Average Error: 13.3 → 0.2
Time: 1.8m
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \left(\frac{\tan z \cdot \sin y}{\cos y} \cdot \tan z\right)} \cdot \left(1 + \tan z \cdot \tan y\right) - \tan a\right)\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.3

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 flip--0.2

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

  6. Applied associate-/r/0.2

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

  8. Applied tan-quot0.2

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

  9. Applied associate-*l/0.2

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

  11. Applied *-commutative0.2

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

  12. Applied associate-*r*0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(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))))