Quotient of products

Average Error: 10.8 → 5.4

Time: 14.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -6.245903185019271 \cdot 10^{+204}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -6.848939379187657 \cdot 10^{-209}:\\ \;\;\;\;\frac{1}{\frac{b2 \cdot b1}{a1 \cdot a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 8.9817388408212 \cdot 10^{-316}:\\ \;\;\;\;\left(\frac{\sqrt[3]{a1}}{b1} \cdot \frac{a2}{b2}\right) \cdot \left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 1.0693629293958468 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r4758471 = a1;
        double r4758472 = a2;
        double r4758473 = r4758471 * r4758472;
        double r4758474 = b1;
        double r4758475 = b2;
        double r4758476 = r4758474 * r4758475;
        double r4758477 = r4758473 / r4758476;
        return r4758477;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r4758478 = a1;
        double r4758479 = a2;
        double r4758480 = r4758478 * r4758479;
        double r4758481 = -6.245903185019271e+204;
        bool r4758482 = r4758480 <= r4758481;
        double r4758483 = b1;
        double r4758484 = r4758478 / r4758483;
        double r4758485 = b2;
        double r4758486 = r4758479 / r4758485;
        double r4758487 = r4758484 * r4758486;
        double r4758488 = -6.848939379187657e-209;
        bool r4758489 = r4758480 <= r4758488;
        double r4758490 = 1.0;
        double r4758491 = r4758485 * r4758483;
        double r4758492 = r4758491 / r4758480;
        double r4758493 = r4758490 / r4758492;
        double r4758494 = 8.9817388408212e-316;
        bool r4758495 = r4758480 <= r4758494;
        double r4758496 = cbrt(r4758478);
        double r4758497 = r4758496 / r4758483;
        double r4758498 = r4758497 * r4758486;
        double r4758499 = r4758496 * r4758496;
        double r4758500 = r4758498 * r4758499;
        double r4758501 = 1.0693629293958468e+158;
        bool r4758502 = r4758480 <= r4758501;
        double r4758503 = r4758480 / r4758483;
        double r4758504 = r4758503 / r4758485;
        double r4758505 = r4758502 ? r4758504 : r4758487;
        double r4758506 = r4758495 ? r4758500 : r4758505;
        double r4758507 = r4758489 ? r4758493 : r4758506;
        double r4758508 = r4758482 ? r4758487 : r4758507;
        return r4758508;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.8
Target	10.6
Herbie	5.4

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

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -6.245903185019271e+204 or 1.0693629293958468e+158 < (* a1 a2)

   1. Initial program 29.3

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

   3. Applied times-frac 11.4

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

   if -6.245903185019271e+204 < (* a1 a2) < -6.848939379187657e-209

   1. Initial program 3.9

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

   3. Applied clear-num 4.2

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

   if -6.848939379187657e-209 < (* a1 a2) < 8.9817388408212e-316

   1. Initial program 17.6

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

   3. Applied times-frac 3.6

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

   5. Applied *-un-lft-identity 3.6

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

   6. Applied add-cube-cbrt 4.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \sqrt[3]{a1}}}{1 \cdot b1} \cdot \frac{a2}{b2}\]

   7. Applied times-frac 4.0

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1} \cdot \frac{\sqrt[3]{a1}}{b1}\right)} \cdot \frac{a2}{b2}\]

   8. Applied associate-*l* 3.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1} \cdot \left(\frac{\sqrt[3]{a1}}{b1} \cdot \frac{a2}{b2}\right)}\]

   if 8.9817388408212e-316 < (* a1 a2) < 1.0693629293958468e+158

   1. Initial program 4.4

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

   3. Applied associate-/r* 5.2

      \[\leadsto \color{blue}{\frac{\frac{a1 \cdot a2}{b1}}{b2}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -6.245903185019271 \cdot 10^{+204}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -6.848939379187657 \cdot 10^{-209}:\\ \;\;\;\;\frac{1}{\frac{b2 \cdot b1}{a1 \cdot a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 8.9817388408212 \cdot 10^{-316}:\\ \;\;\;\;\left(\frac{\sqrt[3]{a1}}{b1} \cdot \frac{a2}{b2}\right) \cdot \left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 1.0693629293958468 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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