Average Error: 13.6 → 0.2
Time: 1.5m
Precision: 64
Internal Precision: 1344
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \left(\frac{\sqrt[3]{\frac{\left(\sin z \cdot \cos y + \cos z \cdot \sin y\right) \cdot \left(\sin z \cdot \cos y + \cos z \cdot \sin y\right)}{\frac{{\left(\cos y\right)}^{3} \cdot {\left(\cos z\right)}^{3}}{\sin z \cdot \cos y + \cos z \cdot \sin y}}}}{1 - \tan y \cdot \tan z} - \tan a\right)\]

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 tan-quot0.2

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

  6. Applied tan-quot0.2

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

  7. Applied frac-add0.2

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

  9. Applied add-cbrt-cube0.2

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

  10. Applied add-cbrt-cube0.3

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

  11. Applied cbrt-unprod0.3

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

  12. Applied add-cbrt-cube0.3

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

  13. Applied cbrt-undiv0.2

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

  14. Applied simplify0.2

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' 
(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))))