Average Error: 15.4 → 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 r16472 = r;
        double r16473 = b;
        double r16474 = sin(r16473);
        double r16475 = r16472 * r16474;
        double r16476 = a;
        double r16477 = r16476 + r16473;
        double r16478 = cos(r16477);
        double r16479 = r16475 / r16478;
        return r16479;
}
↓
double f(double r, double a, double b) {
        double r16480 = r;
        double r16481 = b;
        double r16482 = sin(r16481);
        double r16483 = 1.0;
        double r16484 = cos(r16481);
        double r16485 = a;
        double r16486 = cos(r16485);
        double r16487 = r16484 * r16486;
        double r16488 = sin(r16485);
        double r16489 = r16488 * r16482;
        double r16490 = r16487 - r16489;
        double r16491 = r16483 / r16490;
        double r16492 = r16482 * r16491;
        double r16493 = r16480 * r16492;
        return r16493;
}

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 15.4

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 2020025 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))