Average Error: 25.3 → 6.6
Time: 6.0s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -9.908813752619088 \cdot 10^{+147}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2883203021.7731514:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \left(\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}\right)\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -9.908813752619088 \cdot 10^{+147}:\\
\;\;\;\;x \cdot \left(-y\right)\\

\mathbf{elif}\;z \leq 2883203021.7731514:\\
\;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \left(\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(y \cdot \frac{z}{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}\right)\\

\end{array}
double code(double x, double y, double z, double t, double a) {
	return (((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 <= -9.908813752619088e+147)) {
		VAR = ((double) (x * ((double) -(y))));
	} else {
		double VAR_1;
		if ((z <= 2883203021.7731514)) {
			VAR_1 = ((double) (x * ((double) (y * (z / ((double) sqrt(((double) (((double) cbrt(((double) (((double) (z * z)) - ((double) (t * a)))))) * ((double) (((double) cbrt(((double) (((double) (z * z)) - ((double) (t * a)))))) * ((double) cbrt(((double) (((double) (z * z)) - ((double) (t * a)))))))))))))))));
		} else {
			VAR_1 = ((double) (x * ((double) (y * (z / ((double) (z + ((double) (((double) (t * (a / z))) * -0.5)))))))));
		}
		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
Target7.5
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -9.9088137526190882e147

    1. Initial program 53.4

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

    2. Simplified52.4

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

    3. Taylor expanded around -inf 1.4

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

    4. Simplified1.4

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

    if -9.9088137526190882e147 < z < 2883203021.7731514

    1. Initial program 11.8

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

    2. Simplified8.8

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

    4. Applied add-cube-cbrt9.2

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

    if 2883203021.7731514 < z

    1. Initial program 35.1

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

    2. Simplified32.4

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

    3. Taylor expanded around inf 7.2

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{z - 0.5 \cdot \frac{a \cdot t}{z}}}\right)\]

    4. Simplified4.6

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.908813752619088 \cdot 10^{+147}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2883203021.7731514:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \left(\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(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)))))