Average Error: 24.2 → 6.8
Time: 6.2s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -2.9064435797870745 \cdot 10^{+82}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq -8.759496516960208 \cdot 10^{-250}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{elif}\;z \leq 1.2732782527458712 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 3.819796553361413 \cdot 10^{+121}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -2.9064435797870745 \cdot 10^{+82}:\\
\;\;\;\;-x \cdot y\\

\mathbf{elif}\;z \leq -8.759496516960208 \cdot 10^{-250}:\\
\;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\

\mathbf{elif}\;z \leq 1.2732782527458712 \cdot 10^{+42}:\\
\;\;\;\;\frac{x}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(z \cdot y\right)\\

\mathbf{elif}\;z \leq 3.819796553361413 \cdot 10^{+121}:\\
\;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\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 -2.9064435797870745e+82)
   (- (* x y))
   (if (<= z -8.759496516960208e-250)
     (/ (* x y) (/ (sqrt (- (* z z) (* t a))) z))
     (if (<= z 1.2732782527458712e+42)
       (* (/ x (sqrt (- (* z z) (* t a)))) (* z y))
       (if (<= z 3.819796553361413e+121)
         (* (* x y) (/ z (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 <= -2.9064435797870745e+82) {
		tmp = -(x * y);
	} else if (z <= -8.759496516960208e-250) {
		tmp = (x * y) / (sqrt((z * z) - (t * a)) / z);
	} else if (z <= 1.2732782527458712e+42) {
		tmp = (x / sqrt((z * z) - (t * a))) * (z * y);
	} else if (z <= 3.819796553361413e+121) {
		tmp = (x * y) * (z / 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

Original24.2
Target7.9
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if z < -2.90644357978707446e82

    1. Initial program 40.7

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

    2. Taylor expanded around -inf 3.1

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

    3. Simplified3.1

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

    if -2.90644357978707446e82 < z < -8.7594965169602077e-250

    1. Initial program 9.6

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

    3. Applied associate-/l*_binary648.1

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

    if -8.7594965169602077e-250 < z < 1.2732782527458712e42

    1. Initial program 12.1

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

    3. Applied associate-/l*_binary6411.8

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}}\]
    4. Using strategy rm

    5. Applied div-inv_binary6411.9

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

    6. Applied times-frac_binary6413.3

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

    7. Simplified13.2

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

    if 1.2732782527458712e42 < z < 3.81979655336141291e121

    1. Initial program 8.0

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

    3. Applied *-un-lft-identity_binary648.0

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

    4. Applied sqrt-prod_binary648.0

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

    5. Applied times-frac_binary643.5

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

    6. Simplified3.5

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

    if 3.81979655336141291e121 < z

    1. Initial program 46.6

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

    2. Taylor expanded around inf 1.5

      \[\leadsto \color{blue}{x \cdot y}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9064435797870745 \cdot 10^{+82}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq -8.759496516960208 \cdot 10^{-250}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{elif}\;z \leq 1.2732782527458712 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 3.819796553361413 \cdot 10^{+121}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020273 
(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)))))