Average Error: 25.3 → 7.2
Time: 6.5s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -2.599070637400722 \cdot 10^{+122}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 1.4065604657845695 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -2.599070637400722 \cdot 10^{+122}:\\
\;\;\;\;-x \cdot y\\

\mathbf{elif}\;z \leq 1.4065604657845695 \cdot 10^{+49}:\\
\;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\

\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 -2.599070637400722e+122)
   (- (* x y))
   (if (<= z 1.4065604657845695e+49)
     (* x (/ (* z y) (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 <= -2.599070637400722e+122) {
		tmp = -(x * y);
	} else if (z <= 1.4065604657845695e+49) {
		tmp = x * ((z * y) / 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

Original25.3
Target7.5
Herbie7.2
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.59907063740072209e122

    1. Initial program 47.9

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

    2. Taylor expanded around -inf 1.8

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

    3. Simplified1.8

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

    if -2.59907063740072209e122 < z < 1.4065604657845695e49

    1. Initial program 11.3

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

    3. Applied associate-*l*_binary64_810812.2

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

    4. Simplified12.2

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

    6. Applied *-un-lft-identity_binary64_816512.2

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

    7. Applied sqrt-prod_binary64_818012.2

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

    8. Applied times-frac_binary64_817110.7

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

    9. Simplified10.7

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

    if 1.4065604657845695e49 < z

    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 3.8

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.599070637400722 \cdot 10^{+122}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 1.4065604657845695 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(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)))))