Average Error: 11.6 → 4.6
Time: 40.7s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -9.78621637012035120889222133129827862576 \cdot 10^{209}:\\ \;\;\;\;\frac{1}{b1 \cdot \frac{\frac{b2}{a2}}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \le -9.111280530104685398727222154888263046467 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{1}{\frac{b1 \cdot b2}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}}{\frac{\frac{1}{a2}}{\sqrt[3]{a1}}}\\ \mathbf{elif}\;b1 \cdot b2 \le 4.74006580620091934483000099877771584647 \cdot 10^{-320}:\\ \;\;\;\;\frac{1}{b1 \cdot \frac{\frac{b2}{a2}}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \le 8.888412293585824693935829272063102395365 \cdot 10^{181}:\\ \;\;\;\;\frac{\frac{1}{\frac{b1 \cdot b2}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}}{\frac{\frac{1}{a2}}{\sqrt[3]{a1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 \le -9.78621637012035120889222133129827862576 \cdot 10^{209}:\\
\;\;\;\;\frac{1}{b1 \cdot \frac{\frac{b2}{a2}}{a1}}\\

\mathbf{elif}\;b1 \cdot b2 \le -9.111280530104685398727222154888263046467 \cdot 10^{-184}:\\
\;\;\;\;\frac{\frac{1}{\frac{b1 \cdot b2}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}}{\frac{\frac{1}{a2}}{\sqrt[3]{a1}}}\\

\mathbf{elif}\;b1 \cdot b2 \le 4.74006580620091934483000099877771584647 \cdot 10^{-320}:\\
\;\;\;\;\frac{1}{b1 \cdot \frac{\frac{b2}{a2}}{a1}}\\

\mathbf{elif}\;b1 \cdot b2 \le 8.888412293585824693935829272063102395365 \cdot 10^{181}:\\
\;\;\;\;\frac{\frac{1}{\frac{b1 \cdot b2}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}}{\frac{\frac{1}{a2}}{\sqrt[3]{a1}}}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r128304 = a1;
        double r128305 = a2;
        double r128306 = r128304 * r128305;
        double r128307 = b1;
        double r128308 = b2;
        double r128309 = r128307 * r128308;
        double r128310 = r128306 / r128309;
        return r128310;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r128311 = b1;
        double r128312 = b2;
        double r128313 = r128311 * r128312;
        double r128314 = -9.786216370120351e+209;
        bool r128315 = r128313 <= r128314;
        double r128316 = 1.0;
        double r128317 = a2;
        double r128318 = r128312 / r128317;
        double r128319 = a1;
        double r128320 = r128318 / r128319;
        double r128321 = r128311 * r128320;
        double r128322 = r128316 / r128321;
        double r128323 = -9.111280530104685e-184;
        bool r128324 = r128313 <= r128323;
        double r128325 = cbrt(r128319);
        double r128326 = r128325 * r128325;
        double r128327 = r128313 / r128326;
        double r128328 = r128316 / r128327;
        double r128329 = r128316 / r128317;
        double r128330 = r128329 / r128325;
        double r128331 = r128328 / r128330;
        double r128332 = 4.7400658062009e-320;
        bool r128333 = r128313 <= r128332;
        double r128334 = 8.888412293585825e+181;
        bool r128335 = r128313 <= r128334;
        double r128336 = r128319 / r128311;
        double r128337 = r128336 / r128318;
        double r128338 = r128335 ? r128331 : r128337;
        double r128339 = r128333 ? r128322 : r128338;
        double r128340 = r128324 ? r128331 : r128339;
        double r128341 = r128315 ? r128322 : r128340;
        return r128341;
}

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

Original11.6
Target11.4
Herbie4.6
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* b1 b2) < -9.786216370120351e+209 or -9.111280530104685e-184 < (* b1 b2) < 4.7400658062009e-320

    1. Initial program 25.7

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

    3. Applied associate-/l*25.3

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

    5. Applied clear-num25.4

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

    7. Applied *-un-lft-identity25.4

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

    8. Applied *-un-lft-identity25.4

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

    9. Applied times-frac12.9

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

    10. Applied times-frac6.7

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

    11. Simplified6.7

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

    if -9.786216370120351e+209 < (* b1 b2) < -9.111280530104685e-184 or 4.7400658062009e-320 < (* b1 b2) < 8.888412293585825e+181

    1. Initial program 5.2

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

    3. Applied associate-/l*5.2

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

    5. Applied clear-num5.5

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

    7. Applied add-cube-cbrt6.2

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

    8. Applied div-inv6.2

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

    9. Applied times-frac3.8

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

    10. Applied associate-/r*3.7

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

    if 8.888412293585825e+181 < (* b1 b2)

    1. Initial program 16.3

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

    3. Applied associate-/l*15.8

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

    5. Applied *-un-lft-identity15.8

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

    6. Applied times-frac8.8

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

    7. Applied associate-/r*5.3

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

    8. Simplified5.3

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

  4. Final simplification4.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -9.78621637012035120889222133129827862576 \cdot 10^{209}:\\ \;\;\;\;\frac{1}{b1 \cdot \frac{\frac{b2}{a2}}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \le -9.111280530104685398727222154888263046467 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{1}{\frac{b1 \cdot b2}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}}{\frac{\frac{1}{a2}}{\sqrt[3]{a1}}}\\ \mathbf{elif}\;b1 \cdot b2 \le 4.74006580620091934483000099877771584647 \cdot 10^{-320}:\\ \;\;\;\;\frac{1}{b1 \cdot \frac{\frac{b2}{a2}}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \le 8.888412293585824693935829272063102395365 \cdot 10^{181}:\\ \;\;\;\;\frac{\frac{1}{\frac{b1 \cdot b2}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}}{\frac{\frac{1}{a2}}{\sqrt[3]{a1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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