Average Error: 0.0 → 0.0
Time: 4.3s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[0.25 \cdot \left(\left(\sqrt{2} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
0.25 \cdot \left(\left(\sqrt{2} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}\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
 (* 0.25 (* (* (sqrt 2.0) (- 1.0 (* v v))) (sqrt (- 1.0 (* (* v v) 3.0))))))
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 0.25 * ((sqrt(2.0) * (1.0 - (v * v))) * sqrt(1.0 - ((v * v) * 3.0)));
}

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. Using strategy rm

  3. Applied add-log-exp_binary64_18220.0

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

  4. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\log \left(e^{0.25 \cdot \left(\left(\left(1 - {v}^{2}\right) \cdot \sqrt{2}\right) \cdot \sqrt{1 - 3 \cdot {v}^{2}}\right)}\right)}\]

  5. Simplified0.0

    \[\leadsto \color{blue}{0.25 \cdot \left(\left(\sqrt{2} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}\]

  6. Final simplification0.0

    \[\leadsto 0.25 \cdot \left(\left(\sqrt{2} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}\right)\]

Reproduce

herbie shell --seed 2021045 
(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))))