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

Average Error: 15.2 → 0.4

Time: 25.1s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r947380 = r;
        double r947381 = b;
        double r947382 = sin(r947381);
        double r947383 = r947380 * r947382;
        double r947384 = a;
        double r947385 = r947384 + r947381;
        double r947386 = cos(r947385);
        double r947387 = r947383 / r947386;
        return r947387;
}

↓

double f(double r, double a, double b) {
        double r947388 = r;
        double r947389 = b;
        double r947390 = cos(r947389);
        double r947391 = a;
        double r947392 = cos(r947391);
        double r947393 = r947390 * r947392;
        double r947394 = sin(r947389);
        double r947395 = sin(r947391);
        double r947396 = r947394 * r947395;
        double r947397 = r947393 - r947396;
        double r947398 = r947397 / r947394;
        double r947399 = r947388 / r947398;
        return r947399;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.2

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 associate-/l* 0.4

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))