Average Error: 25.3 → 7.3
Time: 5.9s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.3181126064523061 \cdot 10^{+60}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 9.594770076281723 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.3181126064523061 \cdot 10^{+60}:\\
\;\;\;\;-x \cdot y\\

\mathbf{elif}\;z \leq 9.594770076281723 \cdot 10^{+140}:\\
\;\;\;\;x \cdot \left(\left(y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{z \cdot z - t \cdot a}}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot y\\

\end{array}
(FPCore (x y z t a)
 :precision binary64
 (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.3181126064523061e+60)
   (- (* x y))
   (if (<= z 9.594770076281723e+140)
     (*
      x
      (* (* y (* (cbrt z) (cbrt z))) (/ (cbrt z) (sqrt (- (* z z) (* t a))))))
     (* x y))))
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 tmp;
	if ((z <= -1.3181126064523061e+60)) {
		tmp = ((double) -(((double) (x * y))));
	} else {
		double tmp_1;
		if ((z <= 9.594770076281723e+140)) {
			tmp_1 = ((double) (x * ((double) (((double) (y * ((double) (((double) cbrt(z)) * ((double) cbrt(z)))))) * (((double) cbrt(z)) / ((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a)))))))))));
		} else {
			tmp_1 = ((double) (x * y));
		}
		tmp = tmp_1;
	}
	return tmp;
}

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
Target7.7
Herbie7.3
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.3181126064523061e60

    1. Initial program 39.5

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

    2. Taylor expanded around -inf 3.3

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

    3. Simplified3.3

      \[\leadsto \color{blue}{-x \cdot y}\]

    if -1.3181126064523061e60 < z < 9.5947700762817231e140

    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 *-un-lft-identity_binary6411.6

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

    4. Applied sqrt-prod_binary6411.6

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

    5. Applied times-frac_binary649.9

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

    6. Simplified9.9

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

    8. Applied associate-*l*_binary649.8

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

    10. Applied *-un-lft-identity_binary649.8

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

    11. Applied sqrt-prod_binary649.8

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

    12. Applied add-cube-cbrt_binary6410.5

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

    13. Applied times-frac_binary6410.5

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

    14. Applied associate-*r*_binary6410.8

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

    15. Simplified10.8

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

    if 9.5947700762817231e140 < z

    1. Initial program 50.8

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

    2. Taylor expanded around inf 1.2

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3181126064523061 \cdot 10^{+60}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 9.594770076281723 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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)))))