r*sin(b)/cos(a+b), B

Average Error: 15.3 → 0.3

Time: 23.8s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r611020 = r;
        double r611021 = b;
        double r611022 = sin(r611021);
        double r611023 = a;
        double r611024 = r611023 + r611021;
        double r611025 = cos(r611024);
        double r611026 = r611022 / r611025;
        double r611027 = r611020 * r611026;
        return r611027;
}

↓

double f(double r, double a, double b) {
        double r611028 = r;
        double r611029 = b;
        double r611030 = sin(r611029);
        double r611031 = r611028 * r611030;
        double r611032 = a;
        double r611033 = cos(r611032);
        double r611034 = cos(r611029);
        double r611035 = r611033 * r611034;
        double r611036 = sin(r611032);
        double r611037 = r611030 * r611036;
        double r611038 = r611035 - r611037;
        double r611039 = r611031 / r611038;
        return r611039;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.3

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

3. Applied cos-sum 0.3

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
4. Using strategy rm

5. Applied associate-*r/ 0.3

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

6. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019133 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))