Quotient of products

Average Error: 11.1 → 2.3

Time: 11.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.721065323848875903560093256362069823088 \cdot 10^{-309} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 8.682799991236733148187736602059382241979 \cdot 10^{294}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r105422 = a1;
        double r105423 = a2;
        double r105424 = r105422 * r105423;
        double r105425 = b1;
        double r105426 = b2;
        double r105427 = r105425 * r105426;
        double r105428 = r105424 / r105427;
        return r105428;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r105429 = a1;
        double r105430 = a2;
        double r105431 = r105429 * r105430;
        double r105432 = b1;
        double r105433 = b2;
        double r105434 = r105432 * r105433;
        double r105435 = r105431 / r105434;
        double r105436 = -inf.0;
        bool r105437 = r105435 <= r105436;
        double r105438 = -3.721065323848876e-309;
        bool r105439 = r105435 <= r105438;
        double r105440 = 0.0;
        bool r105441 = r105435 <= r105440;
        double r105442 = !r105441;
        double r105443 = 8.682799991236733e+294;
        bool r105444 = r105435 <= r105443;
        bool r105445 = r105442 && r105444;
        bool r105446 = r105439 || r105445;
        double r105447 = !r105446;
        bool r105448 = r105437 || r105447;
        double r105449 = r105429 / r105432;
        double r105450 = r105430 / r105433;
        double r105451 = r105449 * r105450;
        double r105452 = r105448 ? r105451 : r105435;
        return r105452;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.1
Target	11.7
Herbie	2.3

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

Derivation

1. Split input into 2 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -3.721065323848876e-309 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 8.682799991236733e+294 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 32.8

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

   3. Applied times-frac 5.6

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -3.721065323848876e-309 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 8.682799991236733e+294

   1. Initial program 3.5

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.721065323848875903560093256362069823088 \cdot 10^{-309} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 8.682799991236733148187736602059382241979 \cdot 10^{294}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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