Average Error: 14.9 → 0.4
Time: 6.5s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, -\log \left(e^{\sin a \cdot \sin b}\right)\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, -\log \left(e^{\sin a \cdot \sin b}\right)\right)}
double f(double r, double a, double b) {
        double r17312 = r;
        double r17313 = b;
        double r17314 = sin(r17313);
        double r17315 = r17312 * r17314;
        double r17316 = a;
        double r17317 = r17316 + r17313;
        double r17318 = cos(r17317);
        double r17319 = r17315 / r17318;
        return r17319;
}
↓
double f(double r, double a, double b) {
        double r17320 = r;
        double r17321 = b;
        double r17322 = sin(r17321);
        double r17323 = cos(r17321);
        double r17324 = a;
        double r17325 = cos(r17324);
        double r17326 = sin(r17324);
        double r17327 = r17326 * r17322;
        double r17328 = exp(r17327);
        double r17329 = log(r17328);
        double r17330 = -r17329;
        double r17331 = fma(r17323, r17325, r17330);
        double r17332 = r17322 / r17331;
        double r17333 = r17320 * r17332;
        return r17333;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

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 fma-neg0.3

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

  12. Applied add-log-exp0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))