Average Error: 14.9 → 0.5
Time: 26.5s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\sqrt[3]{\frac{\frac{1}{\cos a \cdot \cos b - \sin b \cdot \sin a}}{\left(\cos a \cdot \cos b - \sin b \cdot \sin a\right) \cdot \left(\cos a \cdot \cos b - \sin b \cdot \sin a\right)}} \cdot \left(r \cdot \sin b\right)\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\sqrt[3]{\frac{\frac{1}{\cos a \cdot \cos b - \sin b \cdot \sin a}}{\left(\cos a \cdot \cos b - \sin b \cdot \sin a\right) \cdot \left(\cos a \cdot \cos b - \sin b \cdot \sin a\right)}} \cdot \left(r \cdot \sin b\right)
double f(double r, double a, double b) {
        double r819609 = r;
        double r819610 = b;
        double r819611 = sin(r819610);
        double r819612 = a;
        double r819613 = r819612 + r819610;
        double r819614 = cos(r819613);
        double r819615 = r819611 / r819614;
        double r819616 = r819609 * r819615;
        return r819616;
}
↓
double f(double r, double a, double b) {
        double r819617 = 1.0;
        double r819618 = a;
        double r819619 = cos(r819618);
        double r819620 = b;
        double r819621 = cos(r819620);
        double r819622 = r819619 * r819621;
        double r819623 = sin(r819620);
        double r819624 = sin(r819618);
        double r819625 = r819623 * r819624;
        double r819626 = r819622 - r819625;
        double r819627 = r819617 / r819626;
        double r819628 = r819626 * r819626;
        double r819629 = r819627 / r819628;
        double r819630 = cbrt(r819629);
        double r819631 = r;
        double r819632 = r819631 * r819623;
        double r819633 = r819630 * r819632;
        return r819633;
}

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 14.9

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 div-inv0.4

    \[\leadsto r \cdot \color{blue}{\left(\sin b \cdot \frac{1}{\cos a \cdot \cos b - \sin a \cdot \sin b}\right)}\]

  6. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube0.5

    \[\leadsto \left(r \cdot \sin b\right) \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(\left(\cos a \cdot \cos b - \sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)\right) \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}}}\]

  9. Applied add-cbrt-cube0.5

    \[\leadsto \left(r \cdot \sin b\right) \cdot \frac{\color{blue}{\sqrt[3]{\left(1 \cdot 1\right) \cdot 1}}}{\sqrt[3]{\left(\left(\cos a \cdot \cos b - \sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)\right) \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}}\]

  10. Applied cbrt-undiv0.5

    \[\leadsto \left(r \cdot \sin b\right) \cdot \color{blue}{\sqrt[3]{\frac{\left(1 \cdot 1\right) \cdot 1}{\left(\left(\cos a \cdot \cos b - \sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)\right) \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}}}\]

  11. Simplified0.5

    \[\leadsto \left(r \cdot \sin b\right) \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{\cos b \cdot \cos a - \sin b \cdot \sin a}}{\left(\cos b \cdot \cos a - \sin b \cdot \sin a\right) \cdot \left(\cos b \cdot \cos a - \sin b \cdot \sin a\right)}}}\]

  12. Final simplification0.5

    \[\leadsto \sqrt[3]{\frac{\frac{1}{\cos a \cdot \cos b - \sin b \cdot \sin a}}{\left(\cos a \cdot \cos b - \sin b \cdot \sin a\right) \cdot \left(\cos a \cdot \cos b - \sin b \cdot \sin a\right)}} \cdot \left(r \cdot \sin b\right)\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))