Average Error: 13.3 → 0.3
Time: 1.5m
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \sqrt[3]{\left(\frac{\tan y + \tan z}{1 - \frac{\left(\sqrt[3]{\tan y} \cdot \sqrt[3]{\tan y}\right) \cdot \left(\sqrt[3]{\tan y} \cdot \sin z\right)}{\cos z}} - \tan a\right) \cdot \left(\left(\frac{\tan y + \tan z}{1 - \frac{\sin z \cdot \tan y}{\cos z}} - \tan a\right) \cdot \left(\frac{\tan y + \tan z}{1 - \frac{\sin z \cdot \tan y}{\cos z}} - \tan a\right)\right)}\]

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 tan-quot0.2

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

  6. Applied associate-*r/0.2

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

  8. Applied add-cbrt-cube0.3

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

  10. Applied add-cube-cbrt0.3

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

  11. Applied associate-*l*0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019053 
(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))))