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

Average Error: 15.1 → 0.3

Time: 24.1s

Precision: 64

Math

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

↓

\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]

C

double f(double r, double a, double b) {
        double r459033 = r;
        double r459034 = b;
        double r459035 = sin(r459034);
        double r459036 = a;
        double r459037 = r459036 + r459034;
        double r459038 = cos(r459037);
        double r459039 = r459035 / r459038;
        double r459040 = r459033 * r459039;
        return r459040;
}

↓

double f(double r, double a, double b) {
        double r459041 = r;
        double r459042 = b;
        double r459043 = sin(r459042);
        double r459044 = r459041 * r459043;
        double r459045 = a;
        double r459046 = cos(r459045);
        double r459047 = cos(r459042);
        double r459048 = r459046 * r459047;
        double r459049 = sin(r459045);
        double r459050 = r459049 * r459043;
        double r459051 = r459048 - r459050;
        double r459052 = r459044 / r459051;
        return r459052;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.1

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
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. Taylor expanded around -inf 0.3

   \[\leadsto \color{blue}{\frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

5. Final simplification 0.3

   \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))