Average Error: 11.1 → 7.3
Time: 2.0s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -8.86452261856546459 \cdot 10^{62}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.4672356117320064 \cdot 10^{-254}:\\ \;\;\;\;\frac{1}{\frac{b1}{\frac{a1 \cdot a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 6.0955568267489453 \cdot 10^{-107}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.3074354815251371 \cdot 10^{202}:\\ \;\;\;\;\frac{1}{\frac{b1}{\frac{a1 \cdot a2}{b2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 \le -8.86452261856546459 \cdot 10^{62}:\\
\;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\

\mathbf{elif}\;a1 \cdot a2 \le -1.4672356117320064 \cdot 10^{-254}:\\
\;\;\;\;\frac{1}{\frac{b1}{\frac{a1 \cdot a2}{b2}}}\\

\mathbf{elif}\;a1 \cdot a2 \le 6.0955568267489453 \cdot 10^{-107}:\\
\;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\

\mathbf{elif}\;a1 \cdot a2 \le 3.3074354815251371 \cdot 10^{202}:\\
\;\;\;\;\frac{1}{\frac{b1}{\frac{a1 \cdot a2}{b2}}}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r174353 = a1;
        double r174354 = a2;
        double r174355 = r174353 * r174354;
        double r174356 = b1;
        double r174357 = b2;
        double r174358 = r174356 * r174357;
        double r174359 = r174355 / r174358;
        return r174359;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r174360 = a1;
        double r174361 = a2;
        double r174362 = r174360 * r174361;
        double r174363 = -8.864522618565465e+62;
        bool r174364 = r174362 <= r174363;
        double r174365 = b1;
        double r174366 = b2;
        double r174367 = r174365 * r174366;
        double r174368 = r174361 / r174367;
        double r174369 = r174360 * r174368;
        double r174370 = -1.4672356117320064e-254;
        bool r174371 = r174362 <= r174370;
        double r174372 = 1.0;
        double r174373 = r174362 / r174366;
        double r174374 = r174365 / r174373;
        double r174375 = r174372 / r174374;
        double r174376 = 6.095556826748945e-107;
        bool r174377 = r174362 <= r174376;
        double r174378 = 3.307435481525137e+202;
        bool r174379 = r174362 <= r174378;
        double r174380 = r174366 / r174361;
        double r174381 = r174365 * r174380;
        double r174382 = r174360 / r174381;
        double r174383 = r174379 ? r174375 : r174382;
        double r174384 = r174377 ? r174369 : r174383;
        double r174385 = r174371 ? r174375 : r174384;
        double r174386 = r174364 ? r174369 : r174385;
        return r174386;
}

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
Target11.7
Herbie7.3
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a1 a2) < -8.864522618565465e+62 or -1.4672356117320064e-254 < (* a1 a2) < 6.095556826748945e-107

    1. Initial program 14.8

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

    3. Applied associate-/l*9.9

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

    5. Applied div-inv10.1

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

    6. Simplified9.9

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

    if -8.864522618565465e+62 < (* a1 a2) < -1.4672356117320064e-254 or 6.095556826748945e-107 < (* a1 a2) < 3.307435481525137e+202

    1. Initial program 4.2

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

    3. Applied clear-num4.5

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

    5. Applied associate-/l*4.4

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

    if 3.307435481525137e+202 < (* a1 a2)

    1. Initial program 34.1

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

    3. Applied associate-/l*17.7

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

    5. Applied *-un-lft-identity17.7

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

    6. Applied times-frac10.1

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

    7. Simplified10.1

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -8.86452261856546459 \cdot 10^{62}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.4672356117320064 \cdot 10^{-254}:\\ \;\;\;\;\frac{1}{\frac{b1}{\frac{a1 \cdot a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 6.0955568267489453 \cdot 10^{-107}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.3074354815251371 \cdot 10^{202}:\\ \;\;\;\;\frac{1}{\frac{b1}{\frac{a1 \cdot a2}{b2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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