Average Error: 24.5 → 6.4
Time: 4.6s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.1318662440682364 \cdot 10^{+125}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 4.433126571344847 \cdot 10^{+73}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.1318662440682364 \cdot 10^{+125}:\\
\;\;\;\;-x \cdot y\\

\mathbf{elif}\;z \leq 4.433126571344847 \cdot 10^{+73}:\\
\;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\

\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 -1.1318662440682364e+125)
   (- (* x y))
   (if (<= z 4.433126571344847e+73)
     (/ (* x y) (/ (sqrt (- (* z z) (* t a))) z))
     (* 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 <= -1.1318662440682364e+125) {
		tmp = -(x * y);
	} else if (z <= 4.433126571344847e+73) {
		tmp = (x * y) / (sqrt((z * z) - (t * a)) / z);
	} 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

Original24.5
Target8.0
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.1318662440682364e125

    1. Initial program 48.6

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

    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 -1.1318662440682364e125 < z < 4.4331265713448471e73

    1. Initial program 10.5

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

    3. Applied associate-/l*_binary649.0

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

    if 4.4331265713448471e73 < z

    1. Initial program 40.5

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

    2. Taylor expanded around inf 3.2

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1318662440682364 \cdot 10^{+125}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 4.433126571344847 \cdot 10^{+73}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(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)))))