Average Error: 13.2 → 0.3
Time: 3.5m
Precision: 64
Internal Precision: 1344
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \sqrt[3]{\frac{(\left(\tan y + \tan z\right) \cdot \left(\cos a\right) + \left((\left(\tan y\right) \cdot \left(\tan z\right) + -1)_* \cdot \sin a\right))_*}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a} \cdot \frac{{\left(\left(\left(\frac{\cos a \cdot \sin y}{\cos y} + \frac{\left(\sin z \cdot \sin y\right) \cdot \sin a}{\cos y \cdot \cos z}\right) + \frac{\sin z \cdot \cos a}{\cos z}\right) - \sin a\right)}^{2}}{{\left(1 - \frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)}^{2} \cdot {\left(\cos a\right)}^{2}}}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.2

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  2. Using strategy rm

  3. Applied tan-quot13.2

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

  4. Applied tan-sum0.2

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

  5. Applied frac-sub0.2

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

  6. Simplified0.2

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

  8. Applied add-cbrt-cube0.3

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

  9. Taylor expanded around inf 0.3

    \[\leadsto x + \sqrt[3]{\color{blue}{\frac{{\left(\left(\frac{\cos a \cdot \sin z}{\cos z} + \left(\frac{\sin a \cdot \left(\sin z \cdot \sin y\right)}{\cos y \cdot \cos z} + \frac{\cos a \cdot \sin y}{\cos y}\right)\right) - \sin a\right)}^{2}}{{\left(\cos a\right)}^{2} \cdot {\left(1 - \frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)}^{2}}} \cdot \frac{(\left(\tan y + \tan z\right) \cdot \left(\cos a\right) + \left((\left(\tan y\right) \cdot \left(\tan z\right) + -1)_* \cdot \sin a\right))_*}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}}\]

  10. Final simplification0.3

    \[\leadsto x + \sqrt[3]{\frac{(\left(\tan y + \tan z\right) \cdot \left(\cos a\right) + \left((\left(\tan y\right) \cdot \left(\tan z\right) + -1)_* \cdot \sin a\right))_*}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a} \cdot \frac{{\left(\left(\left(\frac{\cos a \cdot \sin y}{\cos y} + \frac{\left(\sin z \cdot \sin y\right) \cdot \sin a}{\cos y \cdot \cos z}\right) + \frac{\sin z \cdot \cos a}{\cos z}\right) - \sin a\right)}^{2}}{{\left(1 - \frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)}^{2} \cdot {\left(\cos a\right)}^{2}}}\]

Runtime

Time bar (total: 3.5m)Debug logProfile

herbie shell --seed 2018273 +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))))