Average Error: 0.0 → 0.0

Time: 34.6s

Precision: 64

Internal Precision: 320

\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]

↓

\[e^{\log \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3}\right) + \log \left(\frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}}\right)}\]

Error

The X axis uses an exponential scale — Bits error versus `v`

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Enter valid numbers for all inputs

Derivation

Initial program 0.0
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
Initial simplification0.0
\[\leadsto \frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}} \cdot \sqrt{-3 \cdot \left(v \cdot v\right) + 1}\]
Using strategy rm
Applied add-exp-log0.0
\[\leadsto \frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}} \cdot \color{blue}{e^{\log \left(\sqrt{-3 \cdot \left(v \cdot v\right) + 1}\right)}}\]
Applied add-exp-log0.0
\[\leadsto \color{blue}{e^{\log \left(\frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}}\right)}} \cdot e^{\log \left(\sqrt{-3 \cdot \left(v \cdot v\right) + 1}\right)}\]
Applied prod-exp0.0
\[\leadsto \color{blue}{e^{\log \left(\frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}}\right) + \log \left(\sqrt{-3 \cdot \left(v \cdot v\right) + 1}\right)}}\]
Final simplification0.0
\[\leadsto e^{\log \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3}\right) + \log \left(\frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}}\right)}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.0	0.0	0.0	0.0	0%

herbie shell --seed 2018286 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))