Average Error: 13.3 → 0.2
Time: 55.7s
Precision: 64
Internal Precision: 1344
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\frac{(\left(\frac{\tan z \cdot \tan y + -1}{\tan y + \tan z}\right) \cdot \left(\sin a\right) + \left(\cos a\right))_*}{\cos a \cdot \frac{1 - \tan z \cdot \tan y}{\tan y + \tan z}} + x\]

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-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 *-un-lft-identity0.2

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

  6. Applied *-un-lft-identity0.2

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

  7. Applied distribute-lft-out0.2

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

  8. Applied associate-/l*0.2

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

  10. Applied tan-quot0.2

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

  11. Applied frac-sub0.2

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Runtime

Time bar (total: 55.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.20.20.10.20%
herbie shell --seed 2018296 +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))))