Average Error: 14.9 → 0.4
Time: 6.0s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \left(\sin b \cdot \frac{1}{\cos b \cdot \cos a - \sin a \cdot \sin b}\right)\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \left(\sin b \cdot \frac{1}{\cos b \cdot \cos a - \sin a \cdot \sin b}\right)
double f(double r, double a, double b) {
        double r17457 = r;
        double r17458 = b;
        double r17459 = sin(r17458);
        double r17460 = r17457 * r17459;
        double r17461 = a;
        double r17462 = r17461 + r17458;
        double r17463 = cos(r17462);
        double r17464 = r17460 / r17463;
        return r17464;
}
↓
double f(double r, double a, double b) {
        double r17465 = r;
        double r17466 = b;
        double r17467 = sin(r17466);
        double r17468 = 1.0;
        double r17469 = cos(r17466);
        double r17470 = a;
        double r17471 = cos(r17470);
        double r17472 = r17469 * r17471;
        double r17473 = sin(r17470);
        double r17474 = r17473 * r17467;
        double r17475 = r17472 - r17474;
        double r17476 = r17468 / r17475;
        double r17477 = r17467 * r17476;
        double r17478 = r17465 * r17477;
        return r17478;
}

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 14.9

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 *-un-lft-identity0.3

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

  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. Simplified0.3

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

  10. Applied div-inv0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))