Average Error: 15.2 → 0.4
Time: 6.7s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{1 \cdot \left(\frac{\cos a \cdot \cos b}{\sin b} - \sin a\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r}{1 \cdot \left(\frac{\cos a \cdot \cos b}{\sin b} - \sin a\right)}
double f(double r, double a, double b) {
        double r18142 = r;
        double r18143 = b;
        double r18144 = sin(r18143);
        double r18145 = r18142 * r18144;
        double r18146 = a;
        double r18147 = r18146 + r18143;
        double r18148 = cos(r18147);
        double r18149 = r18145 / r18148;
        return r18149;
}
↓
double f(double r, double a, double b) {
        double r18150 = r;
        double r18151 = 1.0;
        double r18152 = a;
        double r18153 = cos(r18152);
        double r18154 = b;
        double r18155 = cos(r18154);
        double r18156 = r18153 * r18155;
        double r18157 = sin(r18154);
        double r18158 = r18156 / r18157;
        double r18159 = sin(r18152);
        double r18160 = r18158 - r18159;
        double r18161 = r18151 * r18160;
        double r18162 = r18150 / r18161;
        return r18162;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.2

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

  3. Applied cos-sum0.3

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

  5. Applied associate-/l*0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied *-un-lft-identity0.4

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

  9. Applied times-frac0.4

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020065 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))