Average Error: 15.1 → 0.4
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\cos b \cdot \frac{\cos a}{\sin b} - \sin a}\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.1

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

  2. Initial simplification15.1

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

  6. Applied associate-/l*0.4

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

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

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

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

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

  10. Applied times-frac0.4

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

  11. Simplified0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018346 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))