Quotient of products

Average Error: 11.2 → 5.1

Time: 18.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -4.209800947527883 \cdot 10^{+175}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.856528697270096 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -4.151349519826297 \cdot 10^{-252}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a2}}{b2}}{\sqrt[3]{b1}} \cdot \left(a1 \cdot \left(\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}\right)\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 2.5035993978499362 \cdot 10^{-275}:\\ \;\;\;\;\frac{\frac{a2}{\sqrt[3]{b2}}}{b1} \cdot \frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.4951274135627082 \cdot 10^{+288}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r4373206 = a1;
        double r4373207 = a2;
        double r4373208 = r4373206 * r4373207;
        double r4373209 = b1;
        double r4373210 = b2;
        double r4373211 = r4373209 * r4373210;
        double r4373212 = r4373208 / r4373211;
        return r4373212;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r4373213 = a1;
        double r4373214 = a2;
        double r4373215 = r4373213 * r4373214;
        double r4373216 = -4.209800947527883e+175;
        bool r4373217 = r4373215 <= r4373216;
        double r4373218 = b2;
        double r4373219 = r4373214 / r4373218;
        double r4373220 = b1;
        double r4373221 = r4373219 / r4373220;
        double r4373222 = r4373213 * r4373221;
        double r4373223 = -1.856528697270096e-61;
        bool r4373224 = r4373215 <= r4373223;
        double r4373225 = r4373215 / r4373220;
        double r4373226 = r4373225 / r4373218;
        double r4373227 = -4.151349519826297e-252;
        bool r4373228 = r4373215 <= r4373227;
        double r4373229 = cbrt(r4373214);
        double r4373230 = r4373229 / r4373218;
        double r4373231 = cbrt(r4373220);
        double r4373232 = r4373230 / r4373231;
        double r4373233 = r4373229 / r4373231;
        double r4373234 = r4373233 * r4373233;
        double r4373235 = r4373213 * r4373234;
        double r4373236 = r4373232 * r4373235;
        double r4373237 = 2.5035993978499362e-275;
        bool r4373238 = r4373215 <= r4373237;
        double r4373239 = cbrt(r4373218);
        double r4373240 = r4373214 / r4373239;
        double r4373241 = r4373240 / r4373220;
        double r4373242 = r4373239 * r4373239;
        double r4373243 = r4373213 / r4373242;
        double r4373244 = r4373241 * r4373243;
        double r4373245 = 1.4951274135627082e+288;
        bool r4373246 = r4373215 <= r4373245;
        double r4373247 = r4373246 ? r4373226 : r4373222;
        double r4373248 = r4373238 ? r4373244 : r4373247;
        double r4373249 = r4373228 ? r4373236 : r4373248;
        double r4373250 = r4373224 ? r4373226 : r4373249;
        double r4373251 = r4373217 ? r4373222 : r4373250;
        return r4373251;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.2
Target	11.1
Herbie	5.1

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

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -4.209800947527883e+175 or 1.4951274135627082e+288 < (* a1 a2)

   1. Initial program 38.1

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

   3. Applied times-frac 9.9

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

   5. Applied div-inv 10.0

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

   6. Applied associate-*l* 10.1

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

   7. Simplified 10.1

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

   if -4.209800947527883e+175 < (* a1 a2) < -1.856528697270096e-61 or 2.5035993978499362e-275 < (* a1 a2) < 1.4951274135627082e+288

   1. Initial program 4.8

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

   3. Applied associate-/r* 4.6

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

   if -1.856528697270096e-61 < (* a1 a2) < -4.151349519826297e-252

   1. Initial program 4.6

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

   3. Applied times-frac 11.4

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

   5. Applied div-inv 11.4

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

   6. Applied associate-*l* 10.5

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

   7. Simplified 10.4

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

   9. Applied add-cube-cbrt 11.0

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

   10. Applied *-un-lft-identity 11.0

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

   11. Applied add-cube-cbrt 11.1

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

   12. Applied times-frac 11.1

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

   13. Applied times-frac 8.2

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

   14. Applied associate-*r* 5.9

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

   15. Simplified 5.9

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

   if -4.151349519826297e-252 < (* a1 a2) < 2.5035993978499362e-275

   1. Initial program 18.0

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

   3. Applied times-frac 3.0

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

   5. Applied div-inv 3.0

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

   6. Applied associate-*l* 3.4

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

   7. Simplified 3.4

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

   9. Applied *-un-lft-identity 3.4

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

   10. Applied add-cube-cbrt 3.7

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

   11. Applied *-un-lft-identity 3.7

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

   12. Applied times-frac 3.7

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

   13. Applied times-frac 3.5

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

   14. Applied associate-*r* 3.1

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

   15. Simplified 3.1

       \[\leadsto \color{blue}{\frac{a1 \cdot 1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{\frac{a2}{\sqrt[3]{b2}}}{b1}\]
3. Recombined 4 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -4.209800947527883 \cdot 10^{+175}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.856528697270096 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -4.151349519826297 \cdot 10^{-252}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a2}}{b2}}{\sqrt[3]{b1}} \cdot \left(a1 \cdot \left(\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}\right)\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 2.5035993978499362 \cdot 10^{-275}:\\ \;\;\;\;\frac{\frac{a2}{\sqrt[3]{b2}}}{b1} \cdot \frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.4951274135627082 \cdot 10^{+288}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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