Average Error: 11.4 → 4.9
Time: 28.9s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.553414081389674509407045725962943766798 \cdot 10^{290}:\\ \;\;\;\;\frac{1}{\frac{\frac{b2}{\frac{a1}{b1}}}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.803802981211027273100658035489724291435 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.758049283459723037540857018065176298329 \cdot 10^{-256}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.458857710290817792586452271714402343311 \cdot 10^{244}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{b2}{\frac{a1}{b1}}}{a2}}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 \le -2.553414081389674509407045725962943766798 \cdot 10^{290}:\\
\;\;\;\;\frac{1}{\frac{\frac{b2}{\frac{a1}{b1}}}{a2}}\\

\mathbf{elif}\;a1 \cdot a2 \le -1.803802981211027273100658035489724291435 \cdot 10^{-249}:\\
\;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\

\mathbf{elif}\;a1 \cdot a2 \le 2.758049283459723037540857018065176298329 \cdot 10^{-256}:\\
\;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\

\mathbf{elif}\;a1 \cdot a2 \le 1.458857710290817792586452271714402343311 \cdot 10^{244}:\\
\;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\frac{b2}{\frac{a1}{b1}}}{a2}}\\

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r136501 = a1;
        double r136502 = a2;
        double r136503 = r136501 * r136502;
        double r136504 = b1;
        double r136505 = b2;
        double r136506 = r136504 * r136505;
        double r136507 = r136503 / r136506;
        return r136507;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r136508 = a1;
        double r136509 = a2;
        double r136510 = r136508 * r136509;
        double r136511 = -2.5534140813896745e+290;
        bool r136512 = r136510 <= r136511;
        double r136513 = 1.0;
        double r136514 = b2;
        double r136515 = b1;
        double r136516 = r136508 / r136515;
        double r136517 = r136514 / r136516;
        double r136518 = r136517 / r136509;
        double r136519 = r136513 / r136518;
        double r136520 = -1.8038029812110273e-249;
        bool r136521 = r136510 <= r136520;
        double r136522 = r136510 / r136515;
        double r136523 = r136522 / r136514;
        double r136524 = 2.758049283459723e-256;
        bool r136525 = r136510 <= r136524;
        double r136526 = r136509 / r136514;
        double r136527 = r136515 / r136526;
        double r136528 = r136508 / r136527;
        double r136529 = 1.4588577102908178e+244;
        bool r136530 = r136510 <= r136529;
        double r136531 = r136530 ? r136523 : r136519;
        double r136532 = r136525 ? r136528 : r136531;
        double r136533 = r136521 ? r136523 : r136532;
        double r136534 = r136512 ? r136519 : r136533;
        return r136534;
}

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

Original11.4
Target11.3
Herbie4.9
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a1 a2) < -2.5534140813896745e+290 or 1.4588577102908178e+244 < (* a1 a2)

    1. Initial program 48.3

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

    3. Applied associate-/r*48.3

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

    4. Simplified20.1

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

    6. Applied *-un-lft-identity20.1

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

    7. Applied *-un-lft-identity20.1

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

    8. Applied times-frac20.1

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

    9. Applied *-un-lft-identity20.1

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

    10. Applied times-frac20.1

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

    11. Applied associate-/l*20.2

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

    12. Simplified8.1

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

    if -2.5534140813896745e+290 < (* a1 a2) < -1.8038029812110273e-249 or 2.758049283459723e-256 < (* a1 a2) < 1.4588577102908178e+244

    1. Initial program 5.5

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

    3. Applied associate-/r*4.9

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

    4. Simplified10.7

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

    6. Applied frac-2neg10.7

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

    7. Taylor expanded around 0 4.9

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

    if -1.8038029812110273e-249 < (* a1 a2) < 2.758049283459723e-256

    1. Initial program 18.2

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

    3. Applied associate-/r*17.8

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

    4. Simplified8.0

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

    6. Applied div-inv8.0

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

    7. Applied associate-/l*4.0

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

    8. Simplified3.9

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

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.553414081389674509407045725962943766798 \cdot 10^{290}:\\ \;\;\;\;\frac{1}{\frac{\frac{b2}{\frac{a1}{b1}}}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.803802981211027273100658035489724291435 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.758049283459723037540857018065176298329 \cdot 10^{-256}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.458857710290817792586452271714402343311 \cdot 10^{244}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{b2}{\frac{a1}{b1}}}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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