Average Error: 0.5 → 0.1
Time: 3.8m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{1 - v \cdot v}}{\pi \cdot \sqrt{2}}}{\sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*} \cdot t}\]

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. Simplified0.5

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

  4. Applied associate-/r*0.4

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

  6. Applied sqrt-prod0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied times-frac0.4

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

  9. Applied times-frac0.3

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

  10. Simplified0.3

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

  12. Applied frac-times0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019100 +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)))))