Average Error: 15.3 → 0.3
Time: 6.0s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{log1p}\left(\mathsf{expm1}\left(\sin a \cdot \sin b\right)\right)\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{log1p}\left(\mathsf{expm1}\left(\sin a \cdot \sin b\right)\right)\right)}
double f(double r, double a, double b) {
        double r16348 = r;
        double r16349 = b;
        double r16350 = sin(r16349);
        double r16351 = r16348 * r16350;
        double r16352 = a;
        double r16353 = r16352 + r16349;
        double r16354 = cos(r16353);
        double r16355 = r16351 / r16354;
        return r16355;
}
↓
double f(double r, double a, double b) {
        double r16356 = r;
        double r16357 = b;
        double r16358 = sin(r16357);
        double r16359 = r16356 * r16358;
        double r16360 = a;
        double r16361 = cos(r16360);
        double r16362 = cos(r16357);
        double r16363 = sin(r16360);
        double r16364 = r16363 * r16358;
        double r16365 = expm1(r16364);
        double r16366 = log1p(r16365);
        double r16367 = -r16366;
        double r16368 = fma(r16361, r16362, r16367);
        double r16369 = r16359 / r16368;
        return r16369;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.3

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 fma-neg0.3

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

  7. Applied log1p-expm1-u0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))