Average Error: 24.5 → 6.2
Time: 5.1s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -2.0295156182980508 \cdot 10^{+142}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 7.14742597666009 \cdot 10^{+61}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -2.0295156182980508 \cdot 10^{+142}:\\
\;\;\;\;-x \cdot y\\

\mathbf{elif}\;z \leq 7.14742597666009 \cdot 10^{+61}:\\
\;\;\;\;x \cdot \left(y \cdot \frac{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 -2.0295156182980508e+142)
   (- (* x y))
   (if (<= z 7.14742597666009e+61)
     (* x (* y (/ 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 <= -2.0295156182980508e+142)) {
		tmp = ((double) -(((double) (x * y))));
	} else {
		double tmp_1;
		if ((z <= 7.14742597666009e+61)) {
			tmp_1 = ((double) (x * ((double) (y * (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

Original24.5
Target7.7
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.0295156182980508e142

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 1.6

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

    3. Simplified1.6

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

    if -2.0295156182980508e142 < z < 7.14742597666008976e61

    1. Initial program 10.7

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

    3. Applied *-un-lft-identity_binary6410.7

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

    4. Applied sqrt-prod_binary6410.7

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

    5. Applied times-frac_binary648.9

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

    6. Simplified8.9

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

    8. Applied associate-*l*_binary648.7

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

    if 7.14742597666008976e61 < z

    1. Initial program 38.7

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

    2. Taylor expanded around inf 3.2

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.0295156182980508 \cdot 10^{+142}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 7.14742597666009 \cdot 10^{+61}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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)))))