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

Average Error: 15.0 → 0.3

Time: 18.1s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r25941 = r;
        double r25942 = b;
        double r25943 = sin(r25942);
        double r25944 = r25941 * r25943;
        double r25945 = a;
        double r25946 = r25945 + r25942;
        double r25947 = cos(r25946);
        double r25948 = r25944 / r25947;
        return r25948;
}

↓

double f(double r, double a, double b) {
        double r25949 = r;
        double r25950 = b;
        double r25951 = sin(r25950);
        double r25952 = a;
        double r25953 = cos(r25952);
        double r25954 = cos(r25950);
        double r25955 = r25953 * r25954;
        double r25956 = sin(r25952);
        double r25957 = r25956 * r25951;
        double r25958 = r25955 - r25957;
        double r25959 = r25951 / r25958;
        double r25960 = r25949 * r25959;
        return r25960;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.0

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

3. Applied cos-sum 0.3

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

5. Applied *-un-lft-identity 0.3

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{1 \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}}\]

6. Applied times-frac 0.3

   \[\leadsto \color{blue}{\frac{r}{1} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

7. Simplified 0.3

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

8. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019291 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))