Average Error: 13.5 → 0.2
Time: 1.0m
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(\frac{\tan y \cdot \sin z}{\cos z} + 1\right) \cdot \frac{\cos a \cdot \left(\tan y + \tan z\right) - \sin a \cdot \left(1 - \frac{\tan y \cdot \sin z}{\cos z}\right)}{\left(1 - \frac{\tan y \cdot \sin z}{\cos z} \cdot \frac{\tan y \cdot \sin z}{\cos z}\right) \cdot \cos a} + x\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

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Results

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Derivation

  1. Initial program 13.5

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 tan-quot0.2

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

  9. Applied frac-sub0.2

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

  11. Applied flip--0.2

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

  12. Applied associate-*l/0.2

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

  13. Applied associate-/r/0.2

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

  14. Final simplification0.2

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018336 
(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))))