Average Error: 11.3 → 2.4
Time: 15.1s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.414144362639176854803385722786278913018 \cdot 10^{-314} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 1.431802419250923180709933384670076216432 \cdot 10^{298}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\
\;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.414144362639176854803385722786278913018 \cdot 10^{-314} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 1.431802419250923180709933384670076216432 \cdot 10^{298}:\\
\;\;\;\;\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 r79346 = a1;
        double r79347 = a2;
        double r79348 = r79346 * r79347;
        double r79349 = b1;
        double r79350 = b2;
        double r79351 = r79349 * r79350;
        double r79352 = r79348 / r79351;
        return r79352;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r79353 = a1;
        double r79354 = a2;
        double r79355 = r79353 * r79354;
        double r79356 = b1;
        double r79357 = b2;
        double r79358 = r79356 * r79357;
        double r79359 = r79355 / r79358;
        double r79360 = -inf.0;
        bool r79361 = r79359 <= r79360;
        double r79362 = r79354 / r79357;
        double r79363 = r79362 / r79356;
        double r79364 = r79353 * r79363;
        double r79365 = -1.4141443626392e-314;
        bool r79366 = r79359 <= r79365;
        double r79367 = 0.0;
        bool r79368 = r79359 <= r79367;
        double r79369 = !r79368;
        double r79370 = 1.4318024192509232e+298;
        bool r79371 = r79359 <= r79370;
        bool r79372 = r79369 && r79371;
        bool r79373 = r79366 || r79372;
        double r79374 = r79353 / r79356;
        double r79375 = r79374 * r79362;
        double r79376 = r79373 ? r79359 : r79375;
        double r79377 = r79361 ? r79364 : r79376;
        return r79377;
}

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

Original11.3
Target11.2
Herbie2.4
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 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 times-frac11.2

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

    5. Applied div-inv11.3

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

    6. Applied associate-*l*19.0

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

    7. Simplified18.9

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

    if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.4141443626392e-314 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 1.4318024192509232e+298

    1. Initial program 3.6

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]

    if -1.4141443626392e-314 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 1.4318024192509232e+298 < (/ (* a1 a2) (* b1 b2))

    1. Initial program 28.6

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

    3. Applied times-frac3.7

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.414144362639176854803385722786278913018 \cdot 10^{-314} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 1.431802419250923180709933384670076216432 \cdot 10^{298}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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