Average Error: 15.3 → 0.4
Time: 21.6s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \left(\frac{\sin b}{\left(\left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right)\right) \cdot \left(\cos b \cdot \cos a\right) - \left(\sin b \cdot \sin a\right) \cdot \left(\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right)} \cdot \left(\left(\left(\cos b \cdot \cos a\right) \cdot \left(\sin b \cdot \sin a\right) + \left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right) + \left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right)\right)\right)\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \left(\frac{\sin b}{\left(\left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right)\right) \cdot \left(\cos b \cdot \cos a\right) - \left(\sin b \cdot \sin a\right) \cdot \left(\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right)} \cdot \left(\left(\left(\cos b \cdot \cos a\right) \cdot \left(\sin b \cdot \sin a\right) + \left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right) + \left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right)\right)\right)
double f(double r, double a, double b) {
        double r425442 = r;
        double r425443 = b;
        double r425444 = sin(r425443);
        double r425445 = r425442 * r425444;
        double r425446 = a;
        double r425447 = r425446 + r425443;
        double r425448 = cos(r425447);
        double r425449 = r425445 / r425448;
        return r425449;
}
↓
double f(double r, double a, double b) {
        double r425450 = r;
        double r425451 = b;
        double r425452 = sin(r425451);
        double r425453 = cos(r425451);
        double r425454 = a;
        double r425455 = cos(r425454);
        double r425456 = r425453 * r425455;
        double r425457 = r425456 * r425456;
        double r425458 = r425457 * r425456;
        double r425459 = sin(r425454);
        double r425460 = r425452 * r425459;
        double r425461 = r425460 * r425460;
        double r425462 = r425460 * r425461;
        double r425463 = r425458 - r425462;
        double r425464 = r425452 / r425463;
        double r425465 = r425456 * r425460;
        double r425466 = r425465 + r425461;
        double r425467 = r425466 + r425457;
        double r425468 = r425464 * r425467;
        double r425469 = r425450 * r425468;
        return r425469;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.3

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum0.4

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.4

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{1 \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}}\]

  6. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{r}{1} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

  7. Simplified0.3

    \[\leadsto \color{blue}{r} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]
  8. Using strategy rm

  9. Applied flip3--0.4

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}}\]

  10. Applied associate-/r/0.4

    \[\leadsto r \cdot \color{blue}{\left(\frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}} \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)\right)}\]

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))