r*sin(b)/cos(a+b), B

Average Error: 15.2 → 0.4

Time: 6.4s

Precision: 64

Math

\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]

↓

\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \log \left(e^{\sin a \cdot \sin b}\right)}\]

C

double f(double r, double a, double b) {
        double r16859 = r;
        double r16860 = b;
        double r16861 = sin(r16860);
        double r16862 = a;
        double r16863 = r16862 + r16860;
        double r16864 = cos(r16863);
        double r16865 = r16861 / r16864;
        double r16866 = r16859 * r16865;
        return r16866;
}

↓

double f(double r, double a, double b) {
        double r16867 = r;
        double r16868 = b;
        double r16869 = sin(r16868);
        double r16870 = a;
        double r16871 = cos(r16870);
        double r16872 = cos(r16868);
        double r16873 = r16871 * r16872;
        double r16874 = sin(r16870);
        double r16875 = r16874 * r16869;
        double r16876 = exp(r16875);
        double r16877 = log(r16876);
        double r16878 = r16873 - r16877;
        double r16879 = r16869 / r16878;
        double r16880 = r16867 * r16879;
        return r16880;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.2

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
2. Using strategy rm

3. Applied cos-sum 0.3

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
4. Using strategy rm

5. Applied add-log-exp 0.4

   \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\log \left(e^{\sin a \cdot \sin b}\right)}}\]

6. Final simplification 0.4

   \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \log \left(e^{\sin a \cdot \sin b}\right)}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))