Average Error: 15.4 → 0.4
Time: 7.6s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{{\left(\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}^{1}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{{\left(\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}^{1}}
double f(double r, double a, double b) {
        double r21076 = r;
        double r21077 = b;
        double r21078 = sin(r21077);
        double r21079 = a;
        double r21080 = r21079 + r21077;
        double r21081 = cos(r21080);
        double r21082 = r21078 / r21081;
        double r21083 = r21076 * r21082;
        return r21083;
}
↓
double f(double r, double a, double b) {
        double r21084 = r;
        double r21085 = b;
        double r21086 = sin(r21085);
        double r21087 = r21084 * r21086;
        double r21088 = a;
        double r21089 = cos(r21088);
        double r21090 = cos(r21085);
        double r21091 = r21089 * r21090;
        double r21092 = sin(r21088);
        double r21093 = r21092 * r21086;
        double r21094 = log1p(r21093);
        double r21095 = expm1(r21094);
        double r21096 = r21091 - r21095;
        double r21097 = 1.0;
        double r21098 = pow(r21096, r21097);
        double r21099 = r21087 / r21098;
        return r21099;
}

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 15.4

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

  5. Applied expm1-log1p-u0.4

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

  7. Applied pow10.4

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{{\left(\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}^{1}}}\]
  8. Using strategy rm

  9. Applied associate-*r/0.4

    \[\leadsto \color{blue}{\frac{r \cdot \sin b}{{\left(\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}^{1}}}\]

  10. Final simplification0.4

    \[\leadsto \frac{r \cdot \sin b}{{\left(\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}^{1}}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))