Average Error: 15.0 → 0.4
Time: 6.1s
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 r16329 = r;
        double r16330 = b;
        double r16331 = sin(r16330);
        double r16332 = r16329 * r16331;
        double r16333 = a;
        double r16334 = r16333 + r16330;
        double r16335 = cos(r16334);
        double r16336 = r16332 / r16335;
        return r16336;
}
↓
double f(double r, double a, double b) {
        double r16337 = r;
        double r16338 = b;
        double r16339 = sin(r16338);
        double r16340 = 1.0;
        double r16341 = cos(r16338);
        double r16342 = a;
        double r16343 = cos(r16342);
        double r16344 = r16341 * r16343;
        double r16345 = sin(r16342);
        double r16346 = r16345 * r16339;
        double r16347 = r16344 - r16346;
        double r16348 = r16340 / r16347;
        double r16349 = r16339 * r16348;
        double r16350 = r16337 * r16349;
        return r16350;
}

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 15.0

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 2019362 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))