Average Error: 13.0 → 0.3
Time: 42.3s
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(\frac{e^{\log \left(\tan y \cdot \tan y\right)} - \tan z \cdot \tan z}{\left(\tan y - \tan z\right) \cdot \left(1 - \tan z \cdot \tan y\right)} - \tan a\right) + x\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.0

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 flip-+0.2

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

  6. Applied associate-/l/0.2

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

  8. Applied add-exp-log0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019089 
(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))))