Quotient of products

Average Error: 11.5 → 4.9

Time: 14.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -2.690573134936579355188068325109566692365 \cdot 10^{214}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.337432395097420768246040584874582122161 \cdot 10^{-223}:\\ \;\;\;\;\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \frac{\sqrt[3]{a1} \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.239318929799362337308674794520138114567 \cdot 10^{-121}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.135669594507262246668248770076436070243 \cdot 10^{239}:\\ \;\;\;\;\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \frac{\sqrt[3]{a1} \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r101925 = a1;
        double r101926 = a2;
        double r101927 = r101925 * r101926;
        double r101928 = b1;
        double r101929 = b2;
        double r101930 = r101928 * r101929;
        double r101931 = r101927 / r101930;
        return r101931;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r101932 = b1;
        double r101933 = b2;
        double r101934 = r101932 * r101933;
        double r101935 = -2.6905731349365794e+214;
        bool r101936 = r101934 <= r101935;
        double r101937 = a1;
        double r101938 = a2;
        double r101939 = r101938 / r101933;
        double r101940 = r101937 * r101939;
        double r101941 = r101940 / r101932;
        double r101942 = -4.337432395097421e-223;
        bool r101943 = r101934 <= r101942;
        double r101944 = cbrt(r101937);
        double r101945 = r101944 * r101944;
        double r101946 = r101944 * r101938;
        double r101947 = r101946 / r101934;
        double r101948 = r101945 * r101947;
        double r101949 = 1.2393189297993623e-121;
        bool r101950 = r101934 <= r101949;
        double r101951 = r101937 / r101932;
        double r101952 = r101939 * r101951;
        double r101953 = 1.1356695945072622e+239;
        bool r101954 = r101934 <= r101953;
        double r101955 = r101954 ? r101948 : r101941;
        double r101956 = r101950 ? r101952 : r101955;
        double r101957 = r101943 ? r101948 : r101956;
        double r101958 = r101936 ? r101941 : r101957;
        return r101958;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.3
Herbie	4.9

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

Derivation

1. Split input into 3 regimes

2. if (* b1 b2) < -2.6905731349365794e+214 or 1.1356695945072622e+239 < (* b1 b2)

   1. Initial program 16.9

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

   2. Simplified 17.1

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

   4. Applied add-cube-cbrt 17.3

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

   5. Applied times-frac 7.6

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

   6. Simplified 7.6

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

   8. Applied associate-*l/ 7.6

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

   9. Applied associate-*l/ 7.7

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

   10. Applied associate-*r/ 4.3

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

   11. Simplified 4.0

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

   if -2.6905731349365794e+214 < (* b1 b2) < -4.337432395097421e-223 or 1.2393189297993623e-121 < (* b1 b2) < 1.1356695945072622e+239

   1. Initial program 4.6

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

   2. Simplified 4.1

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

   4. Applied add-cube-cbrt 4.9

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

   5. Applied times-frac 9.9

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

   6. Simplified 9.9

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

   8. Applied pow1 9.9

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

   9. Applied pow1 9.9

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

   10. Applied pow1 9.9

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

   11. Applied pow-prod-down 9.9

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

   12. Applied pow-prod-down 9.9

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

   13. Simplified 11.0

       \[\leadsto a1 \cdot {\color{blue}{\left(\frac{\frac{a2}{b2}}{b1}\right)}}^{1}\]
   14. Using strategy rm

   15. Applied add-cube-cbrt 11.7

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

   16. Applied associate-*l* 11.7

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

   17. Simplified 3.3

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

   if -4.337432395097421e-223 < (* b1 b2) < 1.2393189297993623e-121

   1. Initial program 27.1

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

   2. Simplified 28.8

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

   4. Applied *-un-lft-identity 28.8

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

   5. Applied times-frac 18.4

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

   6. Applied associate-*r* 11.9

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

   7. Simplified 11.8

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

4. Final simplification 4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -2.690573134936579355188068325109566692365 \cdot 10^{214}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.337432395097420768246040584874582122161 \cdot 10^{-223}:\\ \;\;\;\;\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \frac{\sqrt[3]{a1} \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.239318929799362337308674794520138114567 \cdot 10^{-121}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.135669594507262246668248770076436070243 \cdot 10^{239}:\\ \;\;\;\;\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \frac{\sqrt[3]{a1} \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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