Quotient of products

Average Error: 11.3 → 2.4

Time: 13.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.414144362639176854803385722786278913018 \cdot 10^{-314} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 1.431802419250923180709933384670076216432 \cdot 10^{298}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r117934 = a1;
        double r117935 = a2;
        double r117936 = r117934 * r117935;
        double r117937 = b1;
        double r117938 = b2;
        double r117939 = r117937 * r117938;
        double r117940 = r117936 / r117939;
        return r117940;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r117941 = a1;
        double r117942 = a2;
        double r117943 = r117941 * r117942;
        double r117944 = b1;
        double r117945 = b2;
        double r117946 = r117944 * r117945;
        double r117947 = r117943 / r117946;
        double r117948 = -inf.0;
        bool r117949 = r117947 <= r117948;
        double r117950 = r117942 / r117945;
        double r117951 = r117950 / r117944;
        double r117952 = r117941 * r117951;
        double r117953 = -1.4141443626392e-314;
        bool r117954 = r117947 <= r117953;
        double r117955 = 0.0;
        bool r117956 = r117947 <= r117955;
        double r117957 = !r117956;
        double r117958 = 1.4318024192509232e+298;
        bool r117959 = r117947 <= r117958;
        bool r117960 = r117957 && r117959;
        bool r117961 = r117954 || r117960;
        double r117962 = r117941 / r117944;
        double r117963 = r117962 * r117950;
        double r117964 = r117961 ? r117947 : r117963;
        double r117965 = r117949 ? r117952 : r117964;
        return r117965;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.2
Herbie	2.4

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   3. Applied times-frac 11.2

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

   5. Applied div-inv 11.3

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

   6. Applied associate-*l* 19.0

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

   7. Simplified 18.9

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.4141443626392e-314 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 1.4318024192509232e+298

   1. Initial program 3.6

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

   if -1.4141443626392e-314 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 1.4318024192509232e+298 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 28.6

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

   3. Applied times-frac 3.7

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.414144362639176854803385722786278913018 \cdot 10^{-314} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 1.431802419250923180709933384670076216432 \cdot 10^{298}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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