Average Error: 15.3 → 0.4
Time: 53.6s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\sin b \cdot \sin a\right) \cdot \left(\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right)}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\sin b \cdot \sin a\right) \cdot \left(\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right)}}
double f(double r, double a, double b) {
        double r2164387 = r;
        double r2164388 = b;
        double r2164389 = sin(r2164388);
        double r2164390 = a;
        double r2164391 = r2164390 + r2164388;
        double r2164392 = cos(r2164391);
        double r2164393 = r2164389 / r2164392;
        double r2164394 = r2164387 * r2164393;
        return r2164394;
}
↓
double f(double r, double a, double b) {
        double r2164395 = r;
        double r2164396 = b;
        double r2164397 = sin(r2164396);
        double r2164398 = r2164395 * r2164397;
        double r2164399 = a;
        double r2164400 = cos(r2164399);
        double r2164401 = cos(r2164396);
        double r2164402 = r2164400 * r2164401;
        double r2164403 = sin(r2164399);
        double r2164404 = r2164397 * r2164403;
        double r2164405 = r2164404 * r2164404;
        double r2164406 = r2164404 * r2164405;
        double r2164407 = cbrt(r2164406);
        double r2164408 = r2164402 - r2164407;
        double r2164409 = r2164398 / r2164408;
        return r2164409;
}

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 15.3

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

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

  7. Applied add-cbrt-cube0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))