Quotient of products

Average Error: 11.1 → 5.0

Time: 6.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 = -\infty:\\ \;\;\;\;\frac{a2}{\sqrt[3]{b2}} \cdot \left(\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{b2}} \cdot \frac{\frac{\sqrt[3]{a1}}{b1}}{\sqrt[3]{b2}}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le -5.841733098692663762903904892944559559654 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 9.870262468729980181103925129699646064015 \cdot 10^{-33}:\\ \;\;\;\;\frac{a2 \cdot \frac{\frac{1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}}{\sqrt[3]{b2} \cdot \frac{\sqrt[3]{b2}}{\frac{\frac{a1}{\sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2}}}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.730858527692710738611248585413969126226 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\frac{b2 \cdot b1}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \frac{\frac{1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}}{\sqrt[3]{b2} \cdot \frac{\sqrt[3]{b2}}{\frac{\frac{a1}{\sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2}}}}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r17996121 = a1;
        double r17996122 = a2;
        double r17996123 = r17996121 * r17996122;
        double r17996124 = b1;
        double r17996125 = b2;
        double r17996126 = r17996124 * r17996125;
        double r17996127 = r17996123 / r17996126;
        return r17996127;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r17996128 = a1;
        double r17996129 = a2;
        double r17996130 = r17996128 * r17996129;
        double r17996131 = -inf.0;
        bool r17996132 = r17996130 <= r17996131;
        double r17996133 = b2;
        double r17996134 = cbrt(r17996133);
        double r17996135 = r17996129 / r17996134;
        double r17996136 = cbrt(r17996128);
        double r17996137 = r17996136 * r17996136;
        double r17996138 = r17996137 / r17996134;
        double r17996139 = b1;
        double r17996140 = r17996136 / r17996139;
        double r17996141 = r17996140 / r17996134;
        double r17996142 = r17996138 * r17996141;
        double r17996143 = r17996135 * r17996142;
        double r17996144 = -5.841733098692664e-155;
        bool r17996145 = r17996130 <= r17996144;
        double r17996146 = r17996130 / r17996139;
        double r17996147 = r17996146 / r17996133;
        double r17996148 = 9.87026246872998e-33;
        bool r17996149 = r17996130 <= r17996148;
        double r17996150 = 1.0;
        double r17996151 = cbrt(r17996139);
        double r17996152 = r17996151 * r17996151;
        double r17996153 = r17996150 / r17996152;
        double r17996154 = r17996134 * r17996134;
        double r17996155 = cbrt(r17996154);
        double r17996156 = r17996153 / r17996155;
        double r17996157 = r17996129 * r17996156;
        double r17996158 = r17996128 / r17996151;
        double r17996159 = cbrt(r17996134);
        double r17996160 = r17996158 / r17996159;
        double r17996161 = r17996134 / r17996160;
        double r17996162 = r17996134 * r17996161;
        double r17996163 = r17996157 / r17996162;
        double r17996164 = 1.7308585276927107e+90;
        bool r17996165 = r17996130 <= r17996164;
        double r17996166 = r17996133 * r17996139;
        double r17996167 = r17996166 / r17996130;
        double r17996168 = r17996150 / r17996167;
        double r17996169 = r17996165 ? r17996168 : r17996163;
        double r17996170 = r17996149 ? r17996163 : r17996169;
        double r17996171 = r17996145 ? r17996147 : r17996170;
        double r17996172 = r17996132 ? r17996143 : r17996171;
        return r17996172;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.1
Target	11.2
Herbie	5.0

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

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -inf.0

   1. Initial program 64.0

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

   3. Applied times-frac 5.8

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

   5. Applied add-cube-cbrt 6.6

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

   6. Applied *-un-lft-identity 6.6

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

   7. Applied times-frac 6.7

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

   8. Applied associate-*r* 6.6

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

   9. Simplified 6.6

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

   11. Applied add-sqr-sqrt 33.3

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

   12. Applied add-sqr-sqrt 33.4

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

   13. Applied *-un-lft-identity 33.4

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

   14. Applied add-cube-cbrt 33.5

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

   15. Applied times-frac 33.5

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

   16. Applied times-frac 31.8

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

   17. Applied times-frac 32.0

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

   18. Simplified 31.9

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

   19. Simplified 3.3

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

   if -inf.0 < (* a1 a2) < -5.841733098692664e-155

   1. Initial program 5.1

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

   3. Applied associate-/r* 4.9

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

   if -5.841733098692664e-155 < (* a1 a2) < 9.87026246872998e-33 or 1.7308585276927107e+90 < (* a1 a2)

   1. Initial program 13.5

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

   3. Applied times-frac 8.9

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

   5. Applied add-cube-cbrt 9.4

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

   6. Applied *-un-lft-identity 9.4

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

   7. Applied times-frac 9.5

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

   8. Applied associate-*r* 8.0

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

   9. Simplified 8.0

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

   11. Applied add-cube-cbrt 8.0

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

   12. Applied cbrt-prod 8.1

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

   13. Applied add-cube-cbrt 8.2

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

   14. Applied *-un-lft-identity 8.2

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

   15. Applied times-frac 8.2

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

   16. Applied times-frac 6.7

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

   17. Applied associate-/l* 6.2

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

   19. Applied frac-times 5.5

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

   if 9.87026246872998e-33 < (* a1 a2) < 1.7308585276927107e+90

   1. Initial program 2.2

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

   3. Applied clear-num 2.4

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

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 = -\infty:\\ \;\;\;\;\frac{a2}{\sqrt[3]{b2}} \cdot \left(\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{b2}} \cdot \frac{\frac{\sqrt[3]{a1}}{b1}}{\sqrt[3]{b2}}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le -5.841733098692663762903904892944559559654 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 9.870262468729980181103925129699646064015 \cdot 10^{-33}:\\ \;\;\;\;\frac{a2 \cdot \frac{\frac{1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}}{\sqrt[3]{b2} \cdot \frac{\sqrt[3]{b2}}{\frac{\frac{a1}{\sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2}}}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.730858527692710738611248585413969126226 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\frac{b2 \cdot b1}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \frac{\frac{1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}}{\sqrt[3]{b2} \cdot \frac{\sqrt[3]{b2}}{\frac{\frac{a1}{\sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2}}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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