Quotient of products

Average Error: 11.2 → 1.9

Time: 2.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.57378486630527 \cdot 10^{-309}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.1129784645222681 \cdot 10^{+303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2}}} \cdot \frac{\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b2}}}{\sqrt[3]{b1} \cdot \sqrt[3]{b2}}}{\frac{\sqrt[3]{\sqrt[3]{b2}}}{\sqrt[3]{a2}} \cdot \frac{\sqrt[3]{\sqrt[3]{b2}}}{\sqrt[3]{a2}}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r54505240 = a1;
        double r54505241 = a2;
        double r54505242 = r54505240 * r54505241;
        double r54505243 = b1;
        double r54505244 = b2;
        double r54505245 = r54505243 * r54505244;
        double r54505246 = r54505242 / r54505245;
        return r54505246;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r54505247 = a1;
        double r54505248 = a2;
        double r54505249 = r54505247 * r54505248;
        double r54505250 = b1;
        double r54505251 = b2;
        double r54505252 = r54505250 * r54505251;
        double r54505253 = r54505249 / r54505252;
        double r54505254 = -inf.0;
        bool r54505255 = r54505253 <= r54505254;
        double r54505256 = r54505247 / r54505250;
        double r54505257 = r54505248 / r54505251;
        double r54505258 = r54505256 * r54505257;
        double r54505259 = -4.57378486630527e-309;
        bool r54505260 = r54505253 <= r54505259;
        double r54505261 = -0.0;
        bool r54505262 = r54505253 <= r54505261;
        double r54505263 = 1.1129784645222681e+303;
        bool r54505264 = r54505253 <= r54505263;
        double r54505265 = cbrt(r54505248);
        double r54505266 = cbrt(r54505250);
        double r54505267 = r54505265 / r54505266;
        double r54505268 = cbrt(r54505251);
        double r54505269 = cbrt(r54505268);
        double r54505270 = r54505267 / r54505269;
        double r54505271 = r54505266 * r54505268;
        double r54505272 = r54505247 / r54505271;
        double r54505273 = r54505272 / r54505271;
        double r54505274 = r54505269 / r54505265;
        double r54505275 = r54505274 * r54505274;
        double r54505276 = r54505273 / r54505275;
        double r54505277 = r54505270 * r54505276;
        double r54505278 = r54505264 ? r54505253 : r54505277;
        double r54505279 = r54505262 ? r54505258 : r54505278;
        double r54505280 = r54505260 ? r54505253 : r54505279;
        double r54505281 = r54505255 ? r54505258 : r54505280;
        return r54505281;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.2
Target	10.9
Herbie	1.9

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -4.57378486630527e-309 < (/ (* a1 a2) (* b1 b2)) < -0.0

   1. Initial program 18.4

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

   3. Applied times-frac 3.0

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -4.57378486630527e-309 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 1.1129784645222681e+303

   1. Initial program 0.8

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

   if 1.1129784645222681e+303 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 60.0

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

   3. Applied associate-/r* 45.0

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

   5. Applied add-cube-cbrt 45.3

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

   6. Applied add-cube-cbrt 45.5

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

   7. Applied times-frac 12.6

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

   8. Applied times-frac 6.7

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

   9. Simplified 5.7

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

   11. Applied add-cube-cbrt 5.9

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

   12. Applied *-un-lft-identity 5.9

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

   13. Applied cbrt-prod 5.9

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

   14. Applied add-cube-cbrt 6.0

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

   15. Applied times-frac 6.0

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

   16. Applied times-frac 5.3

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

   17. Applied associate-*r* 5.6

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

   18. Simplified 5.6

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.57378486630527 \cdot 10^{-309}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.1129784645222681 \cdot 10^{+303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2}}} \cdot \frac{\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b2}}}{\sqrt[3]{b1} \cdot \sqrt[3]{b2}}}{\frac{\sqrt[3]{\sqrt[3]{b2}}}{\sqrt[3]{a2}} \cdot \frac{\sqrt[3]{\sqrt[3]{b2}}}{\sqrt[3]{a2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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