Average Error: 0.3 → 0.3
Time: 57.6s
Precision: 64
Internal Precision: 128
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log t \cdot \left(a - 0.5\right) + \left(\log \left(\sqrt{y + x}\right) + \left(\left(\log z - t\right) + \log \left(\sqrt{y + x}\right)\right)\right)\]

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. Using strategy rm

  5. Applied add-sqr-sqrt0.3

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

  6. Applied log-prod0.3

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

  7. Applied associate-+l+0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019051 
(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))))