Average Error: 12.9 → 0.2
Time: 39.0s
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(\frac{(\left((\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + \left(\tan y \cdot \tan z\right))_*\right) \cdot \left(\tan z + \tan y\right) + \left(\tan z + \tan y\right))_*}{1 - {\left(\tan z\right)}^{3} \cdot \left(\tan y \cdot \left(\left(\tan y \cdot \sqrt[3]{\tan y}\right) \cdot \left(\sqrt[3]{\tan y} \cdot \sqrt[3]{\tan y}\right)\right)\right)} - \tan a\right) + x\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 12.9

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 add-sqr-sqrt31.3

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

  6. Applied flip3--31.3

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

  7. Applied associate-/r/31.3

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

  8. Applied prod-diff31.3

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

  9. Simplified31.2

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

  10. Simplified0.2

    \[\leadsto x + \left(\left(\frac{(\left((\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right))_*\right) \cdot \left(\tan y + \tan z\right) + \left(\tan y + \tan z\right))_*}{1 - {\left(\tan z \cdot \tan y\right)}^{3}} - \tan a\right) + \color{blue}{0}\right)\]
  11. Using strategy rm

  12. Applied add-cbrt-cube0.2

    \[\leadsto x + \left(\left(\frac{(\left((\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right))_*\right) \cdot \left(\tan y + \tan z\right) + \left(\tan y + \tan z\right))_*}{1 - {\left(\tan z \cdot \color{blue}{\sqrt[3]{\left(\tan y \cdot \tan y\right) \cdot \tan y}}\right)}^{3}} - \tan a\right) + 0\right)\]

  13. Applied add-cbrt-cube0.2

    \[\leadsto x + \left(\left(\frac{(\left((\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right))_*\right) \cdot \left(\tan y + \tan z\right) + \left(\tan y + \tan z\right))_*}{1 - {\left(\color{blue}{\sqrt[3]{\left(\tan z \cdot \tan z\right) \cdot \tan z}} \cdot \sqrt[3]{\left(\tan y \cdot \tan y\right) \cdot \tan y}\right)}^{3}} - \tan a\right) + 0\right)\]

  14. Applied cbrt-unprod0.2

    \[\leadsto x + \left(\left(\frac{(\left((\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right))_*\right) \cdot \left(\tan y + \tan z\right) + \left(\tan y + \tan z\right))_*}{1 - {\color{blue}{\left(\sqrt[3]{\left(\left(\tan z \cdot \tan z\right) \cdot \tan z\right) \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan y\right)}\right)}}^{3}} - \tan a\right) + 0\right)\]

  15. Applied rem-cube-cbrt0.2

    \[\leadsto x + \left(\left(\frac{(\left((\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right))_*\right) \cdot \left(\tan y + \tan z\right) + \left(\tan y + \tan z\right))_*}{1 - \color{blue}{\left(\left(\tan z \cdot \tan z\right) \cdot \tan z\right) \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan y\right)}} - \tan a\right) + 0\right)\]

  16. Simplified0.2

    \[\leadsto x + \left(\left(\frac{(\left((\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right))_*\right) \cdot \left(\tan y + \tan z\right) + \left(\tan y + \tan z\right))_*}{1 - \color{blue}{{\left(\tan z\right)}^{3}} \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan y\right)} - \tan a\right) + 0\right)\]
  17. Using strategy rm

  18. Applied add-cube-cbrt0.2

    \[\leadsto x + \left(\left(\frac{(\left((\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right))_*\right) \cdot \left(\tan y + \tan z\right) + \left(\tan y + \tan z\right))_*}{1 - {\left(\tan z\right)}^{3} \cdot \left(\left(\color{blue}{\left(\left(\sqrt[3]{\tan y} \cdot \sqrt[3]{\tan y}\right) \cdot \sqrt[3]{\tan y}\right)} \cdot \tan y\right) \cdot \tan y\right)} - \tan a\right) + 0\right)\]

  19. Applied associate-*l*0.2

    \[\leadsto x + \left(\left(\frac{(\left((\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right))_*\right) \cdot \left(\tan y + \tan z\right) + \left(\tan y + \tan z\right))_*}{1 - {\left(\tan z\right)}^{3} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\tan y} \cdot \sqrt[3]{\tan y}\right) \cdot \left(\sqrt[3]{\tan y} \cdot \tan y\right)\right)} \cdot \tan y\right)} - \tan a\right) + 0\right)\]

  20. Final simplification0.2

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

Reproduce

herbie shell --seed 2019026 +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))))