Average Error: 15.1 → 0.3
Time: 29.0s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]
double f(double r, double a, double b) {
        double r995118 = r;
        double r995119 = b;
        double r995120 = sin(r995119);
        double r995121 = r995118 * r995120;
        double r995122 = a;
        double r995123 = r995122 + r995119;
        double r995124 = cos(r995123);
        double r995125 = r995121 / r995124;
        return r995125;
}
↓
double f(double r, double a, double b) {
        double r995126 = r;
        double r995127 = b;
        double r995128 = sin(r995127);
        double r995129 = r995126 * r995128;
        double r995130 = a;
        double r995131 = cos(r995130);
        double r995132 = cos(r995127);
        double r995133 = r995131 * r995132;
        double r995134 = sin(r995130);
        double r995135 = r995134 * r995128;
        double r995136 = r995133 - r995135;
        double r995137 = r995129 / r995136;
        return r995137;
}


\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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. Taylor expanded around inf 0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019101 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))