Average Error: 13.2 → 0.3
Time: 2.5m
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(\sqrt[3]{\frac{\tan y + \tan z}{1 - \tan z \cdot \tan y} \cdot \left(\frac{\tan y + \tan z}{\sqrt[3]{1 - \tan z \cdot \tan y}} \cdot \left(\frac{\tan y + \tan z}{1 - \tan z \cdot \tan y} \cdot \frac{1}{\sqrt[3]{1 - \tan z \cdot \tan y} \cdot \sqrt[3]{1 - \tan z \cdot \tan y}}\right)\right)} - \tan a\right) + x\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

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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. Using strategy rm

  5. Applied add-cbrt-cube0.2

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

  7. Applied add-cube-cbrt0.3

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

  8. Applied *-un-lft-identity0.3

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

  9. Applied times-frac0.3

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

  10. Applied associate-*r*0.3

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

  11. Final simplification0.3

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018348 +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))))