Average Error: 15.3 → 0.3
Time: 6.3s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin a \cdot \sin b\right)\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\cos b \cdot \cos a - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin a \cdot \sin b\right)\right)}
double f(double r, double a, double b) {
        double r16471 = r;
        double r16472 = b;
        double r16473 = sin(r16472);
        double r16474 = r16471 * r16473;
        double r16475 = a;
        double r16476 = r16475 + r16472;
        double r16477 = cos(r16476);
        double r16478 = r16474 / r16477;
        return r16478;
}
↓
double f(double r, double a, double b) {
        double r16479 = r;
        double r16480 = b;
        double r16481 = sin(r16480);
        double r16482 = cos(r16480);
        double r16483 = a;
        double r16484 = cos(r16483);
        double r16485 = r16482 * r16484;
        double r16486 = sin(r16483);
        double r16487 = r16486 * r16481;
        double r16488 = expm1(r16487);
        double r16489 = log1p(r16488);
        double r16490 = r16485 - r16489;
        double r16491 = r16481 / r16490;
        double r16492 = r16479 * r16491;
        return r16492;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.3

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

  3. Applied cos-sum0.3

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

  5. Applied fma-neg0.3

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

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

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

  8. Applied times-frac0.3

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

  9. Simplified0.3

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

  10. Simplified0.3

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

  12. Applied log1p-expm1-u0.3

    \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin a \cdot \sin b\right)\right)}}\]

  13. Final simplification0.3

    \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin a \cdot \sin b\right)\right)}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))