Average Error: 11.9 → 5.5
Time: 2.1s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.0250245388054168 \cdot 10^{256}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.04962770169091819 \cdot 10^{-94}:\\ \;\;\;\;a1 \cdot \left(a2 \cdot \frac{\frac{1}{b2}}{b1}\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 3.18800829823527456 \cdot 10^{-295}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 2.23031553909857134 \cdot 10^{300}:\\ \;\;\;\;a1 \cdot \left(a2 \cdot \frac{\frac{1}{b2}}{b1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 \le -3.0250245388054168 \cdot 10^{256}:\\
\;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le -4.04962770169091819 \cdot 10^{-94}:\\
\;\;\;\;a1 \cdot \left(a2 \cdot \frac{\frac{1}{b2}}{b1}\right)\\

\mathbf{elif}\;b1 \cdot b2 \le 3.18800829823527456 \cdot 10^{-295}:\\
\;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le 2.23031553909857134 \cdot 10^{300}:\\
\;\;\;\;a1 \cdot \left(a2 \cdot \frac{\frac{1}{b2}}{b1}\right)\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r165152 = a1;
        double r165153 = a2;
        double r165154 = r165152 * r165153;
        double r165155 = b1;
        double r165156 = b2;
        double r165157 = r165155 * r165156;
        double r165158 = r165154 / r165157;
        return r165158;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r165159 = b1;
        double r165160 = b2;
        double r165161 = r165159 * r165160;
        double r165162 = -3.025024538805417e+256;
        bool r165163 = r165161 <= r165162;
        double r165164 = a1;
        double r165165 = r165164 / r165159;
        double r165166 = a2;
        double r165167 = r165165 * r165166;
        double r165168 = r165167 / r165160;
        double r165169 = -4.049627701690918e-94;
        bool r165170 = r165161 <= r165169;
        double r165171 = 1.0;
        double r165172 = r165171 / r165160;
        double r165173 = r165172 / r165159;
        double r165174 = r165166 * r165173;
        double r165175 = r165164 * r165174;
        double r165176 = 3.1880082982352746e-295;
        bool r165177 = r165161 <= r165176;
        double r165178 = 2.2303155390985713e+300;
        bool r165179 = r165161 <= r165178;
        double r165180 = r165166 / r165160;
        double r165181 = r165164 * r165180;
        double r165182 = r165181 / r165159;
        double r165183 = r165179 ? r165175 : r165182;
        double r165184 = r165177 ? r165168 : r165183;
        double r165185 = r165170 ? r165175 : r165184;
        double r165186 = r165163 ? r165168 : r165185;
        return r165186;
}

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

Original11.9
Target11.4
Herbie5.5
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* b1 b2) < -3.025024538805417e+256 or -4.049627701690918e-94 < (* b1 b2) < 3.1880082982352746e-295

    1. Initial program 24.1

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

    3. Applied times-frac8.1

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

    5. Applied associate-*r/7.5

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

    if -3.025024538805417e+256 < (* b1 b2) < -4.049627701690918e-94 or 3.1880082982352746e-295 < (* b1 b2) < 2.2303155390985713e+300

    1. Initial program 5.4

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

    3. Applied times-frac13.8

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

    5. Applied div-inv13.8

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

    6. Applied associate-*l*11.1

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

    7. Simplified11.0

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

    9. Applied *-un-lft-identity11.0

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

    10. Applied div-inv11.1

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

    11. Applied times-frac4.9

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

    12. Simplified4.9

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

    if 2.2303155390985713e+300 < (* b1 b2)

    1. Initial program 22.3

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

    3. Applied times-frac2.8

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

    5. Applied associate-*l/3.2

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.0250245388054168 \cdot 10^{256}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.04962770169091819 \cdot 10^{-94}:\\ \;\;\;\;a1 \cdot \left(a2 \cdot \frac{\frac{1}{b2}}{b1}\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 3.18800829823527456 \cdot 10^{-295}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 2.23031553909857134 \cdot 10^{300}:\\ \;\;\;\;a1 \cdot \left(a2 \cdot \frac{\frac{1}{b2}}{b1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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