Quotient of products

Average Error: 10.8 → 11.0

Time: 14.8s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double b1, double b2) {
        double r141891 = a1;
        double r141892 = a2;
        double r141893 = r141891 * r141892;
        double r141894 = b1;
        double r141895 = b2;
        double r141896 = r141894 * r141895;
        double r141897 = r141893 / r141896;
        return r141897;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r141898 = a1;
        double r141899 = a2;
        double r141900 = r141898 * r141899;
        double r141901 = b1;
        double r141902 = r141900 / r141901;
        double r141903 = b2;
        double r141904 = r141902 / r141903;
        return r141904;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.8
Target	11.7
Herbie	11.0

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

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -3.6160475663182616e+189 or 3.0277503863756097e+132 < (* a1 a2)

   1. Initial program 27.7

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

   3. Applied associate-/r* 27.6

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

   5. Applied *-un-lft-identity 27.6

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

   6. Applied times-frac 16.9

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

   7. Applied associate-/l* 11.0

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

   if -3.6160475663182616e+189 < (* a1 a2) < -4.586910408345106e-202

   1. Initial program 4.1

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

   3. Applied associate-/r* 4.5

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

   5. Applied *-un-lft-identity 4.5

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

   if -4.586910408345106e-202 < (* a1 a2) < 1.6268561300982777e-203

   1. Initial program 14.5

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

   3. Applied associate-/r* 14.7

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

   5. Applied add-cube-cbrt 14.9

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

   6. Applied times-frac 7.5

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

   7. Applied associate-/l* 3.7

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

   if 1.6268561300982777e-203 < (* a1 a2) < 3.0277503863756097e+132

   1. Initial program 3.8

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

   3. Applied associate-/r* 3.7

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

   5. Applied *-un-lft-identity 3.7

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

   6. Applied div-inv 3.8

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

   7. Applied times-frac 4.1

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

   8. Simplified 4.1

      \[\leadsto \color{blue}{\left(a1 \cdot a2\right)} \cdot \frac{\frac{1}{b1}}{b2}\]
3. Recombined 4 regimes into one program.

4. Final simplification 11.0

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

Reproduce

herbie shell --seed 2019298 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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