Average Error: 0.4 → 0.1
Time: 4.1m
Precision: 64
Internal Precision: 128
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{e^{\log \left(\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}\right)}}{\sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_* \cdot 2}}}{t \cdot \left(1 - v \cdot v\right)}\]

Error

Bits error versus v

Bits error versus t

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. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\sqrt{2 \cdot (\left(v \cdot v\right) \cdot -3 + 1)_*} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)}}\]
  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 add-exp-log0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019026 +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)))))