Average Error: 24.9 → 7.5
Time: 4.8s
Precision: binary64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -6.4410615914386545 \cdot 10^{+84}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 3.600147263421176 \cdot 10^{+106}:\\ \;\;\;\;\frac{x \cdot y}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|} \cdot \frac{z}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -6.4410615914386545 \cdot 10^{+84}:\\
\;\;\;\;-x \cdot y\\

\mathbf{elif}\;z \leq 3.600147263421176 \cdot 10^{+106}:\\
\;\;\;\;\frac{x \cdot y}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|} \cdot \frac{z}{\sqrt{\sqrt[3]{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 -6.4410615914386545e+84)
   (- (* x y))
   (if (<= z 3.600147263421176e+106)
     (*
      (/ (* x y) (fabs (cbrt (- (* z z) (* t a)))))
      (/ z (sqrt (cbrt (- (* 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 <= -6.4410615914386545e+84) {
		tmp = -(x * y);
	} else if (z <= 3.600147263421176e+106) {
		tmp = ((x * y) / fabs(cbrt((z * z) - (t * a)))) * (z / sqrt(cbrt((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

Original24.9
Target7.7
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -6.4410615914386545e84

    1. Initial program 41.3

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

    2. Taylor expanded around -inf 2.8

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

    3. Simplified2.8

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

    if -6.4410615914386545e84 < z < 3.600147263421176e106

    1. Initial program 11.5

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

    3. Applied add-cube-cbrt_binary6411.8

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{\left(\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}\right) \cdot \sqrt[3]{z \cdot z - t \cdot a}}}}\]

    4. Applied sqrt-prod_binary6411.8

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}} \cdot \sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}}\]

    5. Applied times-frac_binary6411.2

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}}\]

    6. Simplified11.2

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}} \cdot \frac{z}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\]

    if 3.600147263421176e106 < z

    1. Initial program 45.4

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

    2. Taylor expanded around inf 1.8

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4410615914386545 \cdot 10^{+84}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 3.600147263421176 \cdot 10^{+106}:\\ \;\;\;\;\frac{x \cdot y}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|} \cdot \frac{z}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(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)))))