Average Error: 12.9 → 0.2
Time: 1.7m
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\frac{\left(\cos a \cdot \left(\cos z \cdot \sin y + \sin z \cdot \cos y\right)\right) \cdot \left(1 + \tan z \cdot \tan y\right) - \left(\sin a \cdot \left(1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right)\right) \cdot \left(\cos y \cdot \cos z\right)}{\left(\left(1 + \tan z \cdot \tan y\right) \cdot \left(\cos y \cdot \cos z\right)\right) \cdot \left(\left(1 - \tan z \cdot \tan y\right) \cdot \cos a\right)} + x\]

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-quot12.9

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

  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 flip--0.2

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

  8. Applied associate-*l/0.2

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

  9. Applied tan-quot0.2

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

  10. Applied tan-quot0.2

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

  11. Applied frac-add0.2

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

  12. Applied associate-*l/0.2

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

  13. Applied frac-sub0.2

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

  14. Applied associate-/l/0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2019093 +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))))