Average Error: 13.3 → 0.3
Time: 2.3m
Precision: 64
Internal Precision: 1408
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\frac{\tan y + \tan z}{1 - \frac{\sin z}{\cos y} \cdot \frac{\sin y}{\cos z}} + \left(x - \tan a\right)\]

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 add-cbrt-cube0.2

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

  6. Applied add-cbrt-cube0.3

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

  7. Applied cbrt-undiv0.2

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

  8. Applied simplify0.2

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

  9. Taylor expanded around inf 0.3

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

  10. Applied simplify0.3

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

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +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))))