Average Error: 13.2 → 0.2
Time: 6.1m
Precision: 64
Internal Precision: 1344
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(\frac{1}{1 - \tan y \cdot \tan z} \cdot \tan y + \left(\frac{\tan z}{1 - \tan y \cdot \tan z} + \left(-\tan a\right)\right)\right) + x\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

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Your Program's Arguments

Results

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Derivation

  1. Initial program 13.2

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

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Runtime

Time bar (total: 6.1m)Debug logProfile

herbie shell --seed 2018214 +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))))