Average Error: 15.2 → 0.3
Time: 20.0s
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 r1197335 = r;
        double r1197336 = b;
        double r1197337 = sin(r1197336);
        double r1197338 = r1197335 * r1197337;
        double r1197339 = a;
        double r1197340 = r1197339 + r1197336;
        double r1197341 = cos(r1197340);
        double r1197342 = r1197338 / r1197341;
        return r1197342;
}
↓
double f(double r, double a, double b) {
        double r1197343 = r;
        double r1197344 = b;
        double r1197345 = sin(r1197344);
        double r1197346 = r1197343 * r1197345;
        double r1197347 = a;
        double r1197348 = cos(r1197347);
        double r1197349 = cos(r1197344);
        double r1197350 = sin(r1197347);
        double r1197351 = r1197350 * r1197345;
        double r1197352 = expm1(r1197351);
        double r1197353 = log1p(r1197352);
        double r1197354 = -r1197353;
        double r1197355 = fma(r1197348, r1197349, r1197354);
        double r1197356 = r1197346 / r1197355;
        return r1197356;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.2

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 2019172 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))