Average Error: 24.6 → 6.7
Time: 10.2s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.1701128116145023 \cdot 10^{+88}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 6.724224666826136 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 2.456100249180412 \cdot 10^{+118}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.1701128116145023 \cdot 10^{+88}:\\
\;\;\;\;-x \cdot y\\

\mathbf{elif}\;z \leq 6.724224666826136 \cdot 10^{-307}:\\
\;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\

\mathbf{elif}\;z \leq 2.456100249180412 \cdot 10^{+118}:\\
\;\;\;\;\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\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 -1.1701128116145023e+88)
   (- (* x y))
   (if (<= z 6.724224666826136e-307)
     (* x (/ (* z y) (sqrt (- (* z z) (* t a)))))
     (if (<= z 2.456100249180412e+118)
       (*
        (/ (* x y) (sqrt (sqrt (- (* z z) (* t a)))))
        (/ z (sqrt (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 <= -1.1701128116145023e+88) {
		tmp = -(x * y);
	} else if (z <= 6.724224666826136e-307) {
		tmp = x * ((z * y) / sqrt((z * z) - (t * a)));
	} else if (z <= 2.456100249180412e+118) {
		tmp = ((x * y) / sqrt(sqrt((z * z) - (t * a)))) * (z / sqrt(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

Original24.6
Target7.6
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if z < -1.1701128116145023e88

    1. Initial program 42.0

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

    2. Taylor expanded around -inf 2.0

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

    3. Simplified2.0

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

    if -1.1701128116145023e88 < z < 6.724224666826136e-307

    1. Initial program 11.0

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

    3. Applied *-un-lft-identity_binary64_962611.0

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

    4. Applied sqrt-prod_binary64_964211.0

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

    5. Applied times-frac_binary64_96329.6

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

    6. Simplified9.6

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

    8. Applied associate-*l*_binary64_95679.0

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

    10. Applied associate-*r/_binary64_956810.2

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

    11. Simplified10.2

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

    if 6.724224666826136e-307 < z < 2.4561002491804121e118

    1. Initial program 11.0

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

    3. Applied add-sqr-sqrt_binary64_964811.1

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

    4. Applied times-frac_binary64_96329.8

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

    if 2.4561002491804121e118 < z

    1. Initial program 46.7

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

    2. Taylor expanded around inf 1.9

      \[\leadsto \color{blue}{x \cdot y}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1701128116145023 \cdot 10^{+88}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 6.724224666826136 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 2.456100249180412 \cdot 10^{+118}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(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)))))