Average Error: 12.9 → 0.3
Time: 2.1m
Precision: 64
Internal Precision: 1344
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \left(\left(\frac{\frac{-\left(\tan z + \tan y\right)}{\sqrt[3]{(\left(\tan z\right) \cdot \left(\tan y\right) + \left(-1\right))_*}}}{\sqrt[3]{(\left(\tan z\right) \cdot \left(\tan y\right) + \left(-1\right))_*} \cdot \sqrt[3]{(\left(\tan z\right) \cdot \left(\tan y\right) + \left(-1\right))_*}} - \tan a\right) + 0\right)\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 12.9

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 frac-2neg0.2

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

  6. Applied simplify0.2

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

  8. Applied add-cube-cbrt0.3

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

  9. Applied add-cube-cbrt0.4

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

  10. Applied add-sqr-sqrt31.2

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

  11. Applied times-frac31.2

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

  12. Applied prod-diff31.2

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

  13. Applied simplify0.3

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

  14. Applied simplify0.3

    \[\leadsto x + \left(\left(\frac{\frac{-\left(\tan z + \tan y\right)}{\sqrt[3]{(\left(\tan z\right) \cdot \left(\tan y\right) + \left(-1\right))_*}}}{\sqrt[3]{(\left(\tan z\right) \cdot \left(\tan y\right) + \left(-1\right))_*} \cdot \sqrt[3]{(\left(\tan z\right) \cdot \left(\tan y\right) + \left(-1\right))_*}} - \tan a\right) + \color{blue}{0}\right)\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' +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))))