Average Error: 25.6 → 7.5
Time: 4.7s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.0706587308293988 \cdot 10^{58}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 9.37579073063252577 \cdot 10^{51}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.0706587308293988 \cdot 10^{58}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le 9.37579073063252577 \cdot 10^{51}:\\
\;\;\;\;\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{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 ((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 <= -5.070658730829399e+58)) {
		VAR = ((double) (-1.0 * ((double) (x * y))));
	} else {
		double VAR_1;
		if ((z <= 9.375790730632526e+51)) {
			VAR_1 = ((double) (((double) (((double) (x * y)) / ((double) sqrt(((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a)))))))))) * ((double) (z / ((double) sqrt(((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

Original25.6
Target7.8
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -5.070658730829399e+58

    1. Initial program 38.3

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

    2. Taylor expanded around -inf 3.2

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

    if -5.070658730829399e+58 < z < 9.375790730632526e+51

    1. Initial program 12.6

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

    3. Applied add-sqr-sqrt12.6

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

    4. Applied sqrt-prod12.8

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

    5. Applied times-frac11.7

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

    if 9.375790730632526e+51 < z

    1. Initial program 37.0

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

    2. Taylor expanded around inf 3.8

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.0706587308293988 \cdot 10^{58}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 9.37579073063252577 \cdot 10^{51}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +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)))))