Average Error: 15.2 → 0.3
Time: 27.2s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(\left(-\sin b\right) \cdot \sin a\right))_*}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(\left(-\sin b\right) \cdot \sin a\right))_*}
double f(double r, double a, double b) {
        double r958407 = r;
        double r958408 = b;
        double r958409 = sin(r958408);
        double r958410 = r958407 * r958409;
        double r958411 = a;
        double r958412 = r958411 + r958408;
        double r958413 = cos(r958412);
        double r958414 = r958410 / r958413;
        return r958414;
}
↓
double f(double r, double a, double b) {
        double r958415 = r;
        double r958416 = b;
        double r958417 = sin(r958416);
        double r958418 = r958415 * r958417;
        double r958419 = a;
        double r958420 = cos(r958419);
        double r958421 = cos(r958416);
        double r958422 = -r958417;
        double r958423 = sin(r958419);
        double r958424 = r958422 * r958423;
        double r958425 = fma(r958420, r958421, r958424);
        double r958426 = r958418 / r958425;
        return r958426;
}

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

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(-\sin a \cdot \sin b\right))_*}}\]

  6. Taylor expanded around -inf 0.3

    \[\leadsto \frac{r \cdot \sin b}{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(-\color{blue}{\sin a \cdot \sin b}\right))_*}\]

  7. Final simplification0.3

    \[\leadsto \frac{r \cdot \sin b}{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(\left(-\sin b\right) \cdot \sin a\right))_*}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))