Average Error: 24.9 → 6.3
Time: 9.7s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.195058800664299 \cdot 10^{+154}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \leq 1.6097388499861037 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.195058800664299 \cdot 10^{+154}:\\
\;\;\;\;-y \cdot x\\

\mathbf{elif}\;z \leq 1.6097388499861037 \cdot 10^{+83}:\\
\;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\

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


\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 -1.195058800664299e+154)
   (- (* y x))
   (if (<= z 1.6097388499861037e+83)
     (* x (/ y (/ (sqrt (- (* z z) (* a t))) z)))
     (* y x))))
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 <= -1.195058800664299e+154) {
		tmp = -(y * x);
	} else if (z <= 1.6097388499861037e+83) {
		tmp = x * (y / (sqrt((z * z) - (a * t)) / z));
	} else {
		tmp = y * x;
	}
	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

Original24.9
Target7.5
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 < -1.1950588006642989e154

    1. Initial program 52.7

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

    2. Taylor expanded in z around -inf 1.3

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

    3. Simplified1.3

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

    if -1.1950588006642989e154 < z < 1.6097388499861037e83

    1. Initial program 11.5

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

    2. Applied *-un-lft-identity_binary6411.5

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

    3. Applied times-frac_binary649.0

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

    4. Applied *-un-lft-identity_binary649.0

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

    5. Applied times-frac_binary649.0

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

    6. Applied associate-*l*_binary648.9

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

    7. Simplified8.9

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

    8. Applied associate-*r/_binary6410.7

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

    9. Applied associate-/l*_binary649.0

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

    if 1.6097388499861037e83 < z

    1. Initial program 41.1

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

    2. Taylor expanded in z around inf 2.4

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.195058800664299 \cdot 10^{+154}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \leq 1.6097388499861037 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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)))))