Average Error: 13.6 → 0.2
Time: 1.2m
Precision: 64
Internal Precision: 1344
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \left((\left(\sqrt[3]{\left(\tan z \cdot \tan y\right) \cdot \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right)}\right) \cdot \left((\left(\tan y\right) \cdot \left(\tan z\right) + 1)_*\right) + 1)_* \cdot \frac{\tan y + \tan z}{1 - {\left(\frac{\sin z \cdot \sin y}{\cos z \cdot \cos y}\right)}^{3}} - \tan a\right)\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.6

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 flip3--0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\frac{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}}{1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)}}} - \tan a\right)\]

  6. Applied associate-/r/0.2

    \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)\right)} - \tan a\right)\]

  7. Simplified0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \color{blue}{(\left(\tan z \cdot \tan y\right) \cdot \left((\left(\tan y\right) \cdot \left(\tan z\right) + 1)_*\right) + 1)_*} - \tan a\right)\]

  8. Taylor expanded around -inf 0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{{1}^{3} - {\color{blue}{\left(\frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)}}^{3}} \cdot (\left(\tan z \cdot \tan y\right) \cdot \left((\left(\tan y\right) \cdot \left(\tan z\right) + 1)_*\right) + 1)_* - \tan a\right)\]
  9. Using strategy rm

  10. Applied add-cbrt-cube0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)}^{3}} \cdot (\color{blue}{\left(\sqrt[3]{\left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right) \cdot \left(\tan z \cdot \tan y\right)}\right)} \cdot \left((\left(\tan y\right) \cdot \left(\tan z\right) + 1)_*\right) + 1)_* - \tan a\right)\]

  11. Final simplification0.2

    \[\leadsto x + \left((\left(\sqrt[3]{\left(\tan z \cdot \tan y\right) \cdot \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right)}\right) \cdot \left((\left(\tan y\right) \cdot \left(\tan z\right) + 1)_*\right) + 1)_* \cdot \frac{\tan y + \tan z}{1 - {\left(\frac{\sin z \cdot \sin y}{\cos z \cdot \cos y}\right)}^{3}} - \tan a\right)\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018251 +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))))