Average Error: 14.8 → 0.5
Time: 40.2s
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\sin b \cdot \sin a + \cos a \cdot \cos b\right) \cdot \left(\sin b \cdot \sin a\right)\right) \cdot \frac{r \cdot \sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin b \cdot \sin a\right)}^{3}}\]

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 14.8

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

  2. Initial simplification14.8

    \[\leadsto \frac{r \cdot \sin b}{\cos \left(b + a\right)}\]
  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 flip3--0.4

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

  7. Applied associate-/r/0.5

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Runtime

Time bar (total: 40.2s)Debug logProfile

herbie shell --seed 2018336 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))