Quotient of products

Average Error: 11.5 → 4.5

Time: 5.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.97251895095999178047329774749525954037 \cdot 10^{245}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.538508659456969327282848851815247527073 \cdot 10^{-281} \lor \neg \left(a1 \cdot a2 \le 4.178012939374055441343683101567782411238 \cdot 10^{-287} \lor \neg \left(a1 \cdot a2 \le 3.859905230799150421067740794787184289132 \cdot 10^{236}\right)\right):\\ \;\;\;\;\frac{a1 \cdot a2}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{\frac{1}{\sqrt[3]{b1}}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{b1 \cdot \sqrt[3]{b2}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r110920 = a1;
        double r110921 = a2;
        double r110922 = r110920 * r110921;
        double r110923 = b1;
        double r110924 = b2;
        double r110925 = r110923 * r110924;
        double r110926 = r110922 / r110925;
        return r110926;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r110927 = a1;
        double r110928 = a2;
        double r110929 = r110927 * r110928;
        double r110930 = -2.972518950959992e+245;
        bool r110931 = r110929 <= r110930;
        double r110932 = b1;
        double r110933 = r110927 / r110932;
        double r110934 = b2;
        double r110935 = r110928 / r110934;
        double r110936 = r110933 * r110935;
        double r110937 = -2.5385086594569693e-281;
        bool r110938 = r110929 <= r110937;
        double r110939 = 4.1780129393740554e-287;
        bool r110940 = r110929 <= r110939;
        double r110941 = 3.8599052307991504e+236;
        bool r110942 = r110929 <= r110941;
        double r110943 = !r110942;
        bool r110944 = r110940 || r110943;
        double r110945 = !r110944;
        bool r110946 = r110938 || r110945;
        double r110947 = cbrt(r110932);
        double r110948 = r110947 * r110947;
        double r110949 = r110929 / r110948;
        double r110950 = 1.0;
        double r110951 = r110950 / r110947;
        double r110952 = r110951 / r110934;
        double r110953 = r110949 * r110952;
        double r110954 = cbrt(r110934);
        double r110955 = r110954 * r110954;
        double r110956 = r110927 / r110955;
        double r110957 = r110932 * r110954;
        double r110958 = r110928 / r110957;
        double r110959 = r110956 * r110958;
        double r110960 = r110946 ? r110953 : r110959;
        double r110961 = r110931 ? r110936 : r110960;
        return r110961;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.5
Herbie	4.5

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

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -2.972518950959992e+245

   1. Initial program 44.4

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

   3. Applied times-frac 10.0

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

   if -2.972518950959992e+245 < (* a1 a2) < -2.5385086594569693e-281 or 4.1780129393740554e-287 < (* a1 a2) < 3.8599052307991504e+236

   1. Initial program 5.1

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

   3. Applied clear-num 5.4

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

   5. Applied associate-/l* 5.6

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

   7. Applied div-inv 5.7

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

   8. Applied add-cube-cbrt 6.3

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

   9. Applied times-frac 4.2

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

   10. Applied add-cube-cbrt 4.2

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

   11. Applied times-frac 4.3

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

   12. Simplified 4.1

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

   13. Simplified 4.0

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

   if -2.5385086594569693e-281 < (* a1 a2) < 4.1780129393740554e-287 or 3.8599052307991504e+236 < (* a1 a2)

   1. Initial program 24.4

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

   3. Applied clear-num 24.4

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

   5. Applied associate-/l* 24.0

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

   7. Applied div-inv 24.0

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

   8. Applied *-un-lft-identity 24.0

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

   9. Applied times-frac 25.1

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

   10. Applied associate-/r* 25.0

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

   11. Simplified 24.4

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

   13. Applied add-cube-cbrt 24.5

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

   14. Applied add-cube-cbrt 24.5

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

   15. Applied times-frac 24.5

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

   16. Applied *-un-lft-identity 24.5

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

   17. Applied times-frac 24.5

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

   18. Applied times-frac 4.9

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

   19. Simplified 4.9

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

   20. Simplified 4.9

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

4. Final simplification 4.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.97251895095999178047329774749525954037 \cdot 10^{245}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.538508659456969327282848851815247527073 \cdot 10^{-281} \lor \neg \left(a1 \cdot a2 \le 4.178012939374055441343683101567782411238 \cdot 10^{-287} \lor \neg \left(a1 \cdot a2 \le 3.859905230799150421067740794787184289132 \cdot 10^{236}\right)\right):\\ \;\;\;\;\frac{a1 \cdot a2}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{\frac{1}{\sqrt[3]{b1}}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{b1 \cdot \sqrt[3]{b2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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