Average Error: 13.0 → 0.2
Time: 59.8s
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\frac{\cos a \cdot \left(\tan y + \tan z\right) - \left(1 - \tan z \cdot \tan y\right) \cdot \sin a}{\left(1 - \left(\tan z \cdot \left(\tan z \cdot \tan z\right)\right) \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan y\right)\right) \cdot \cos a} \cdot \left((\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right))_* + 1\right) + x\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.0

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

  3. Applied tan-quot13.0

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

  4. Applied tan-sum0.2

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

  5. Applied frac-sub0.2

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

  7. Applied flip3--0.2

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

  8. Applied associate-*l/0.2

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

  9. Applied associate-/r/0.2

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

  10. Simplified0.2

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

  12. Applied add-cbrt-cube0.3

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

  13. Applied add-cbrt-cube0.3

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

  14. Applied cbrt-unprod0.3

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

  15. Applied rem-cube-cbrt0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2019089 +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))))