Average Error: 61.6 → 61.6
Time: 58.1s
Precision: 64
Internal Precision: 128
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\log \left(e^{\frac{\frac{4}{v}}{v}}\right) + 5\right) \cdot \frac{\frac{1}{t}}{\sqrt{2} \cdot \pi}\]

Error

Bits error versus v

Bits error versus t

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 61.6

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  2. Taylor expanded around 0 61.6

    \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(t \cdot \left(\sqrt{2} \cdot \pi\right)\right)} \cdot \left(1 - v \cdot v\right)}\]

  3. Taylor expanded around -inf 61.6

    \[\leadsto \color{blue}{5 \cdot \frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + 4 \cdot \frac{1}{t \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)}}\]

  4. Simplified61.6

    \[\leadsto \color{blue}{\frac{\frac{1}{t}}{\sqrt{2} \cdot \pi} \cdot \left(\frac{\frac{4}{v}}{v} + 5\right)}\]
  5. Using strategy rm

  6. Applied add-log-exp61.6

    \[\leadsto \frac{\frac{1}{t}}{\sqrt{2} \cdot \pi} \cdot \left(\color{blue}{\log \left(e^{\frac{\frac{4}{v}}{v}}\right)} + 5\right)\]

  7. Final simplification61.6

    \[\leadsto \left(\log \left(e^{\frac{\frac{4}{v}}{v}}\right) + 5\right) \cdot \frac{\frac{1}{t}}{\sqrt{2} \cdot \pi}\]

Runtime

Time bar (total: 58.1s)Debug logProfile

herbie shell --seed 2018255 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))