Average Error: 14.7 → 0.4
Time: 24.2s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \left(\sin b \cdot \frac{1}{\cos a \cdot \cos b - \sin a \cdot \sin b}\right)\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \left(\sin b \cdot \frac{1}{\cos a \cdot \cos b - \sin a \cdot \sin b}\right)
double f(double r, double a, double b) {
        double r24585 = r;
        double r24586 = b;
        double r24587 = sin(r24586);
        double r24588 = r24585 * r24587;
        double r24589 = a;
        double r24590 = r24589 + r24586;
        double r24591 = cos(r24590);
        double r24592 = r24588 / r24591;
        return r24592;
}
↓
double f(double r, double a, double b) {
        double r24593 = r;
        double r24594 = b;
        double r24595 = sin(r24594);
        double r24596 = 1.0;
        double r24597 = a;
        double r24598 = cos(r24597);
        double r24599 = cos(r24594);
        double r24600 = r24598 * r24599;
        double r24601 = sin(r24597);
        double r24602 = r24601 * r24595;
        double r24603 = r24600 - r24602;
        double r24604 = r24596 / r24603;
        double r24605 = r24595 * r24604;
        double r24606 = r24593 * r24605;
        return r24606;
}

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 14.7

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

  9. Applied div-inv0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))