Average Error: 0.5 → 0.3
Time: 2.3m
Precision: 64
Internal Precision: 128
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{(v \cdot \left(v \cdot -5\right) + 1)_*}{t} \cdot \frac{\frac{\frac{1}{\pi}}{1 - v \cdot v}}{\sqrt{(\left(v \cdot v\right) \cdot -6 + 2)_*}}\]

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 0.5

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

  2. Initial simplification0.3

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

  4. Applied associate-/r*0.1

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

  6. Applied *-un-lft-identity0.1

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

  7. Applied div-inv0.1

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

  8. Applied times-frac0.1

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

  9. Applied times-frac0.3

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

  10. Simplified0.3

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

  11. Simplified0.3

    \[\leadsto \frac{(v \cdot \left(v \cdot -5\right) + 1)_*}{t} \cdot \color{blue}{\frac{\frac{\frac{1}{\pi}}{1 - v \cdot v}}{\sqrt{(\left(v \cdot v\right) \cdot -6 + 2)_*}}}\]

  12. Final simplification0.3

    \[\leadsto \frac{(v \cdot \left(v \cdot -5\right) + 1)_*}{t} \cdot \frac{\frac{\frac{1}{\pi}}{1 - v \cdot v}}{\sqrt{(\left(v \cdot v\right) \cdot -6 + 2)_*}}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018349 +o rules:numerics
(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)))))