Average Error: 15.3 → 0.3
Time: 21.5s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}
double f(double r, double a, double b) {
        double r1133014 = r;
        double r1133015 = b;
        double r1133016 = sin(r1133015);
        double r1133017 = r1133014 * r1133016;
        double r1133018 = a;
        double r1133019 = r1133018 + r1133015;
        double r1133020 = cos(r1133019);
        double r1133021 = r1133017 / r1133020;
        return r1133021;
}
↓
double f(double r, double a, double b) {
        double r1133022 = r;
        double r1133023 = b;
        double r1133024 = sin(r1133023);
        double r1133025 = r1133022 * r1133024;
        double r1133026 = a;
        double r1133027 = cos(r1133026);
        double r1133028 = cos(r1133023);
        double r1133029 = sin(r1133026);
        double r1133030 = r1133029 * r1133024;
        double r1133031 = log1p(r1133030);
        double r1133032 = expm1(r1133031);
        double r1133033 = -r1133032;
        double r1133034 = fma(r1133027, r1133028, r1133033);
        double r1133035 = r1133025 / r1133034;
        return r1133035;
}

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 prod-diff0.3

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

  7. Applied expm1-log1p-u0.3

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

  8. Taylor expanded around 0 0.3

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)\right) + \color{blue}{0}}\]

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))