Average Error: 24.5 → 5.5
Time: 10.2s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \[t, a] = \mathsf{sort}([t, a]) \\]
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -3.9587781002086833 \cdot 10^{+152}:\\ \;\;\;\;-y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{y \cdot x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{if}\;z \leq -1.6005527008883997 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.0712262234135167 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{\mathsf{hypot}\left(\sqrt{-t} \cdot \sqrt{a}, z\right)} \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 3.2862688268057826 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\\ \end{array} \]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -3.9587781002086833 \cdot 10^{+152}:\\
\;\;\;\;-y \cdot x\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_1 := \frac{y \cdot x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\
\mathbf{if}\;z \leq -1.6005527008883997 \cdot 10^{-83}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;z \leq 1.0712262234135167 \cdot 10^{-132}:\\
\;\;\;\;\frac{x}{\mathsf{hypot}\left(\sqrt{-t} \cdot \sqrt{a}, z\right)} \cdot \left(z \cdot y\right)\\

\mathbf{elif}\;z \leq 3.2862688268057826 \cdot 10^{+148}:\\
\;\;\;\;t_1\\

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


\end{array}\\


\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 -3.9587781002086833e+152)
   (- (* y x))
   (let* ((t_1 (/ (* y x) (/ (sqrt (- (* z z) (* t a))) z))))
     (if (<= z -1.6005527008883997e-83)
       t_1
       (if (<= z 1.0712262234135167e-132)
         (* (/ x (hypot (* (sqrt (- t)) (sqrt a)) z)) (* z y))
         (if (<= z 3.2862688268057826e+148) t_1 (* 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 <= -3.9587781002086833e+152) {
		tmp = -(y * x);
	} else {
		double t_1 = (y * x) / (sqrt((z * z) - (t * a)) / z);
		double tmp_1;
		if (z <= -1.6005527008883997e-83) {
			tmp_1 = t_1;
		} else if (z <= 1.0712262234135167e-132) {
			tmp_1 = (x / hypot((sqrt(-t) * sqrt(a)), z)) * (z * y);
		} else if (z <= 3.2862688268057826e+148) {
			tmp_1 = t_1;
		} else {
			tmp_1 = y * x;
		}
		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
Target7.8
Herbie5.5
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Derivation

  1. Split input into 4 regimes

  2. if z < -3.9587781002086833e152

    1. Initial program 53.5

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

    2. Taylor expanded in z around -inf 1.4

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

    3. Simplified1.4

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

    if -3.9587781002086833e152 < z < -1.60055270088839973e-83 or 1.0712262234135167e-132 < z < 3.28626882680578261e148

    1. Initial program 8.1

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

    2. Applied associate-/l*_binary644.6

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

    if -1.60055270088839973e-83 < z < 1.0712262234135167e-132

    1. Initial program 15.3

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

    2. Applied associate-/l*_binary6415.8

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

    3. Simplified12.9

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

    4. Applied distribute-rgt-neg-in_binary6412.9

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

    5. Applied sqrt-prod_binary6413.3

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

    6. Applied div-inv_binary6413.3

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

    7. Applied times-frac_binary6412.4

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

    8. Simplified12.4

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

    9. Simplified12.4

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

    if 3.28626882680578261e148 < z

    1. Initial program 52.4

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

    2. Taylor expanded in z around inf 1.1

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9587781002086833 \cdot 10^{+152}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \leq -1.6005527008883997 \cdot 10^{-83}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{elif}\;z \leq 1.0712262234135167 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{\mathsf{hypot}\left(\sqrt{-t} \cdot \sqrt{a}, z\right)} \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 3.2862688268057826 \cdot 10^{+148}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Reproduce

herbie shell --seed 2022097 
(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)))))