Quotient of products

Average Error: 11.3 → 5.4

Time: 49.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.741067738673537939167608822187447910972 \cdot 10^{131}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le -1.160438436830115127650758241442701919848 \cdot 10^{-242}:\\ \;\;\;\;\frac{a2}{b1 \cdot b2} \cdot a1\\ \mathbf{elif}\;b1 \cdot b2 \le 1.207660505247444569564751128982598390884 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{b1}}}{\sqrt[3]{b1}} \cdot \left(\left(\left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right) \cdot \frac{\frac{1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}{\sqrt[3]{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}\right) \cdot \frac{\frac{a1}{\sqrt[3]{b2}}}{\frac{\sqrt[3]{\sqrt[3]{b1}}}{\sqrt[3]{a2}}}\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 5.01380293571370106165791780789048119467 \cdot 10^{135}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r228204 = a1;
        double r228205 = a2;
        double r228206 = r228204 * r228205;
        double r228207 = b1;
        double r228208 = b2;
        double r228209 = r228207 * r228208;
        double r228210 = r228206 / r228209;
        return r228210;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r228211 = b1;
        double r228212 = b2;
        double r228213 = r228211 * r228212;
        double r228214 = -1.741067738673538e+131;
        bool r228215 = r228213 <= r228214;
        double r228216 = a1;
        double r228217 = r228216 / r228212;
        double r228218 = a2;
        double r228219 = r228218 / r228211;
        double r228220 = r228217 * r228219;
        double r228221 = -1.1604384368301151e-242;
        bool r228222 = r228213 <= r228221;
        double r228223 = r228218 / r228213;
        double r228224 = r228223 * r228216;
        double r228225 = 1.207660505247445e-309;
        bool r228226 = r228213 <= r228225;
        double r228227 = 1.0;
        double r228228 = cbrt(r228211);
        double r228229 = r228227 / r228228;
        double r228230 = r228229 / r228228;
        double r228231 = cbrt(r228218);
        double r228232 = r228231 * r228231;
        double r228233 = cbrt(r228212);
        double r228234 = r228233 * r228233;
        double r228235 = r228227 / r228234;
        double r228236 = r228228 * r228228;
        double r228237 = cbrt(r228236);
        double r228238 = r228235 / r228237;
        double r228239 = r228232 * r228238;
        double r228240 = r228216 / r228233;
        double r228241 = cbrt(r228228);
        double r228242 = r228241 / r228231;
        double r228243 = r228240 / r228242;
        double r228244 = r228239 * r228243;
        double r228245 = r228230 * r228244;
        double r228246 = 5.013802935713701e+135;
        bool r228247 = r228213 <= r228246;
        double r228248 = r228218 * r228216;
        double r228249 = r228227 / r228213;
        double r228250 = r228248 * r228249;
        double r228251 = r228247 ? r228250 : r228220;
        double r228252 = r228226 ? r228245 : r228251;
        double r228253 = r228222 ? r228224 : r228252;
        double r228254 = r228215 ? r228220 : r228253;
        return r228254;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.3
Herbie	5.4

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

Derivation

1. Split input into 4 regimes

2. if (* b1 b2) < -1.741067738673538e+131 or 5.013802935713701e+135 < (* b1 b2)

   1. Initial program 13.6

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

   3. Applied associate-/r* 8.8

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

   4. Simplified 6.6

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

   6. Applied *-un-lft-identity 6.6

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

   7. Applied *-un-lft-identity 6.6

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

   8. Applied add-cube-cbrt 7.1

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

   9. Applied times-frac 7.1

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

   10. Applied *-un-lft-identity 7.1

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

   11. Applied times-frac 7.4

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

   12. Applied times-frac 7.3

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

   13. Simplified 7.3

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

   14. Simplified 5.7

       \[\leadsto \frac{\frac{1}{\sqrt[3]{b1}}}{\sqrt[3]{b1}} \cdot \color{blue}{\frac{\frac{a1}{b2}}{\frac{\sqrt[3]{b1}}{a2}}}\]
   15. Using strategy rm

   16. Applied add-cube-cbrt 5.8

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

   17. Applied add-cube-cbrt 5.8

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

   18. Applied cbrt-prod 5.9

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

   19. Applied times-frac 5.8

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

   20. Applied add-cube-cbrt 5.9

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

   21. Applied *-un-lft-identity 5.9

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

   22. Applied times-frac 5.9

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

   23. Applied times-frac 3.4

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

   24. Simplified 3.4

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

   25. Taylor expanded around 0 13.6

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

   26. Simplified 6.2

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

   if -1.741067738673538e+131 < (* b1 b2) < -1.1604384368301151e-242

   1. Initial program 3.7

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

   2. Simplified 5.0

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

   if -1.1604384368301151e-242 < (* b1 b2) < 1.207660505247445e-309

   1. Initial program 48.3

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

   3. Applied associate-/r* 17.4

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

   4. Simplified 8.7

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

   6. Applied *-un-lft-identity 8.7

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

   7. Applied *-un-lft-identity 8.7

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

   8. Applied add-cube-cbrt 9.7

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

   9. Applied times-frac 9.7

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

   10. Applied *-un-lft-identity 9.7

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

   11. Applied times-frac 13.0

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

   12. Applied times-frac 13.0

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

   13. Simplified 13.1

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

   14. Simplified 8.4

       \[\leadsto \frac{\frac{1}{\sqrt[3]{b1}}}{\sqrt[3]{b1}} \cdot \color{blue}{\frac{\frac{a1}{b2}}{\frac{\sqrt[3]{b1}}{a2}}}\]
   15. Using strategy rm

   16. Applied add-cube-cbrt 8.6

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

   17. Applied add-cube-cbrt 8.6

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

   18. Applied cbrt-prod 8.7

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

   19. Applied times-frac 8.7

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

   20. Applied add-cube-cbrt 8.9

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

   21. Applied *-un-lft-identity 8.9

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

   22. Applied times-frac 8.9

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

   23. Applied times-frac 4.5

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

   24. Simplified 4.5

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

   if 1.207660505247445e-309 < (* b1 b2) < 5.013802935713701e+135

   1. Initial program 4.8

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

   3. Applied div-inv 5.0

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

   4. Simplified 5.0

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

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.741067738673537939167608822187447910972 \cdot 10^{131}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le -1.160438436830115127650758241442701919848 \cdot 10^{-242}:\\ \;\;\;\;\frac{a2}{b1 \cdot b2} \cdot a1\\ \mathbf{elif}\;b1 \cdot b2 \le 1.207660505247444569564751128982598390884 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{b1}}}{\sqrt[3]{b1}} \cdot \left(\left(\left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right) \cdot \frac{\frac{1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}{\sqrt[3]{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}\right) \cdot \frac{\frac{a1}{\sqrt[3]{b2}}}{\frac{\sqrt[3]{\sqrt[3]{b1}}}{\sqrt[3]{a2}}}\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 5.01380293571370106165791780789048119467 \cdot 10^{135}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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