Average Error: 1.0 → 0.0
Time: 6.3m
Precision: 64
Internal Precision: 384
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\sqrt[3]{\frac{\frac{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{2 - v \cdot \left(6 \cdot v\right)} \cdot \frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{\frac{\sqrt{2 - v \cdot \left(6 \cdot v\right)}}{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}}}\]

Error

Bits error versus v

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube0.0

    \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\right) \cdot \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}}\]

  4. Applied simplify0.0

    \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{2 - v \cdot \left(6 \cdot v\right)} \cdot \frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{\frac{\sqrt{2 - v \cdot \left(6 \cdot v\right)}}{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}}}}\]
  5. Removed slow pow expressions.

Runtime

Time bar (total: 6.3m)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))