Average Error: 15.3 → 0.3
Time: 23.3s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\left(\cos a\right), \left(\cos b\right), \left(-\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin a \cdot \sin b\right)\right)\right)\right)\right)\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(\left(\cos a\right), \left(\cos b\right), \left(-\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin a \cdot \sin b\right)\right)\right)\right)\right)\right)}
double f(double r, double a, double b) {
        double r783336 = r;
        double r783337 = b;
        double r783338 = sin(r783337);
        double r783339 = r783336 * r783338;
        double r783340 = a;
        double r783341 = r783340 + r783337;
        double r783342 = cos(r783341);
        double r783343 = r783339 / r783342;
        return r783343;
}
↓
double f(double r, double a, double b) {
        double r783344 = r;
        double r783345 = b;
        double r783346 = sin(r783345);
        double r783347 = r783344 * r783346;
        double r783348 = a;
        double r783349 = cos(r783348);
        double r783350 = cos(r783345);
        double r783351 = sin(r783348);
        double r783352 = r783351 * r783346;
        double r783353 = log1p(r783352);
        double r783354 = expm1(r783353);
        double r783355 = -r783354;
        double r783356 = fma(r783349, r783350, r783355);
        double r783357 = r783347 / r783356;
        return r783357;
}

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

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

  7. Applied expm1-log1p-u0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))