Average Error: 24.6 → 6.8
Time: 4.8s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.8983821495978231 \cdot 10^{54}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 1.33907476510945172 \cdot 10^{23}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}{\frac{\sqrt{z \cdot z - t \cdot a}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.8983821495978231 \cdot 10^{54}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le 1.33907476510945172 \cdot 10^{23}:\\
\;\;\;\;x \cdot \frac{y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}{\frac{\sqrt{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 <= -7.898382149597823e+54)) {
		VAR = ((double) (-1.0 * ((double) (x * y))));
	} else {
		double VAR_1;
		if ((z <= 1.3390747651094517e+23)) {
			VAR_1 = ((double) (x * ((double) (((double) (y * ((double) (((double) cbrt(z)) * ((double) cbrt(z)))))) / ((double) (((double) sqrt(((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

Original24.6
Target7.4
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -7.8983821495978231e54

    1. Initial program 37.8

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

    2. Taylor expanded around -inf 2.9

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

    if -7.8983821495978231e54 < z < 1.33907476510945172e23

    1. Initial program 11.2

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

    3. Applied associate-/l*10.5

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

    5. Applied *-un-lft-identity10.5

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

    6. Applied *-un-lft-identity10.5

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

    7. Applied sqrt-prod10.5

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

    8. Applied times-frac10.5

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

    9. Applied times-frac10.4

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

    10. Simplified10.4

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

    12. Applied add-cube-cbrt11.0

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

    13. Applied *-un-lft-identity11.0

      \[\leadsto x \cdot \frac{y}{\frac{\sqrt{\color{blue}{1 \cdot \left(z \cdot z - t \cdot a\right)}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

    14. Applied sqrt-prod11.0

      \[\leadsto x \cdot \frac{y}{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{z \cdot z - t \cdot a}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

    15. Applied times-frac11.0

      \[\leadsto x \cdot \frac{y}{\color{blue}{\frac{\sqrt{1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt{z \cdot z - t \cdot a}}{\sqrt[3]{z}}}}\]

    16. Applied associate-/r*10.3

      \[\leadsto x \cdot \color{blue}{\frac{\frac{y}{\frac{\sqrt{1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}}{\frac{\sqrt{z \cdot z - t \cdot a}}{\sqrt[3]{z}}}}\]

    17. Simplified10.3

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

    if 1.33907476510945172e23 < z

    1. Initial program 35.2

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

    2. Taylor expanded around inf 4.6

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.8983821495978231 \cdot 10^{54}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 1.33907476510945172 \cdot 10^{23}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}{\frac{\sqrt{z \cdot z - t \cdot a}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(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) (neg (* 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)))))