Average Error: 0.4 → 0.1
Time: 3.1m
Precision: 64
Internal Precision: 576
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{\sqrt{2 - \left(v \cdot v\right) \cdot \left(3 \cdot 2\right)}}}{t \cdot \left(1 - v \cdot v\right)}\]

Error

Bits error versus v

Bits error versus t

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Your Program's Arguments

Results

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Derivation

  1. Initial program 0.4

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

  2. Initial simplification0.4

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied times-frac0.4

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

  6. Applied associate-/l*0.3

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{t}}{\frac{\sqrt{2 \cdot \left(1 - \left(v \cdot 3\right) \cdot v\right)} \cdot \left(1 - v \cdot v\right)}{\frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{\pi}}}}\]
  7. Using strategy rm

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

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

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

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

  10. Applied times-frac0.3

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

  11. Simplified0.3

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018225 
(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)))))