Average Error: 0.4 → 0.5
Time: 5.5s
Precision: 64
Internal Precision: 576
\[\frac{1}{1 + \cos x}\]
\[e^{-\left(\log \left(1 + {\left(\cos x\right)}^{3}\right) - \log \left(\left(\cos x \cdot \cos x - \cos x\right) + 1\right)\right)}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\frac{1}{1 + \cos x}\]
  2. Using strategy rm
  3. Applied add-exp-log0.4

    \[\leadsto \frac{1}{\color{blue}{e^{\log \left(1 + \cos x\right)}}}\]
  4. Applied rec-exp0.4

    \[\leadsto \color{blue}{e^{-\log \left(1 + \cos x\right)}}\]
  5. Using strategy rm
  6. Applied flip3-+0.5

    \[\leadsto e^{-\log \color{blue}{\left(\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x - 1 \cdot \cos x\right)}\right)}}\]
  7. Applied log-div0.5

    \[\leadsto e^{-\color{blue}{\left(\log \left({1}^{3} + {\left(\cos x\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\cos x \cdot \cos x - 1 \cdot \cos x\right)\right)\right)}}\]
  8. Simplified0.5

    \[\leadsto e^{-\left(\color{blue}{\log \left(1 + {\left(\cos x\right)}^{3}\right)} - \log \left(1 \cdot 1 + \left(\cos x \cdot \cos x - 1 \cdot \cos x\right)\right)\right)}\]
  9. Final simplification0.5

    \[\leadsto e^{-\left(\log \left(1 + {\left(\cos x\right)}^{3}\right) - \log \left(\left(\cos x \cdot \cos x - \cos x\right) + 1\right)\right)}\]

Runtime

Time bar (total: 5.5s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "1 / (1 + cos(x))"
  (/ 1 (+ 1 (cos x))))