Average Error: 0.4 → 0.3
Time: 3.7m
Precision: 64
Internal Precision: 384
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1 + -5 \cdot \left(v \cdot v\right)}{\pi}}{\sqrt{\left(v \cdot v\right) \cdot -6 + 2} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)}\]

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. Taylor expanded around 0 0.4

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

  3. Applied simplify0.3

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

  4. Applied simplify0.3

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

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(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)))))