Average Error: 15.0 → 0.3
Time: 35.7s
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 r1136241 = r;
        double r1136242 = b;
        double r1136243 = sin(r1136242);
        double r1136244 = r1136241 * r1136243;
        double r1136245 = a;
        double r1136246 = r1136245 + r1136242;
        double r1136247 = cos(r1136246);
        double r1136248 = r1136244 / r1136247;
        return r1136248;
}
↓
double f(double r, double a, double b) {
        double r1136249 = r;
        double r1136250 = b;
        double r1136251 = sin(r1136250);
        double r1136252 = a;
        double r1136253 = cos(r1136252);
        double r1136254 = cos(r1136250);
        double r1136255 = r1136253 * r1136254;
        double r1136256 = sin(r1136252);
        double r1136257 = r1136256 * r1136251;
        double r1136258 = r1136255 - r1136257;
        double r1136259 = r1136251 / r1136258;
        double r1136260 = r1136249 * r1136259;
        return r1136260;
}

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 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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))