Quotient of products

Average Error: 10.9 → 4.3

Time: 29.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.2127685659922545 \cdot 10^{+214}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -4.4979291324398776 \cdot 10^{-178}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b2 \cdot b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.196927419235656 \cdot 10^{-145}:\\ \;\;\;\;\left(\frac{a1}{\frac{b1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}\right) \cdot \frac{1}{\frac{1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 9.61446956110427 \cdot 10^{+171}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r7230868 = a1;
        double r7230869 = a2;
        double r7230870 = r7230868 * r7230869;
        double r7230871 = b1;
        double r7230872 = b2;
        double r7230873 = r7230871 * r7230872;
        double r7230874 = r7230870 / r7230873;
        return r7230874;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r7230875 = a1;
        double r7230876 = a2;
        double r7230877 = r7230875 * r7230876;
        double r7230878 = -1.2127685659922545e+214;
        bool r7230879 = r7230877 <= r7230878;
        double r7230880 = b1;
        double r7230881 = r7230876 / r7230880;
        double r7230882 = b2;
        double r7230883 = r7230881 / r7230882;
        double r7230884 = r7230875 * r7230883;
        double r7230885 = -4.4979291324398776e-178;
        bool r7230886 = r7230877 <= r7230885;
        double r7230887 = 1.0;
        double r7230888 = r7230882 * r7230880;
        double r7230889 = r7230887 / r7230888;
        double r7230890 = r7230877 * r7230889;
        double r7230891 = 1.196927419235656e-145;
        bool r7230892 = r7230877 <= r7230891;
        double r7230893 = cbrt(r7230876);
        double r7230894 = cbrt(r7230882);
        double r7230895 = r7230893 / r7230894;
        double r7230896 = r7230880 / r7230895;
        double r7230897 = r7230875 / r7230896;
        double r7230898 = r7230897 * r7230895;
        double r7230899 = r7230887 / r7230895;
        double r7230900 = r7230887 / r7230899;
        double r7230901 = r7230898 * r7230900;
        double r7230902 = 9.61446956110427e+171;
        bool r7230903 = r7230877 <= r7230902;
        double r7230904 = r7230895 * r7230895;
        double r7230905 = r7230880 / r7230904;
        double r7230906 = r7230875 / r7230905;
        double r7230907 = r7230906 * r7230895;
        double r7230908 = r7230903 ? r7230890 : r7230907;
        double r7230909 = r7230892 ? r7230901 : r7230908;
        double r7230910 = r7230886 ? r7230890 : r7230909;
        double r7230911 = r7230879 ? r7230884 : r7230910;
        return r7230911;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.9
Target	11.0
Herbie	4.3

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

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -1.2127685659922545e+214

   1. Initial program 33.2

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   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* 11.4

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

   7. Simplified 8.9

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

   if -1.2127685659922545e+214 < (* a1 a2) < -4.4979291324398776e-178 or 1.196927419235656e-145 < (* a1 a2) < 9.61446956110427e+171

   1. Initial program 4.1

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

   3. Applied div-inv 4.4

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

   if -4.4979291324398776e-178 < (* a1 a2) < 1.196927419235656e-145

   1. Initial program 12.8

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

   3. Applied times-frac 5.7

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

   5. Applied add-cube-cbrt 6.1

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

   6. Applied add-cube-cbrt 6.3

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

   7. Applied times-frac 6.3

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

   8. Applied associate-*r* 4.0

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

   9. Simplified 3.3

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

   11. Applied *-un-lft-identity 3.3

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

   12. Applied times-frac 3.0

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

   13. Applied *-un-lft-identity 3.0

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

   14. Applied times-frac 2.9

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

   15. Applied associate-*l* 2.9

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

   if 9.61446956110427e+171 < (* a1 a2)

   1. Initial program 31.1

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

   3. Applied times-frac 11.5

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

   5. Applied add-cube-cbrt 12.3

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

   6. Applied add-cube-cbrt 12.5

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

   7. Applied times-frac 12.5

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

   8. Applied associate-*r* 7.8

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

   9. Simplified 6.7

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

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.2127685659922545 \cdot 10^{+214}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -4.4979291324398776 \cdot 10^{-178}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b2 \cdot b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.196927419235656 \cdot 10^{-145}:\\ \;\;\;\;\left(\frac{a1}{\frac{b1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}\right) \cdot \frac{1}{\frac{1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 9.61446956110427 \cdot 10^{+171}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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