Average Error: 14.5 → 0.4
Time: 6.4s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \left(\frac{\sin b}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)} \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)\right)\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \left(\frac{\sin b}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)} \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)\right)
double f(double r, double a, double b) {
        double r17020 = r;
        double r17021 = b;
        double r17022 = sin(r17021);
        double r17023 = a;
        double r17024 = r17023 + r17021;
        double r17025 = cos(r17024);
        double r17026 = r17022 / r17025;
        double r17027 = r17020 * r17026;
        return r17027;
}
↓
double f(double r, double a, double b) {
        double r17028 = r;
        double r17029 = b;
        double r17030 = sin(r17029);
        double r17031 = a;
        double r17032 = cos(r17031);
        double r17033 = cos(r17029);
        double r17034 = r17032 * r17033;
        double r17035 = r17034 * r17034;
        double r17036 = sin(r17031);
        double r17037 = r17036 * r17030;
        double r17038 = r17037 * r17037;
        double r17039 = r17035 - r17038;
        double r17040 = r17030 / r17039;
        double r17041 = r17034 + r17037;
        double r17042 = r17040 * r17041;
        double r17043 = r17028 * r17042;
        return r17043;
}

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 14.5

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

  3. Applied cos-sum0.3

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

  5. Applied flip--0.4

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

  6. Applied associate-/r/0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2020059 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))