Average Error: 12.9 → 0.2
Time: 2.0m
Precision: 64
Internal Precision: 1344
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(\left(\tan z \cdot \tan y + 1\right) \cdot \frac{\tan y + \tan z}{1 - \frac{\left(\sin z \cdot \sin y\right) \cdot \left(\sin z \cdot \sin y\right)}{\left(\cos z \cdot \cos y\right) \cdot \left(\cos z \cdot \cos y\right)}} - \tan a\right) + x\]

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 12.9

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

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

  6. Applied associate-/r/0.2

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

  8. Applied tan-quot0.2

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

  9. Applied tan-quot0.2

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

  10. Applied frac-times0.2

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

  11. Applied tan-quot0.2

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

  12. Applied tan-quot0.2

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

  13. Applied frac-times0.2

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

  14. Applied frac-times0.2

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

  15. Final simplification0.2

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018217 
(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))))