Average Error: 11.6 → 3.5
Time: 13.0s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{1}{\frac{\frac{b1 \cdot b2}{a2}}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -9.782499787656681574696062098790783172828 \cdot 10^{-322}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.421277284710987870204738960879832034274 \cdot 10^{-315}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{b1} \cdot a2\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.286175403506427577716209346328092058444 \cdot 10^{301}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\
\;\;\;\;\frac{1}{\frac{\frac{b1 \cdot b2}{a2}}{a1}}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -9.782499787656681574696062098790783172828 \cdot 10^{-322}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.421277284710987870204738960879832034274 \cdot 10^{-315}:\\
\;\;\;\;\frac{\frac{a1}{b2}}{b1} \cdot a2\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.286175403506427577716209346328092058444 \cdot 10^{301}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r56416 = a1;
        double r56417 = a2;
        double r56418 = r56416 * r56417;
        double r56419 = b1;
        double r56420 = b2;
        double r56421 = r56419 * r56420;
        double r56422 = r56418 / r56421;
        return r56422;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r56423 = a1;
        double r56424 = a2;
        double r56425 = r56423 * r56424;
        double r56426 = b1;
        double r56427 = b2;
        double r56428 = r56426 * r56427;
        double r56429 = r56425 / r56428;
        double r56430 = -inf.0;
        bool r56431 = r56429 <= r56430;
        double r56432 = 1.0;
        double r56433 = r56428 / r56424;
        double r56434 = r56433 / r56423;
        double r56435 = r56432 / r56434;
        double r56436 = -9.7824997876567e-322;
        bool r56437 = r56429 <= r56436;
        double r56438 = 1.421277284711e-315;
        bool r56439 = r56429 <= r56438;
        double r56440 = r56423 / r56427;
        double r56441 = r56440 / r56426;
        double r56442 = r56441 * r56424;
        double r56443 = 2.2861754035064276e+301;
        bool r56444 = r56429 <= r56443;
        double r56445 = r56423 / r56426;
        double r56446 = r56424 / r56427;
        double r56447 = r56445 * r56446;
        double r56448 = r56444 ? r56429 : r56447;
        double r56449 = r56439 ? r56442 : r56448;
        double r56450 = r56437 ? r56429 : r56449;
        double r56451 = r56431 ? r56435 : r56450;
        return r56451;
}

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
Herbie3.5
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (* a1 a2) (* b1 b2)) < -inf.0

    1. Initial program 64.0

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

    3. Applied associate-/l*30.8

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

    5. Applied clear-num30.9

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

    if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -9.7824997876567e-322 or 1.421277284711e-315 < (/ (* a1 a2) (* b1 b2)) < 2.2861754035064276e+301

    1. Initial program 0.9

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

    3. Applied associate-/l*8.3

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

    5. Applied *-un-lft-identity8.3

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

    6. Applied times-frac14.7

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

    7. Applied *-un-lft-identity14.7

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

    8. Applied times-frac14.5

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

    9. Simplified14.5

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

    10. Simplified14.5

      \[\leadsto \frac{1}{b1} \cdot \color{blue}{\left(\frac{a1}{b2} \cdot a2\right)}\]
    11. Using strategy rm

    12. Applied *-un-lft-identity14.5

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

    13. Applied *-un-lft-identity14.5

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

    14. Applied times-frac14.5

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

    15. Applied associate-*l*14.5

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

    16. Simplified0.9

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

    if -9.7824997876567e-322 < (/ (* a1 a2) (* b1 b2)) < 1.421277284711e-315

    1. Initial program 13.6

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

    3. Applied associate-/l*7.3

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

    5. Applied *-un-lft-identity7.3

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

    6. Applied times-frac3.9

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

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

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

    8. Applied times-frac3.4

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

    9. Simplified3.4

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

    10. Simplified3.5

      \[\leadsto \frac{1}{b1} \cdot \color{blue}{\left(\frac{a1}{b2} \cdot a2\right)}\]
    11. Using strategy rm

    12. Applied associate-*r*3.8

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

    13. Simplified3.8

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

    if 2.2861754035064276e+301 < (/ (* a1 a2) (* b1 b2))

    1. Initial program 62.0

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

    3. Applied times-frac8.0

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{1}{\frac{\frac{b1 \cdot b2}{a2}}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -9.782499787656681574696062098790783172828 \cdot 10^{-322}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.421277284710987870204738960879832034274 \cdot 10^{-315}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{b1} \cdot a2\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.286175403506427577716209346328092058444 \cdot 10^{301}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

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

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

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