Average Error: 13.1 → 0.2
Time: 41.4s
Precision: 64
Internal Precision: 1344
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(\left(\frac{\sin y}{\left(1 - \frac{\sin z \cdot \sin y}{\cos z \cdot \cos y}\right) \cdot \cos y} + \frac{\sin z}{\cos z \cdot \left(1 - \sqrt[3]{\frac{{\left(\sin y\right)}^{3} \cdot {\left(\sin z\right)}^{3}}{{\left(\cos z \cdot \cos y\right)}^{3}}}\right)}\right) - \frac{\sin a}{\cos a}\right) + x\]

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 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. Taylor expanded around -inf 0.2

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

  6. Applied add-cbrt-cube0.2

    \[\leadsto x + \left(\left(\frac{\sin z}{\cos z \cdot \left(1 - \frac{\sin z \cdot \sin y}{\color{blue}{\sqrt[3]{\left(\left(\cos y \cdot \cos z\right) \cdot \left(\cos y \cdot \cos z\right)\right) \cdot \left(\cos y \cdot \cos z\right)}}}\right)} + \frac{\sin y}{\cos y \cdot \left(1 - \frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)}\right) - \frac{\sin a}{\cos a}\right)\]

  7. Applied add-cbrt-cube0.2

    \[\leadsto x + \left(\left(\frac{\sin z}{\cos z \cdot \left(1 - \frac{\sin z \cdot \color{blue}{\sqrt[3]{\left(\sin y \cdot \sin y\right) \cdot \sin y}}}{\sqrt[3]{\left(\left(\cos y \cdot \cos z\right) \cdot \left(\cos y \cdot \cos z\right)\right) \cdot \left(\cos y \cdot \cos z\right)}}\right)} + \frac{\sin y}{\cos y \cdot \left(1 - \frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)}\right) - \frac{\sin a}{\cos a}\right)\]

  8. Applied add-cbrt-cube0.2

    \[\leadsto x + \left(\left(\frac{\sin z}{\cos z \cdot \left(1 - \frac{\color{blue}{\sqrt[3]{\left(\sin z \cdot \sin z\right) \cdot \sin z}} \cdot \sqrt[3]{\left(\sin y \cdot \sin y\right) \cdot \sin y}}{\sqrt[3]{\left(\left(\cos y \cdot \cos z\right) \cdot \left(\cos y \cdot \cos z\right)\right) \cdot \left(\cos y \cdot \cos z\right)}}\right)} + \frac{\sin y}{\cos y \cdot \left(1 - \frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)}\right) - \frac{\sin a}{\cos a}\right)\]

  9. Applied cbrt-unprod0.2

    \[\leadsto x + \left(\left(\frac{\sin z}{\cos z \cdot \left(1 - \frac{\color{blue}{\sqrt[3]{\left(\left(\sin z \cdot \sin z\right) \cdot \sin z\right) \cdot \left(\left(\sin y \cdot \sin y\right) \cdot \sin y\right)}}}{\sqrt[3]{\left(\left(\cos y \cdot \cos z\right) \cdot \left(\cos y \cdot \cos z\right)\right) \cdot \left(\cos y \cdot \cos z\right)}}\right)} + \frac{\sin y}{\cos y \cdot \left(1 - \frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)}\right) - \frac{\sin a}{\cos a}\right)\]

  10. Applied cbrt-undiv0.2

    \[\leadsto x + \left(\left(\frac{\sin z}{\cos z \cdot \left(1 - \color{blue}{\sqrt[3]{\frac{\left(\left(\sin z \cdot \sin z\right) \cdot \sin z\right) \cdot \left(\left(\sin y \cdot \sin y\right) \cdot \sin y\right)}{\left(\left(\cos y \cdot \cos z\right) \cdot \left(\cos y \cdot \cos z\right)\right) \cdot \left(\cos y \cdot \cos z\right)}}}\right)} + \frac{\sin y}{\cos y \cdot \left(1 - \frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)}\right) - \frac{\sin a}{\cos a}\right)\]

  11. Simplified0.2

    \[\leadsto x + \left(\left(\frac{\sin z}{\cos z \cdot \left(1 - \sqrt[3]{\color{blue}{\frac{{\left(\sin z\right)}^{3} \cdot {\left(\sin y\right)}^{3}}{{\left(\cos y \cdot \cos z\right)}^{3}}}}\right)} + \frac{\sin y}{\cos y \cdot \left(1 - \frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)}\right) - \frac{\sin a}{\cos a}\right)\]

  12. Final simplification0.2

    \[\leadsto \left(\left(\frac{\sin y}{\left(1 - \frac{\sin z \cdot \sin y}{\cos z \cdot \cos y}\right) \cdot \cos y} + \frac{\sin z}{\cos z \cdot \left(1 - \sqrt[3]{\frac{{\left(\sin y\right)}^{3} \cdot {\left(\sin z\right)}^{3}}{{\left(\cos z \cdot \cos y\right)}^{3}}}\right)}\right) - \frac{\sin a}{\cos a}\right) + x\]

Runtime

Time bar (total: 41.4s)Debug logProfile

herbie shell --seed 2018255 
(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))))