Average Error: 24.7 → 6.3
Time: 4.7s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -2.921133149674579 \cdot 10^{+103}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 3.68902160281237 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -2.921133149674579 \cdot 10^{+103}:\\
\;\;\;\;-x \cdot y\\

\mathbf{elif}\;z \leq 3.68902160281237 \cdot 10^{+114}:\\
\;\;\;\;x \cdot \left(\left(y \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\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 -2.921133149674579e+103)
   (- (* x y))
   (if (<= z 3.68902160281237e+114)
     (*
      x
      (*
       (* y (/ (* (cbrt z) (cbrt z)) (sqrt (sqrt (- (* z z) (* t a))))))
       (/ (cbrt z) (sqrt (sqrt (- (* z z) (* t a)))))))
     (* x y))))
double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / sqrt((z * z) - (t * a));
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.921133149674579e+103) {
		tmp = -(x * y);
	} else if (z <= 3.68902160281237e+114) {
		tmp = x * ((y * ((cbrt(z) * cbrt(z)) / sqrt(sqrt((z * z) - (t * a))))) * (cbrt(z) / sqrt(sqrt((z * z) - (t * a)))));
	} else {
		tmp = x * y;
	}
	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

Original24.7
Target7.6
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.92113314967457889e103

    1. Initial program 43.7

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

    2. Taylor expanded around -inf 2.2

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

    3. Simplified2.2

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

    if -2.92113314967457889e103 < z < 3.68902160281237e114

    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 *-un-lft-identity_binary6410.8

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

    4. Applied sqrt-prod_binary6410.8

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

    5. Applied times-frac_binary649.0

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

    6. Simplified9.0

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

    8. Applied associate-*l*_binary648.9

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

    10. Applied add-sqr-sqrt_binary649.1

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

    11. Applied add-cube-cbrt_binary649.6

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

    12. Applied times-frac_binary649.6

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

    13. Applied associate-*r*_binary649.1

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

    if 3.68902160281237e114 < z

    1. Initial program 46.2

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

    2. Taylor expanded around inf 2.3

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.921133149674579 \cdot 10^{+103}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 3.68902160281237 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(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)))))