Quotient of products

Average Error: 11.4 → 4.9

Time: 11.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -6.898490520999014 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{1}{b1}}{\frac{\frac{1}{a2}}{\frac{a1}{b2}}}\\ \mathbf{elif}\;b1 \cdot b2 \le -9.006559789728174 \cdot 10^{-250}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.30715975193335 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 7.676958632971629 \cdot 10^{+257}:\\ \;\;\;\;\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\sqrt[3]{a2}}} \cdot \frac{1}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r2229920 = a1;
        double r2229921 = a2;
        double r2229922 = r2229920 * r2229921;
        double r2229923 = b1;
        double r2229924 = b2;
        double r2229925 = r2229923 * r2229924;
        double r2229926 = r2229922 / r2229925;
        return r2229926;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r2229927 = b1;
        double r2229928 = b2;
        double r2229929 = r2229927 * r2229928;
        double r2229930 = -6.898490520999014e+277;
        bool r2229931 = r2229929 <= r2229930;
        double r2229932 = 1.0;
        double r2229933 = r2229932 / r2229927;
        double r2229934 = a2;
        double r2229935 = r2229932 / r2229934;
        double r2229936 = a1;
        double r2229937 = r2229936 / r2229928;
        double r2229938 = r2229935 / r2229937;
        double r2229939 = r2229933 / r2229938;
        double r2229940 = -9.006559789728174e-250;
        bool r2229941 = r2229929 <= r2229940;
        double r2229942 = r2229934 / r2229929;
        double r2229943 = r2229936 * r2229942;
        double r2229944 = 1.30715975193335e-196;
        bool r2229945 = r2229929 <= r2229944;
        double r2229946 = r2229936 / r2229927;
        double r2229947 = r2229928 / r2229934;
        double r2229948 = r2229946 / r2229947;
        double r2229949 = 7.676958632971629e+257;
        bool r2229950 = r2229929 <= r2229949;
        double r2229951 = r2229936 / r2229929;
        double r2229952 = r2229951 / r2229935;
        double r2229953 = cbrt(r2229934);
        double r2229954 = r2229928 / r2229953;
        double r2229955 = r2229936 / r2229954;
        double r2229956 = r2229953 * r2229953;
        double r2229957 = r2229927 / r2229956;
        double r2229958 = r2229932 / r2229957;
        double r2229959 = r2229955 * r2229958;
        double r2229960 = r2229950 ? r2229952 : r2229959;
        double r2229961 = r2229945 ? r2229948 : r2229960;
        double r2229962 = r2229941 ? r2229943 : r2229961;
        double r2229963 = r2229931 ? r2229939 : r2229962;
        return r2229963;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.0
Herbie	4.9

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

Derivation

1. Split input into 5 regimes

2. if (* b1 b2) < -6.898490520999014e+277

   1. Initial program 20.9

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

   3. Applied associate-/l* 20.8

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

   5. Applied div-inv 20.8

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

   6. Applied associate-/r* 20.9

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

   8. Applied *-un-lft-identity 20.9

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

   9. Applied times-frac 7.2

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

   10. Applied associate-/l* 2.4

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

   if -6.898490520999014e+277 < (* b1 b2) < -9.006559789728174e-250

   1. Initial program 5.0

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

   3. Applied associate-/l* 5.2

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

   5. Applied div-inv 5.2

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

   6. Applied associate-/r* 4.9

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

   8. Applied *-un-lft-identity 4.9

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

   9. Applied add-cube-cbrt 4.9

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

   10. Applied times-frac 4.9

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

   11. Applied div-inv 4.9

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

   12. Applied times-frac 5.3

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

   13. Simplified 5.3

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

   14. Simplified 5.2

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

   if -9.006559789728174e-250 < (* b1 b2) < 1.30715975193335e-196

   1. Initial program 35.6

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

   3. Applied associate-/l* 35.7

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

   5. Applied *-un-lft-identity 35.7

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

   6. Applied times-frac 17.2

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

   7. Applied associate-/r* 10.0

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

   8. Simplified 10.0

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

   if 1.30715975193335e-196 < (* b1 b2) < 7.676958632971629e+257

   1. Initial program 5.0

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

   3. Applied associate-/l* 4.7

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

   5. Applied div-inv 4.7

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

   6. Applied associate-/r* 4.3

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

   if 7.676958632971629e+257 < (* b1 b2)

   1. Initial program 18.3

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

   3. Applied associate-/l* 18.4

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

   5. Applied add-cube-cbrt 18.5

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

   6. Applied times-frac 7.9

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

   7. Applied *-un-lft-identity 7.9

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

   8. Applied times-frac 2.4

      \[\leadsto \color{blue}{\frac{1}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -6.898490520999014 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{1}{b1}}{\frac{\frac{1}{a2}}{\frac{a1}{b2}}}\\ \mathbf{elif}\;b1 \cdot b2 \le -9.006559789728174 \cdot 10^{-250}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.30715975193335 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 7.676958632971629 \cdot 10^{+257}:\\ \;\;\;\;\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\sqrt[3]{a2}}} \cdot \frac{1}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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