Average Error: 11.3 → 7.3
Time: 3.2s
Precision: binary64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -\infty:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \leq -1.638489677251224 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \leq 1.7835250034523412 \cdot 10^{-274}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a1 \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)\right) \cdot \frac{\sqrt[3]{a2}}{b2}}{b1}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 \leq -\infty:\\
\;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\

\mathbf{elif}\;a1 \cdot a2 \leq -1.638489677251224 \cdot 10^{-228}:\\
\;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\

\mathbf{elif}\;a1 \cdot a2 \leq 1.7835250034523412 \cdot 10^{-274}:\\
\;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(a1 \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)\right) \cdot \frac{\sqrt[3]{a2}}{b2}}{b1}\\

\end{array}
(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))
↓
(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* a1 a2) (- INFINITY))
   (* a1 (/ (/ a2 b2) b1))
   (if (<= (* a1 a2) -1.638489677251224e-228)
     (/ (/ (* a1 a2) b2) b1)
     (if (<= (* a1 a2) 1.7835250034523412e-274)
       (* a1 (/ (/ a2 b2) b1))
       (/ (* (* a1 (* (cbrt a2) (cbrt a2))) (/ (cbrt a2) b2)) b1)))))
double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}
↓
double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if ((a1 * a2) <= -((double) INFINITY)) {
		tmp = a1 * ((a2 / b2) / b1);
	} else if ((a1 * a2) <= -1.638489677251224e-228) {
		tmp = ((a1 * a2) / b2) / b1;
	} else if ((a1 * a2) <= 1.7835250034523412e-274) {
		tmp = a1 * ((a2 / b2) / b1);
	} else {
		tmp = ((a1 * (cbrt(a2) * cbrt(a2))) * (cbrt(a2) / b2)) / b1;
	}
	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
Target11.1
Herbie7.3
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 a1 a2) < -inf.0 or -1.6384896772512241e-228 < (*.f64 a1 a2) < 1.78352500345234121e-274

    1. Initial program 21.8

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

    3. Applied times-frac_binary643.5

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

    5. Applied associate-*l/_binary649.3

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

    7. Applied *-un-lft-identity_binary649.3

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

    8. Applied times-frac_binary644.3

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

    9. Simplified4.3

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

    if -inf.0 < (*.f64 a1 a2) < -1.6384896772512241e-228

    1. Initial program 5.7

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

    3. Applied times-frac_binary6414.1

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

    5. Applied associate-*l/_binary6411.4

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

    7. Applied associate-*r/_binary645.9

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

    if 1.78352500345234121e-274 < (*.f64 a1 a2)

    1. Initial program 9.6

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

    3. Applied times-frac_binary6413.2

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

    5. Applied associate-*l/_binary6412.7

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

    7. Applied *-un-lft-identity_binary6412.7

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

    8. Applied add-cube-cbrt_binary6413.3

      \[\leadsto \frac{a1 \cdot \frac{\color{blue}{\left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right) \cdot \sqrt[3]{a2}}}{1 \cdot b2}}{b1}\]

    9. Applied times-frac_binary6413.3

      \[\leadsto \frac{a1 \cdot \color{blue}{\left(\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{1} \cdot \frac{\sqrt[3]{a2}}{b2}\right)}}{b1}\]

    10. Applied associate-*r*_binary6410.3

      \[\leadsto \frac{\color{blue}{\left(a1 \cdot \frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{1}\right) \cdot \frac{\sqrt[3]{a2}}{b2}}}{b1}\]

    11. Simplified10.3

      \[\leadsto \frac{\color{blue}{\left(a1 \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)\right)} \cdot \frac{\sqrt[3]{a2}}{b2}}{b1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -\infty:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \leq -1.638489677251224 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \leq 1.7835250034523412 \cdot 10^{-274}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a1 \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)\right) \cdot \frac{\sqrt[3]{a2}}{b2}}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020275 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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