdot 10^{137}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.76145830290999486 \cdot 10^{153}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le -8.5049186523994108 \cdot 10^{-253}:\\
\;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\

\mathbf{elif}\;z \le 1.9983883551142221 \cdot 10^{137}:\\
\;\;\;\;\frac{x \cdot y}{\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 <= -6.761458302909995e+153)) {
		VAR = (-1.0 * (x * y));
	} else {
		double VAR_1;
		if ((z <= -8.504918652399411e-253)) {
			VAR_1 = (x * (y / (sqrt(((z * z) - (t * a))) / z)));
		} else {
			double VAR_2;
			if ((z <= 1.998388355114222e+137)) {
				VAR_2 = ((x * y) / (sqrt(((z * z) - (t * a))) / z));
			} 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

Original25.3
Target7.7
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if z < -6.761458302909995e+153

    1. Initial program 54.4

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

    2. Taylor expanded around -inf 1.4

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

    if -6.761458302909995e+153 < z < -8.504918652399411e-253

    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 associate-/l*8.1

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

    5. Applied *-un-lft-identity8.1

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

    6. Applied *-un-lft-identity8.1

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

    7. Applied sqrt-prod8.1

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

    8. Applied times-frac8.1

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

    9. Applied times-frac7.3

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

    10. Simplified7.3

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

    if -8.504918652399411e-253 < z < 1.998388355114222e+137

    1. Initial program 11.8

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

    3. Applied associate-/l*10.2

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

    if 1.998388355114222e+137 < z

    1. Initial program 50.6

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

    2. Taylor expanded around inf 1.6

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.76145830290999486 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le -8.5049186523994108 \cdot 10^{-253}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{elif}\;z \le 1.9983883551142221 \cdot 10^{137}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(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)))))