Average Error: 15.1 → 0.4
Time: 7.0s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\cos b \cdot \cos a - \sin a \cdot \sin b} \cdot \sin b\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\cos b \cdot \cos a - \sin a \cdot \sin b} \cdot \sin b
double f(double r, double a, double b) {
        double r18814 = r;
        double r18815 = b;
        double r18816 = sin(r18815);
        double r18817 = r18814 * r18816;
        double r18818 = a;
        double r18819 = r18818 + r18815;
        double r18820 = cos(r18819);
        double r18821 = r18817 / r18820;
        return r18821;
}
↓
double f(double r, double a, double b) {
        double r18822 = r;
        double r18823 = b;
        double r18824 = cos(r18823);
        double r18825 = a;
        double r18826 = cos(r18825);
        double r18827 = r18824 * r18826;
        double r18828 = sin(r18825);
        double r18829 = sin(r18823);
        double r18830 = r18828 * r18829;
        double r18831 = r18827 - r18830;
        double r18832 = r18822 / r18831;
        double r18833 = r18832 * r18829;
        return r18833;
}

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 15.1

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

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

  7. Applied add-log-exp0.5

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

  9. Applied associate-/r/0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))