Average Error: 13.4 → 0.2
Time: 28.0s
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\frac{\left(\left(\tan y + \tan z\right) \cdot \cos a - \sin a\right) + \tan y \cdot \left(\sin a \cdot \tan z\right)}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a} + x\]

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 13.4

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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-log-exp0.2

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

  6. Applied add-log-exp0.3

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

  7. Applied sum-log0.3

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

  8. Simplified0.3

    \[\leadsto x + \left(\frac{\log \color{blue}{\left(e^{\tan y + \tan z}\right)}}{1 - \tan y \cdot \tan z} - \tan a\right)\]
  9. Using strategy rm

  10. Applied tan-quot0.3

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

  11. Applied frac-sub0.3

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019021 
(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))))