Average Error: 14.7 → 0.4
Time: 18.2s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{3}}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{3}}}
double f(double r, double a, double b) {
        double r26035 = r;
        double r26036 = b;
        double r26037 = sin(r26036);
        double r26038 = a;
        double r26039 = r26038 + r26036;
        double r26040 = cos(r26039);
        double r26041 = r26037 / r26040;
        double r26042 = r26035 * r26041;
        return r26042;
}
↓
double f(double r, double a, double b) {
        double r26043 = r;
        double r26044 = b;
        double r26045 = sin(r26044);
        double r26046 = a;
        double r26047 = cos(r26046);
        double r26048 = cos(r26044);
        double r26049 = r26047 * r26048;
        double r26050 = sin(r26046);
        double r26051 = r26045 * r26050;
        double r26052 = 3.0;
        double r26053 = pow(r26051, r26052);
        double r26054 = cbrt(r26053);
        double r26055 = r26049 - r26054;
        double r26056 = r26045 / r26055;
        double r26057 = r26043 * r26056;
        return r26057;
}

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 14.7

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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. Simplified0.3

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

  6. Applied add-cbrt-cube0.4

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

  7. Applied add-cbrt-cube0.4

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

  8. Applied cbrt-unprod0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))