Average Error: 11.3 → 2.4
Time: 14.4s
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 r82292 = a1;
        double r82293 = a2;
        double r82294 = r82292 * r82293;
        double r82295 = b1;
        double r82296 = b2;
        double r82297 = r82295 * r82296;
        double r82298 = r82294 / r82297;
        return r82298;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r82299 = a1;
        double r82300 = a2;
        double r82301 = r82299 * r82300;
        double r82302 = b1;
        double r82303 = b2;
        double r82304 = r82302 * r82303;
        double r82305 = r82301 / r82304;
        double r82306 = -inf.0;
        bool r82307 = r82305 <= r82306;
        double r82308 = r82300 / r82303;
        double r82309 = r82308 / r82302;
        double r82310 = r82299 * r82309;
        double r82311 = -1.4141443626392e-314;
        bool r82312 = r82305 <= r82311;
        double r82313 = 0.0;
        bool r82314 = r82305 <= r82313;
        double r82315 = !r82314;
        double r82316 = 1.4318024192509232e+298;
        bool r82317 = r82305 <= r82316;
        bool r82318 = r82315 && r82317;
        bool r82319 = r82312 || r82318;
        double r82320 = r82299 / r82302;
        double r82321 = r82320 * r82308;
        double r82322 = r82319 ? r82305 : r82321;
        double r82323 = r82307 ? r82310 : r82322;
        return r82323;
}

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 0.9

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

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

    1. Initial program 22.7

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

    3. Applied times-frac3.2

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