Average Error: 13.1 → 0.2
Time: 43.8s
Precision: 64
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\frac{\left(\cos a \cdot \left(\tan y + \tan z\right)\right) \cdot \left(\cos a \cdot \left(\tan y + \tan z\right)\right) - \left(\sin a \cdot \left(1 - \tan z \cdot \tan y\right)\right) \cdot \left(\sin a \cdot \left(1 - \tan z \cdot \tan y\right)\right)}{\left(\cos a \cdot \left(\tan y + \tan z\right) + \sin a \cdot \left(1 - \tan z \cdot \tan y\right)\right) \cdot \left(\left(1 - \tan z \cdot \tan y\right) \cdot \cos a\right)} + x\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.1

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  2. Using strategy rm

  3. Applied tan-sum0.2

    \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right)\]
  4. Using strategy rm

  5. Applied tan-quot0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \color{blue}{\frac{\sin a}{\cos a}}\right)\]

  6. Applied frac-sub0.2

    \[\leadsto x + \color{blue}{\frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}}\]
  7. Using strategy rm

  8. Applied flip--0.2

    \[\leadsto x + \frac{\color{blue}{\frac{\left(\left(\tan y + \tan z\right) \cdot \cos a\right) \cdot \left(\left(\tan y + \tan z\right) \cdot \cos a\right) - \left(\left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right) \cdot \left(\left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)}{\left(\tan y + \tan z\right) \cdot \cos a + \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}}}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}\]

  9. Applied associate-/l/0.2

    \[\leadsto x + \color{blue}{\frac{\left(\left(\tan y + \tan z\right) \cdot \cos a\right) \cdot \left(\left(\tan y + \tan z\right) \cdot \cos a\right) - \left(\left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right) \cdot \left(\left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)}{\left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right) \cdot \left(\left(\tan y + \tan z\right) \cdot \cos a + \left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)}}\]

  10. Final simplification0.2

    \[\leadsto \frac{\left(\cos a \cdot \left(\tan y + \tan z\right)\right) \cdot \left(\cos a \cdot \left(\tan y + \tan z\right)\right) - \left(\sin a \cdot \left(1 - \tan z \cdot \tan y\right)\right) \cdot \left(\sin a \cdot \left(1 - \tan z \cdot \tan y\right)\right)}{\left(\cos a \cdot \left(\tan y + \tan z\right) + \sin a \cdot \left(1 - \tan z \cdot \tan y\right)\right) \cdot \left(\left(1 - \tan z \cdot \tan y\right) \cdot \cos a\right)} + x\]

Reproduce

herbie shell --seed 2019100 
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :pre (and (or (== x 0) (<= 0.5884142 x 505.5909)) (or (<= -1.796658e+308 y -9.425585e-310) (<= 1.284938e-309 y 1.751224e+308)) (or (<= -1.776707e+308 z -8.599796e-310) (<= 3.293145e-311 z 1.725154e+308)) (or (<= -1.796658e+308 a -9.425585e-310) (<= 1.284938e-309 a 1.751224e+308)))
  (+ x (- (tan (+ y z)) (tan a))))