Average Error: 0.4 → 0.3
Time: 3.8m
Precision: 64
Internal Precision: 576
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{t} \cdot \frac{\frac{\frac{1}{\pi}}{1 - v \cdot v}}{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}\]

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. Applied simplify0.3

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

  4. Applied div-inv0.3

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

  5. Applied simplify0.3

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

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +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)))))