Average Error: 11.3 → 5.8
Time: 3.5s
Precision: binary64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -6.559473697767163 \cdot 10^{+134}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \leq -1.2409094968965834 \cdot 10^{-152}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq -0:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \leq 1.1711099455545799 \cdot 10^{+290}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 \leq -6.559473697767163 \cdot 10^{+134}:\\
\;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\

\mathbf{elif}\;a1 \cdot a2 \leq -1.2409094968965834 \cdot 10^{-152}:\\
\;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\

\mathbf{elif}\;a1 \cdot a2 \leq -0:\\
\;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\

\mathbf{elif}\;a1 \cdot a2 \leq 1.1711099455545799 \cdot 10^{+290}:\\
\;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\

\end{array}
(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))
↓
(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* a1 a2) -6.559473697767163e+134)
   (/ a1 (/ b1 (/ a2 b2)))
   (if (<= (* a1 a2) -1.2409094968965834e-152)
     (* (* a1 a2) (/ 1.0 (* b1 b2)))
     (if (<= (* a1 a2) -0.0)
       (/ a1 (/ b1 (/ a2 b2)))
       (if (<= (* a1 a2) 1.1711099455545799e+290)
         (* (* a1 a2) (/ 1.0 (* b1 b2)))
         (/ (/ a1 b1) (/ b2 a2)))))))
double code(double a1, double a2, double b1, double b2) {
	return (((double) (a1 * a2)) / ((double) (b1 * b2)));
}
↓
double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if ((((double) (a1 * a2)) <= -6.559473697767163e+134)) {
		tmp = (a1 / (b1 / (a2 / b2)));
	} else {
		double tmp_1;
		if ((((double) (a1 * a2)) <= -1.2409094968965834e-152)) {
			tmp_1 = ((double) (((double) (a1 * a2)) * (1.0 / ((double) (b1 * b2)))));
		} else {
			double tmp_2;
			if ((((double) (a1 * a2)) <= -0.0)) {
				tmp_2 = (a1 / (b1 / (a2 / b2)));
			} else {
				double tmp_3;
				if ((((double) (a1 * a2)) <= 1.1711099455545799e+290)) {
					tmp_3 = ((double) (((double) (a1 * a2)) * (1.0 / ((double) (b1 * b2)))));
				} else {
					tmp_3 = ((a1 / b1) / (b2 / a2));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.3
Target10.8
Herbie5.8
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 a1 a2) < -6.5594736977671627e134 or -1.2409094968965834e-152 < (*.f64 a1 a2) < -0.0

    1. Initial program 18.4

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    2. Using strategy rm

    3. Applied associate-/l*_binary6410.0

      \[\leadsto \color{blue}{\frac{a1}{\frac{b1 \cdot b2}{a2}}}\]

    4. Simplified6.8

      \[\leadsto \frac{a1}{\color{blue}{\frac{b1}{\frac{a2}{b2}}}}\]

    if -6.5594736977671627e134 < (*.f64 a1 a2) < -1.2409094968965834e-152 or -0.0 < (*.f64 a1 a2) < 1.1711099455545799e290

    1. Initial program 4.7

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    2. Using strategy rm

    3. Applied div-inv_binary645.1

      \[\leadsto \color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}}\]

    if 1.1711099455545799e290 < (*.f64 a1 a2)

    1. Initial program 56.9

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    2. Using strategy rm

    3. Applied clear-num_binary6456.9

      \[\leadsto \color{blue}{\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}}\]
    4. Using strategy rm

    5. Applied times-frac_binary648.9

      \[\leadsto \frac{1}{\color{blue}{\frac{b1}{a1} \cdot \frac{b2}{a2}}}\]

    6. Applied associate-/r*_binary649.3

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{b1}{a1}}}{\frac{b2}{a2}}}\]

    7. Simplified9.2

      \[\leadsto \frac{\color{blue}{\frac{a1}{b1}}}{\frac{b2}{a2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -6.559473697767163 \cdot 10^{+134}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \leq -1.2409094968965834 \cdot 10^{-152}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq -0:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \leq 1.1711099455545799 \cdot 10^{+290}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

  (/ (* a1 a2) (* b1 b2)))