Average Error: 0.5 → 0.7

Time: 33.3s

Precision: 64

Internal Precision: 128

\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]

↓

\[\cos^{-1} \left(-1 + 4 \cdot \left(v \cdot v + {v}^{4}\right)\right)\]

Error

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Enter valid numbers for all inputs

Initial program 0.5
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
Taylor expanded around 0 0.7
\[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{4} + 4 \cdot {v}^{2}\right) - 1\right)}\]
Simplified0.7
\[\leadsto \cos^{-1} \color{blue}{\left(-1 + \left(v \cdot v + {v}^{4}\right) \cdot 4\right)}\]
Final simplification0.7
\[\leadsto \cos^{-1} \left(-1 + 4 \cdot \left(v \cdot v + {v}^{4}\right)\right)\]

	Baseline	Herbie	Oracle	Span	%
Regimes	0.7	0.7	0.5	0.2	0%

herbie shell --seed 2018353 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))