Average Error: 24.3 → 6.1
Time: 4.1m
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.62164551794021002 \cdot 10^{125}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 9.68141924057567665 \cdot 10^{95}:\\ \;\;\;\;\frac{x \cdot 1}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -8.62164551794021002 \cdot 10^{125}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le 9.68141924057567665 \cdot 10^{95}:\\
\;\;\;\;\frac{x \cdot 1}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}\\

\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 <= -8.62164551794021e+125)) {
		VAR = ((double) (-1.0 * ((double) (x * y))));
	} else {
		double VAR_1;
		if ((z <= 9.681419240575677e+95)) {
			VAR_1 = ((double) (((double) (x * 1.0)) / ((double) (((double) (((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a)))))) / z)) / y))));
		} 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

Original24.3
Target7.5
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -8.62164551794021e+125

    1. Initial program 48.1

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

    2. Taylor expanded around -inf 1.7

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

    if -8.62164551794021e+125 < z < 9.681419240575677e+95

    1. Initial program 10.8

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

    3. Applied associate-/l*9.4

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

    5. Applied *-un-lft-identity9.4

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

    6. Applied associate-*r*9.4

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

    7. Applied associate-/l*8.7

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

    if 9.681419240575677e+95 < z

    1. Initial program 41.9

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

    2. Taylor expanded around inf 2.5

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.62164551794021002 \cdot 10^{125}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 9.68141924057567665 \cdot 10^{95}:\\ \;\;\;\;\frac{x \cdot 1}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +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)))))