Average Error: 13.3 → 0.2
Time: 1.1m
Precision: 64
Internal Precision: 1408
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \left(\tan y \cdot \left(\left(\sqrt[3]{\tan z} \cdot \left(\sqrt[3]{\sqrt[3]{\tan z}} \cdot \sqrt[3]{\sqrt[3]{\tan z}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\tan z}}\right)\right) \cdot \sqrt[3]{\tan z}\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.3

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

  3. Applied tan-quot13.3

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

  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. Using strategy rm

  7. Applied add-cube-cbrt0.2

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

  8. Applied associate-*r*0.2

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

  10. Applied add-cube-cbrt0.2

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

  11. Applied associate-*r*0.2

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' 
(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))))