Quotient of products

Average Error: 11.0 → 2.6

Time: 20.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.378626986605869 \cdot 10^{-297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}} \cdot \frac{a1}{\frac{b1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 6.0388420886701275 \cdot 10^{+293}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a1}{\frac{b1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}\right) \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r6842677 = a1;
        double r6842678 = a2;
        double r6842679 = r6842677 * r6842678;
        double r6842680 = b1;
        double r6842681 = b2;
        double r6842682 = r6842680 * r6842681;
        double r6842683 = r6842679 / r6842682;
        return r6842683;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r6842684 = a1;
        double r6842685 = a2;
        double r6842686 = r6842684 * r6842685;
        double r6842687 = b1;
        double r6842688 = b2;
        double r6842689 = r6842687 * r6842688;
        double r6842690 = r6842686 / r6842689;
        double r6842691 = -inf.0;
        bool r6842692 = r6842690 <= r6842691;
        double r6842693 = r6842684 / r6842687;
        double r6842694 = r6842685 * r6842693;
        double r6842695 = r6842694 / r6842688;
        double r6842696 = -3.378626986605869e-297;
        bool r6842697 = r6842690 <= r6842696;
        double r6842698 = -0.0;
        bool r6842699 = r6842690 <= r6842698;
        double r6842700 = cbrt(r6842685);
        double r6842701 = cbrt(r6842688);
        double r6842702 = r6842700 / r6842701;
        double r6842703 = r6842702 * r6842702;
        double r6842704 = r6842687 / r6842703;
        double r6842705 = r6842684 / r6842704;
        double r6842706 = r6842702 * r6842705;
        double r6842707 = 6.0388420886701275e+293;
        bool r6842708 = r6842690 <= r6842707;
        double r6842709 = r6842687 / r6842702;
        double r6842710 = r6842684 / r6842709;
        double r6842711 = r6842710 * r6842702;
        double r6842712 = r6842711 * r6842702;
        double r6842713 = r6842708 ? r6842690 : r6842712;
        double r6842714 = r6842699 ? r6842706 : r6842713;
        double r6842715 = r6842697 ? r6842690 : r6842714;
        double r6842716 = r6842692 ? r6842695 : r6842715;
        return r6842716;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.0
Target	11.1
Herbie	2.6

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

Derivation

1. Split input into 4 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0

   1. Initial program 59.9

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

   3. Applied times-frac 10.7

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

   5. Applied associate-*r/ 14.3

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -3.378626986605869e-297 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 6.0388420886701275e+293

   1. Initial program 0.9

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

   if -3.378626986605869e-297 < (/ (* a1 a2) (* b1 b2)) < -0.0

   1. Initial program 14.1

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

   3. Applied times-frac 2.7

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

   5. Applied add-cube-cbrt 2.9

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

   6. Applied add-cube-cbrt 3.0

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

   7. Applied times-frac 3.0

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

   8. Applied associate-*r* 1.8

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

   9. Simplified 3.4

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

   if 6.0388420886701275e+293 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 57.0

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

   3. Applied times-frac 7.7

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

   5. Applied add-cube-cbrt 8.6

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

   6. Applied add-cube-cbrt 8.8

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

   7. Applied times-frac 8.7

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

   8. Applied associate-*r* 5.9

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

   9. Simplified 11.8

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

   11. Applied *-un-lft-identity 11.8

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

   12. Applied times-frac 11.3

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

   13. Applied *-un-lft-identity 11.3

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

   14. Applied times-frac 7.5

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

   15. Simplified 7.5

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

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.378626986605869 \cdot 10^{-297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}} \cdot \frac{a1}{\frac{b1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 6.0388420886701275 \cdot 10^{+293}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a1}{\frac{b1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}\right) \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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