Average Error: 11.5 → 5.2
Time: 13.9s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -4.369619197843261223988403902442698260098 \cdot 10^{-287}:\\ \;\;\;\;\frac{1}{\frac{b2 \cdot b1}{a1 \cdot a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.983618642889101104598899002423951855124 \cdot 10^{-209}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.979773594632940735324722698003690676961 \cdot 10^{199}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt[3]{b2}} \cdot \left(\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{1}{b1}\right)\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 = -\infty:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;a1 \cdot a2 \le -4.369619197843261223988403902442698260098 \cdot 10^{-287}:\\
\;\;\;\;\frac{1}{\frac{b2 \cdot b1}{a1 \cdot a2}}\\

\mathbf{elif}\;a1 \cdot a2 \le 2.983618642889101104598899002423951855124 \cdot 10^{-209}:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;a1 \cdot a2 \le 5.979773594632940735324722698003690676961 \cdot 10^{199}:\\
\;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\

\mathbf{else}:\\
\;\;\;\;\frac{a2}{\sqrt[3]{b2}} \cdot \left(\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{1}{b1}\right)\\

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r5478364 = a1;
        double r5478365 = a2;
        double r5478366 = r5478364 * r5478365;
        double r5478367 = b1;
        double r5478368 = b2;
        double r5478369 = r5478367 * r5478368;
        double r5478370 = r5478366 / r5478369;
        return r5478370;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r5478371 = a1;
        double r5478372 = a2;
        double r5478373 = r5478371 * r5478372;
        double r5478374 = -inf.0;
        bool r5478375 = r5478373 <= r5478374;
        double r5478376 = b1;
        double r5478377 = r5478371 / r5478376;
        double r5478378 = b2;
        double r5478379 = r5478372 / r5478378;
        double r5478380 = r5478377 * r5478379;
        double r5478381 = -4.369619197843261e-287;
        bool r5478382 = r5478373 <= r5478381;
        double r5478383 = 1.0;
        double r5478384 = r5478378 * r5478376;
        double r5478385 = r5478384 / r5478373;
        double r5478386 = r5478383 / r5478385;
        double r5478387 = 2.983618642889101e-209;
        bool r5478388 = r5478373 <= r5478387;
        double r5478389 = 5.979773594632941e+199;
        bool r5478390 = r5478373 <= r5478389;
        double r5478391 = r5478373 / r5478376;
        double r5478392 = r5478391 / r5478378;
        double r5478393 = cbrt(r5478378);
        double r5478394 = r5478372 / r5478393;
        double r5478395 = r5478393 * r5478393;
        double r5478396 = r5478371 / r5478395;
        double r5478397 = r5478383 / r5478376;
        double r5478398 = r5478396 * r5478397;
        double r5478399 = r5478394 * r5478398;
        double r5478400 = r5478390 ? r5478392 : r5478399;
        double r5478401 = r5478388 ? r5478380 : r5478400;
        double r5478402 = r5478382 ? r5478386 : r5478401;
        double r5478403 = r5478375 ? r5478380 : r5478402;
        return r5478403;
}

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

Original11.5
Target11.1
Herbie5.2
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (* a1 a2) < -inf.0 or -4.369619197843261e-287 < (* a1 a2) < 2.983618642889101e-209

    1. Initial program 21.8

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

    3. Applied times-frac4.0

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

    if -inf.0 < (* a1 a2) < -4.369619197843261e-287

    1. Initial program 5.9

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

    3. Applied clear-num6.2

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

    if 2.983618642889101e-209 < (* a1 a2) < 5.979773594632941e+199

    1. Initial program 4.5

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

    3. Applied associate-/r*4.3

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

    if 5.979773594632941e+199 < (* a1 a2)

    1. Initial program 34.1

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

    3. Applied times-frac11.5

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

    5. Applied add-cube-cbrt12.3

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

    6. Applied *-un-lft-identity12.3

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

    7. Applied times-frac12.3

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

    8. Applied associate-*r*10.2

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

    9. Simplified8.9

      \[\leadsto \color{blue}{\frac{\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}{b1}} \cdot \frac{a2}{\sqrt[3]{b2}}\]
    10. Using strategy rm

    11. Applied div-inv8.9

      \[\leadsto \color{blue}{\left(\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{1}{b1}\right)} \cdot \frac{a2}{\sqrt[3]{b2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -4.369619197843261223988403902442698260098 \cdot 10^{-287}:\\ \;\;\;\;\frac{1}{\frac{b2 \cdot b1}{a1 \cdot a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.983618642889101104598899002423951855124 \cdot 10^{-209}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.979773594632940735324722698003690676961 \cdot 10^{199}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt[3]{b2}} \cdot \left(\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{1}{b1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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