Average Error: 0.3 → 0.3
Time: 1.9m
Precision: 64
Internal Precision: 576
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\log \left(x + y\right) - (\left(\log t\right) \cdot \left(0.5 - a\right) + t)_*\right) + \log z\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Initial program 0.3

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
  2. Using strategy rm

  3. Applied associate--l+0.3

    \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right)} + \left(a - 0.5\right) \cdot \log t\]

  4. Applied associate-+l+0.3

    \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)}\]
  5. Using strategy rm

  6. Applied associate-+l-0.3

    \[\leadsto \log \left(x + y\right) + \color{blue}{\left(\log z - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)}\]

  7. Applied associate-+r-0.3

    \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)}\]

  8. Simplified0.3

    \[\leadsto \left(\log \left(x + y\right) + \log z\right) - \color{blue}{(\left(0.5 - a\right) \cdot \left(\log t\right) + t)_*}\]
  9. Using strategy rm

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

    \[\leadsto \left(\log \left(x + y\right) + \log z\right) - \color{blue}{1 \cdot (\left(0.5 - a\right) \cdot \left(\log t\right) + t)_*}\]

  11. Applied add-sqr-sqrt21.4

    \[\leadsto \color{blue}{\sqrt{\log \left(x + y\right) + \log z} \cdot \sqrt{\log \left(x + y\right) + \log z}} - 1 \cdot (\left(0.5 - a\right) \cdot \left(\log t\right) + t)_*\]

  12. Applied prod-diff21.4

    \[\leadsto \color{blue}{(\left(\sqrt{\log \left(x + y\right) + \log z}\right) \cdot \left(\sqrt{\log \left(x + y\right) + \log z}\right) + \left(-(\left(0.5 - a\right) \cdot \left(\log t\right) + t)_* \cdot 1\right))_* + (\left(-(\left(0.5 - a\right) \cdot \left(\log t\right) + t)_*\right) \cdot 1 + \left((\left(0.5 - a\right) \cdot \left(\log t\right) + t)_* \cdot 1\right))_*}\]

  13. Simplified0.3

    \[\leadsto \color{blue}{\left(\log z + \left(\log \left(y + x\right) - (\left(\log t\right) \cdot \left(0.5 - a\right) + t)_*\right)\right)} + (\left(-(\left(0.5 - a\right) \cdot \left(\log t\right) + t)_*\right) \cdot 1 + \left((\left(0.5 - a\right) \cdot \left(\log t\right) + t)_* \cdot 1\right))_*\]

  14. Simplified0.3

    \[\leadsto \left(\log z + \left(\log \left(y + x\right) - (\left(\log t\right) \cdot \left(0.5 - a\right) + t)_*\right)\right) + \color{blue}{0}\]

  15. Final simplification0.3

    \[\leadsto \left(\log \left(x + y\right) - (\left(\log t\right) \cdot \left(0.5 - a\right) + t)_*\right) + \log z\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))