Average Error: 13.1 → 0.2
Time: 44.9s
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left((\left(\tan y + \tan z\right) \cdot \left(\frac{1}{1 - \log \left(e^{\tan z \cdot \tan y}\right)}\right) + \left(-\tan a\right))_* + x\right) + (-1 \cdot \left(\tan a\right) + \left(\tan a\right))_*\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.1

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

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

  7. Applied *-un-lft-identity0.2

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

  8. Applied div-inv0.2

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

  9. Applied prod-diff0.2

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

  10. Applied associate-+r+0.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019030 +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))))