Average Error: 15.2 → 0.4
Time: 1.3m
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{\frac{\left(\cos b \cdot \left(\cos b \cdot \cos b\right)\right) \cdot \left(\cos a \cdot \left(\cos a \cdot \cos a\right)\right) - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\cos b \cdot \cos a\right) + \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}} \cdot r\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{\sin b}{\frac{\left(\cos b \cdot \left(\cos b \cdot \cos b\right)\right) \cdot \left(\cos a \cdot \left(\cos a \cdot \cos a\right)\right) - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\cos b \cdot \cos a\right) + \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}} \cdot r
double f(double r, double a, double b) {
        double r4725040 = r;
        double r4725041 = b;
        double r4725042 = sin(r4725041);
        double r4725043 = a;
        double r4725044 = r4725043 + r4725041;
        double r4725045 = cos(r4725044);
        double r4725046 = r4725042 / r4725045;
        double r4725047 = r4725040 * r4725046;
        return r4725047;
}
↓
double f(double r, double a, double b) {
        double r4725048 = b;
        double r4725049 = sin(r4725048);
        double r4725050 = cos(r4725048);
        double r4725051 = r4725050 * r4725050;
        double r4725052 = r4725050 * r4725051;
        double r4725053 = a;
        double r4725054 = cos(r4725053);
        double r4725055 = r4725054 * r4725054;
        double r4725056 = r4725054 * r4725055;
        double r4725057 = r4725052 * r4725056;
        double r4725058 = sin(r4725053);
        double r4725059 = r4725058 * r4725049;
        double r4725060 = 3.0;
        double r4725061 = pow(r4725059, r4725060);
        double r4725062 = r4725057 - r4725061;
        double r4725063 = r4725050 * r4725054;
        double r4725064 = r4725063 * r4725063;
        double r4725065 = r4725059 * r4725063;
        double r4725066 = r4725059 * r4725059;
        double r4725067 = r4725065 + r4725066;
        double r4725068 = r4725064 + r4725067;
        double r4725069 = r4725062 / r4725068;
        double r4725070 = r4725049 / r4725069;
        double r4725071 = r;
        double r4725072 = r4725070 * r4725071;
        return r4725072;
}

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 15.2

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 flip3--0.4

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}}\]
  6. Using strategy rm

  7. Applied add-cbrt-cube0.7

    \[\leadsto r \cdot \frac{\sin b}{\frac{{\left(\cos a \cdot \color{blue}{\sqrt[3]{\left(\cos b \cdot \cos b\right) \cdot \cos b}}\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}\]

  8. Applied add-cbrt-cube0.9

    \[\leadsto r \cdot \frac{\sin b}{\frac{{\left(\color{blue}{\sqrt[3]{\left(\cos a \cdot \cos a\right) \cdot \cos a}} \cdot \sqrt[3]{\left(\cos b \cdot \cos b\right) \cdot \cos b}\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}\]

  9. Applied cbrt-unprod0.9

    \[\leadsto r \cdot \frac{\sin b}{\frac{{\color{blue}{\left(\sqrt[3]{\left(\left(\cos a \cdot \cos a\right) \cdot \cos a\right) \cdot \left(\left(\cos b \cdot \cos b\right) \cdot \cos b\right)}\right)}}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}\]

  10. Applied rem-cube-cbrt0.4

    \[\leadsto r \cdot \frac{\sin b}{\frac{\color{blue}{\left(\left(\cos a \cdot \cos a\right) \cdot \cos a\right) \cdot \left(\left(\cos b \cdot \cos b\right) \cdot \cos b\right)} - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}\]

  11. Final simplification0.4

    \[\leadsto \frac{\sin b}{\frac{\left(\cos b \cdot \left(\cos b \cdot \cos b\right)\right) \cdot \left(\cos a \cdot \left(\cos a \cdot \cos a\right)\right) - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\cos b \cdot \cos a\right) + \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}} \cdot r\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))