Average Error: 14.7 → 0.4
Time: 1.1m
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right) \cdot \left(\left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right) \cdot \left(\sin a \cdot \sqrt[3]{\sin b}\right)\right)}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right) \cdot \left(\left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right) \cdot \left(\sin a \cdot \sqrt[3]{\sin b}\right)\right)}}
double f(double r, double a, double b) {
        double r3045179 = r;
        double r3045180 = b;
        double r3045181 = sin(r3045180);
        double r3045182 = a;
        double r3045183 = r3045182 + r3045180;
        double r3045184 = cos(r3045183);
        double r3045185 = r3045181 / r3045184;
        double r3045186 = r3045179 * r3045185;
        return r3045186;
}
↓
double f(double r, double a, double b) {
        double r3045187 = r;
        double r3045188 = b;
        double r3045189 = sin(r3045188);
        double r3045190 = a;
        double r3045191 = cos(r3045190);
        double r3045192 = cos(r3045188);
        double r3045193 = r3045191 * r3045192;
        double r3045194 = sin(r3045190);
        double r3045195 = r3045194 * r3045189;
        double r3045196 = r3045195 * r3045195;
        double r3045197 = cbrt(r3045189);
        double r3045198 = r3045197 * r3045197;
        double r3045199 = r3045194 * r3045197;
        double r3045200 = r3045198 * r3045199;
        double r3045201 = r3045196 * r3045200;
        double r3045202 = cbrt(r3045201);
        double r3045203 = r3045193 - r3045202;
        double r3045204 = r3045189 / r3045203;
        double r3045205 = r3045187 * r3045204;
        return r3045205;
}

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. Using strategy rm

  5. Applied add-cbrt-cube0.4

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

  6. Applied add-cbrt-cube0.4

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

  7. Applied cbrt-unprod0.4

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

  8. Simplified0.4

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

  10. Applied add-cube-cbrt0.4

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

  11. Applied associate-*l*0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))