Average Error: 14.9 → 0.3
Time: 24.3s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}
double f(double r, double a, double b) {
        double r778572 = r;
        double r778573 = b;
        double r778574 = sin(r778573);
        double r778575 = r778572 * r778574;
        double r778576 = a;
        double r778577 = r778576 + r778573;
        double r778578 = cos(r778577);
        double r778579 = r778575 / r778578;
        return r778579;
}
↓
double f(double r, double a, double b) {
        double r778580 = r;
        double r778581 = b;
        double r778582 = sin(r778581);
        double r778583 = a;
        double r778584 = cos(r778583);
        double r778585 = cos(r778581);
        double r778586 = r778584 * r778585;
        double r778587 = sin(r778583);
        double r778588 = r778587 * r778582;
        double r778589 = r778586 - r778588;
        double r778590 = r778582 / r778589;
        double r778591 = r778580 * r778590;
        return r778591;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

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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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))