Quotient of products

Average Error: 11.4 → 2.0

Time: 15.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.5091135102985783 \cdot 10^{-293}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\left(\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}} \cdot \sqrt[3]{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}}{\frac{1}{\frac{\sqrt[3]{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}{\sqrt[3]{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}}}\right) \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}}{\frac{\sqrt[3]{b2}}{\frac{\sqrt[3]{\sqrt[3]{a2}}}{\sqrt[3]{\sqrt[3]{b1}}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.7924622985567088 \cdot 10^{265}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r172273 = a1;
        double r172274 = a2;
        double r172275 = r172273 * r172274;
        double r172276 = b1;
        double r172277 = b2;
        double r172278 = r172276 * r172277;
        double r172279 = r172275 / r172278;
        return r172279;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r172280 = a1;
        double r172281 = a2;
        double r172282 = r172280 * r172281;
        double r172283 = b1;
        double r172284 = b2;
        double r172285 = r172283 * r172284;
        double r172286 = r172282 / r172285;
        double r172287 = -inf.0;
        bool r172288 = r172286 <= r172287;
        double r172289 = cbrt(r172284);
        double r172290 = r172280 / r172289;
        double r172291 = r172290 / r172289;
        double r172292 = r172281 / r172283;
        double r172293 = r172292 / r172289;
        double r172294 = r172291 * r172293;
        double r172295 = -1.5091135102985783e-293;
        bool r172296 = r172286 <= r172295;
        double r172297 = 0.0;
        bool r172298 = r172286 <= r172297;
        double r172299 = cbrt(r172281);
        double r172300 = r172299 * r172299;
        double r172301 = cbrt(r172283);
        double r172302 = r172301 * r172301;
        double r172303 = r172300 / r172302;
        double r172304 = cbrt(r172303);
        double r172305 = r172304 * r172304;
        double r172306 = 1.0;
        double r172307 = cbrt(r172300);
        double r172308 = cbrt(r172302);
        double r172309 = r172307 / r172308;
        double r172310 = r172306 / r172309;
        double r172311 = r172305 / r172310;
        double r172312 = r172291 * r172311;
        double r172313 = cbrt(r172299);
        double r172314 = cbrt(r172301);
        double r172315 = r172313 / r172314;
        double r172316 = r172289 / r172315;
        double r172317 = r172304 / r172316;
        double r172318 = r172312 * r172317;
        double r172319 = 2.7924622985567088e+265;
        bool r172320 = r172286 <= r172319;
        double r172321 = r172280 / r172283;
        double r172322 = r172281 / r172284;
        double r172323 = r172321 * r172322;
        double r172324 = r172320 ? r172286 : r172323;
        double r172325 = r172298 ? r172318 : r172324;
        double r172326 = r172296 ? r172286 : r172325;
        double r172327 = r172288 ? r172294 : r172326;
        return r172327;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.4
Herbie	2.0

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

Derivation

1. Split input into 4 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 associate-/r* 34.4

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

   5. Applied add-cube-cbrt 34.9

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

   6. Applied *-un-lft-identity 34.9

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

   7. Applied times-frac 16.1

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

   8. Applied times-frac 9.7

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

   9. Simplified 9.7

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.5091135102985783e-293 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 2.7924622985567088e+265

   1. Initial program 0.9

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

   if -1.5091135102985783e-293 < (/ (* a1 a2) (* b1 b2)) < 0.0

   1. Initial program 13.6

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

   3. Applied associate-/r* 6.7

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

   5. Applied add-cube-cbrt 6.8

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

   6. Applied *-un-lft-identity 6.8

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

   7. Applied times-frac 4.0

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

   8. Applied times-frac 2.7

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

   9. Simplified 2.7

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

   11. Applied add-cube-cbrt 2.7

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

   12. Applied add-cube-cbrt 2.8

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

   13. Applied times-frac 2.8

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

   14. Applied associate-/l* 2.5

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

   16. Applied add-cube-cbrt 2.5

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

   17. Applied cbrt-prod 2.5

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

   18. Applied add-cube-cbrt 2.5

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

   19. Applied cbrt-prod 2.5

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

   20. Applied times-frac 2.5

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

   21. Applied *-un-lft-identity 2.5

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

   22. Applied times-frac 2.5

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

   23. Applied add-cube-cbrt 2.5

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

   24. Applied times-frac 2.5

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

   25. Applied associate-*r* 1.1

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

   if 2.7924622985567088e+265 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 54.3

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

   3. Applied times-frac 10.1

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.5091135102985783 \cdot 10^{-293}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\left(\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}} \cdot \sqrt[3]{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}}{\frac{1}{\frac{\sqrt[3]{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}{\sqrt[3]{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}}}\right) \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}}{\frac{\sqrt[3]{b2}}{\frac{\sqrt[3]{\sqrt[3]{a2}}}{\sqrt[3]{\sqrt[3]{b1}}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.7924622985567088 \cdot 10^{265}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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