Average Error: 11.1 → 5.1
Time: 12.0s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -6.112419174821975432135783811036282806513 \cdot 10^{173}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.251241574118469563607708541008145541133 \cdot 10^{-252}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 5.86129416444517535704513449031412392708 \cdot 10^{-283}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 9.3869286443022839600436283384046295134 \cdot 10^{260}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 \le -6.112419174821975432135783811036282806513 \cdot 10^{173}:\\
\;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\

\mathbf{elif}\;b1 \cdot b2 \le -4.251241574118469563607708541008145541133 \cdot 10^{-252}:\\
\;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\

\mathbf{elif}\;b1 \cdot b2 \le 5.86129416444517535704513449031412392708 \cdot 10^{-283}:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le 9.3869286443022839600436283384046295134 \cdot 10^{260}:\\
\;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r137332 = a1;
        double r137333 = a2;
        double r137334 = r137332 * r137333;
        double r137335 = b1;
        double r137336 = b2;
        double r137337 = r137335 * r137336;
        double r137338 = r137334 / r137337;
        return r137338;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r137339 = b1;
        double r137340 = b2;
        double r137341 = r137339 * r137340;
        double r137342 = -6.112419174821975e+173;
        bool r137343 = r137341 <= r137342;
        double r137344 = a1;
        double r137345 = cbrt(r137339);
        double r137346 = r137345 * r137345;
        double r137347 = r137344 / r137346;
        double r137348 = a2;
        double r137349 = cbrt(r137348);
        double r137350 = r137349 * r137349;
        double r137351 = r137347 * r137350;
        double r137352 = cbrt(r137340);
        double r137353 = r137352 * r137352;
        double r137354 = r137351 / r137353;
        double r137355 = r137349 / r137345;
        double r137356 = r137355 / r137352;
        double r137357 = r137354 * r137356;
        double r137358 = -4.2512415741184696e-252;
        bool r137359 = r137341 <= r137358;
        double r137360 = r137348 / r137341;
        double r137361 = r137344 * r137360;
        double r137362 = 5.861294164445175e-283;
        bool r137363 = r137341 <= r137362;
        double r137364 = r137344 / r137339;
        double r137365 = r137348 / r137340;
        double r137366 = r137364 * r137365;
        double r137367 = 9.386928644302284e+260;
        bool r137368 = r137341 <= r137367;
        double r137369 = r137368 ? r137361 : r137357;
        double r137370 = r137363 ? r137366 : r137369;
        double r137371 = r137359 ? r137361 : r137370;
        double r137372 = r137343 ? r137357 : r137371;
        return r137372;
}

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

Original11.1
Target11.6
Herbie5.1
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* b1 b2) < -6.112419174821975e+173 or 9.386928644302284e+260 < (* b1 b2)

    1. Initial program 16.3

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

    3. Applied associate-/r*8.6

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

    5. Applied add-cube-cbrt8.9

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

    6. Applied add-cube-cbrt9.0

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

    7. Applied times-frac4.4

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

    8. Applied times-frac3.9

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

    10. Applied *-un-lft-identity3.9

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

    11. Applied *-un-lft-identity3.9

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

    12. Applied add-cube-cbrt4.0

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

    13. Applied times-frac4.0

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

    14. Applied times-frac3.9

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

    15. Applied associate-*r*3.4

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

    16. Simplified3.4

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

    if -6.112419174821975e+173 < (* b1 b2) < -4.2512415741184696e-252 or 5.861294164445175e-283 < (* b1 b2) < 9.386928644302284e+260

    1. Initial program 4.9

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

    3. Applied associate-/r*11.0

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

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

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

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

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

    7. Applied times-frac13.8

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

    8. Applied times-frac11.6

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

    9. Simplified11.6

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

    10. Simplified5.1

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

    if -4.2512415741184696e-252 < (* b1 b2) < 5.861294164445175e-283

    1. Initial program 44.0

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

    3. Applied times-frac10.3

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -6.112419174821975432135783811036282806513 \cdot 10^{173}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.251241574118469563607708541008145541133 \cdot 10^{-252}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 5.86129416444517535704513449031412392708 \cdot 10^{-283}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 9.3869286443022839600436283384046295134 \cdot 10^{260}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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