Average Error: 0.0 → 0.2
Time: 4.6s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt{2} \cdot \left(\left(0.25 + {v}^{4} \cdot 0.09375\right) - \left(\left(v \cdot v\right) \cdot 0.625 + {v}^{6} \cdot 0.140625\right)\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt{2} \cdot \left(\left(0.25 + {v}^{4} \cdot 0.09375\right) - \left(\left(v \cdot v\right) \cdot 0.625 + {v}^{6} \cdot 0.140625\right)\right)
(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v)
 :precision binary64
 (*
  (sqrt 2.0)
  (-
   (+ 0.25 (* (pow v 4.0) 0.09375))
   (+ (* (* v v) 0.625) (* (pow v 6.0) 0.140625)))))
double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt(1.0 - (3.0 * (v * v)))) * (1.0 - (v * v));
}
↓
double code(double v) {
	return sqrt(2.0) * ((0.25 + (pow(v, 4.0) * 0.09375)) - (((v * v) * 0.625) + (pow(v, 6.0) * 0.140625)));
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{\left(0.09375 \cdot \left({v}^{4} \cdot \sqrt{2}\right) + 0.25 \cdot \sqrt{2}\right) - \left(0.625 \cdot \left({v}^{2} \cdot \sqrt{2}\right) + 0.140625 \cdot \left({v}^{6} \cdot \sqrt{2}\right)\right)}\]

  3. Simplified0.2

    \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\left(0.25 + {v}^{4} \cdot 0.09375\right) - \left(\left(v \cdot v\right) \cdot 0.625 + {v}^{6} \cdot 0.140625\right)\right)}\]

  4. Final simplification0.2

    \[\leadsto \sqrt{2} \cdot \left(\left(0.25 + {v}^{4} \cdot 0.09375\right) - \left(\left(v \cdot v\right) \cdot 0.625 + {v}^{6} \cdot 0.140625\right)\right)\]

Reproduce

herbie shell --seed 2021028 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))