Average Error: 14.9 → 0.4
Time: 6.7s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos b \cdot \cos a\right)\right) - \sin a \cdot \sin b}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos b \cdot \cos a\right)\right) - \sin a \cdot \sin b}
double f(double r, double a, double b) {
        double r17002 = r;
        double r17003 = b;
        double r17004 = sin(r17003);
        double r17005 = r17002 * r17004;
        double r17006 = a;
        double r17007 = r17006 + r17003;
        double r17008 = cos(r17007);
        double r17009 = r17005 / r17008;
        return r17009;
}
↓
double f(double r, double a, double b) {
        double r17010 = r;
        double r17011 = b;
        double r17012 = sin(r17011);
        double r17013 = cos(r17011);
        double r17014 = a;
        double r17015 = cos(r17014);
        double r17016 = r17013 * r17015;
        double r17017 = expm1(r17016);
        double r17018 = log1p(r17017);
        double r17019 = sin(r17014);
        double r17020 = r17019 * r17012;
        double r17021 = r17018 - r17020;
        double r17022 = r17012 / r17021;
        double r17023 = r17010 * r17022;
        return r17023;
}

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 14.9

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 *-un-lft-identity0.3

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

  6. Applied times-frac0.3

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

  7. Simplified0.3

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

  8. Simplified0.3

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

  10. Applied log1p-expm1-u0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))