Average Error: 0.0 → 0.0
Time: 7.9s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} - {v}^{6}\right)\right)}{4 \cdot \left(\left(v \cdot v\right) \cdot \left(1 + v \cdot v\right) + 1 \cdot 1\right)}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\sqrt{2} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} - {v}^{6}\right)\right)}{4 \cdot \left(\left(v \cdot v\right) \cdot \left(1 + v \cdot v\right) + 1 \cdot 1\right)}
double code(double v) {
	return ((double) (((double) ((((double) sqrt(2.0)) / 4.0) * ((double) sqrt(((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))))) * ((double) (1.0 - ((double) (v * v))))));
}
↓
double code(double v) {
	return (((double) (((double) sqrt(2.0)) * ((double) (((double) sqrt(((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))) * ((double) (((double) pow(1.0, 3.0)) - ((double) pow(v, 6.0)))))))) / ((double) (4.0 * ((double) (((double) (((double) (v * v)) * ((double) (1.0 + ((double) (v * v)))))) + ((double) (1.0 * 1.0)))))));
}

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 flip3--0.0

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

  4. Applied associate-*l/0.0

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

  5. Applied frac-times0.0

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

  6. Simplified0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020182 
(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))))