Average Error: 15.5 → 0.4
Time: 23.7s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\mathsf{fma}\left(\cos a, \frac{\cos b}{\sin b}, -\sin a\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\mathsf{fma}\left(\cos a, \frac{\cos b}{\sin b}, -\sin a\right)}
double f(double r, double a, double b) {
        double r24138 = r;
        double r24139 = b;
        double r24140 = sin(r24139);
        double r24141 = a;
        double r24142 = r24141 + r24139;
        double r24143 = cos(r24142);
        double r24144 = r24140 / r24143;
        double r24145 = r24138 * r24144;
        return r24145;
}
↓
double f(double r, double a, double b) {
        double r24146 = r;
        double r24147 = a;
        double r24148 = cos(r24147);
        double r24149 = b;
        double r24150 = cos(r24149);
        double r24151 = sin(r24149);
        double r24152 = r24150 / r24151;
        double r24153 = sin(r24147);
        double r24154 = -r24153;
        double r24155 = fma(r24148, r24152, r24154);
        double r24156 = r24146 / r24155;
        return r24156;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.5

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 pow10.3

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

  6. Applied pow10.3

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

  7. Applied pow-prod-down0.3

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

  8. Simplified0.4

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

  10. Applied *-un-lft-identity0.4

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

  11. Applied times-frac0.4

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

  12. Applied fma-neg0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))