Average Error: 15.1 → 0.3
Time: 41.6s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b
double f(double r, double a, double b) {
        double r776503 = r;
        double r776504 = b;
        double r776505 = sin(r776504);
        double r776506 = a;
        double r776507 = r776506 + r776504;
        double r776508 = cos(r776507);
        double r776509 = r776505 / r776508;
        double r776510 = r776503 * r776509;
        return r776510;
}
↓
double f(double r, double a, double b) {
        double r776511 = r;
        double r776512 = a;
        double r776513 = cos(r776512);
        double r776514 = b;
        double r776515 = cos(r776514);
        double r776516 = r776513 * r776515;
        double r776517 = sin(r776514);
        double r776518 = sin(r776512);
        double r776519 = r776517 * r776518;
        double r776520 = r776516 - r776519;
        double r776521 = r776511 / r776520;
        double r776522 = r776521 * r776517;
        return r776522;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.1

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

  3. Applied cos-sum0.3

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

  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. Using strategy rm

  6. Applied fma-neg0.3

    \[\leadsto \frac{\sin b \cdot r}{\color{blue}{\mathsf{fma}\left(\cos a, \cos b, -\sin a \cdot \sin b\right)}}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity0.3

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

  9. Applied times-frac0.3

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

  10. Simplified0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))