Average Error: 15.3 → 0.4
Time: 22.4s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\left(r \cdot \sin b\right) \cdot \left(\frac{\frac{1}{\cos a \cdot \cos b + \sin a \cdot \sin b}}{\cos a \cdot \cos b - \sin a \cdot \sin b} \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)\right)\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\left(r \cdot \sin b\right) \cdot \left(\frac{\frac{1}{\cos a \cdot \cos b + \sin a \cdot \sin b}}{\cos a \cdot \cos b - \sin a \cdot \sin b} \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)\right)
double f(double r, double a, double b) {
        double r26656 = r;
        double r26657 = b;
        double r26658 = sin(r26657);
        double r26659 = r26656 * r26658;
        double r26660 = a;
        double r26661 = r26660 + r26657;
        double r26662 = cos(r26661);
        double r26663 = r26659 / r26662;
        return r26663;
}
↓
double f(double r, double a, double b) {
        double r26664 = r;
        double r26665 = b;
        double r26666 = sin(r26665);
        double r26667 = r26664 * r26666;
        double r26668 = 1.0;
        double r26669 = a;
        double r26670 = cos(r26669);
        double r26671 = cos(r26665);
        double r26672 = r26670 * r26671;
        double r26673 = sin(r26669);
        double r26674 = r26673 * r26666;
        double r26675 = r26672 + r26674;
        double r26676 = r26668 / r26675;
        double r26677 = r26672 - r26674;
        double r26678 = r26676 / r26677;
        double r26679 = r26678 * r26675;
        double r26680 = r26667 * r26679;
        return r26680;
}

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

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

  2. Simplified15.3

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

  4. Applied cos-sum0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}}\]

  5. Simplified0.3

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

  7. Applied div-inv0.4

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

  9. Applied flip--0.4

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

  10. Applied associate-/r/0.5

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019179 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))