Average Error: 25.2 → 6.4
Time: 17.6s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.6441179798628076 \cdot 10^{111}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 8.6766762052655863 \cdot 10^{51}:\\ \;\;\;\;\frac{\frac{x}{\frac{\frac{\frac{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}{\sqrt[3]{z}}}{\sqrt[3]{z}}}{y}}}{\frac{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.6441179798628076 \cdot 10^{111}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le 8.6766762052655863 \cdot 10^{51}:\\
\;\;\;\;\frac{\frac{x}{\frac{\frac{\frac{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}{\sqrt[3]{z}}}{\sqrt[3]{z}}}{y}}}{\frac{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\

\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 <= -7.644117979862808e+111)) {
		VAR = ((double) (-1.0 * ((double) (x * y))));
	} else {
		double VAR_1;
		if ((z <= 8.676676205265586e+51)) {
			VAR_1 = ((double) (((double) (x / ((double) (((double) (((double) (((double) fabs(((double) cbrt(((double) (((double) (z * z)) - ((double) (t * a)))))))) / ((double) cbrt(z)))) / ((double) cbrt(z)))) / y)))) / ((double) (((double) sqrt(((double) cbrt(((double) (((double) (z * z)) - ((double) (t * a)))))))) / ((double) cbrt(z))))));
		} 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

Original25.2
Target7.8
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 3 regimes

  2. if z < -7.644117979862808e+111

    1. Initial program 46.1

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

    2. Taylor expanded around -inf 2.1

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

    if -7.644117979862808e+111 < z < 8.676676205265586e+51

    1. Initial program 11.6

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

    3. Applied associate-/l*10.4

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

    5. Applied add-cube-cbrt11.1

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

    6. Applied add-cube-cbrt11.1

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

    7. Applied sqrt-prod11.0

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

    8. Applied times-frac11.1

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

    9. Applied associate-/r*9.9

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

    10. Simplified9.3

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

    if 8.676676205265586e+51 < z

    1. Initial program 37.9

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.6441179798628076 \cdot 10^{111}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 8.6766762052655863 \cdot 10^{51}:\\ \;\;\;\;\frac{\frac{x}{\frac{\frac{\frac{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}{\sqrt[3]{z}}}{\sqrt[3]{z}}}{y}}}{\frac{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020126 
(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)))))