Average Error: 0.0 → 0.0
Time: 15.1s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(\frac{\sqrt{2}}{\sqrt{4}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(v \cdot v, v \cdot v, 1 \cdot \left(v \cdot v\right)\right)\right) \cdot \sqrt{4}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left(\frac{\sqrt{2}}{\sqrt{4}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(v \cdot v, v \cdot v, 1 \cdot \left(v \cdot v\right)\right)\right) \cdot \sqrt{4}}
double code(double v) {
	return ((double) (((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) (((double) (((double) (((double) sqrt(2.0)) / ((double) sqrt(4.0)))) * ((double) sqrt(((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))))) * ((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) (v * v)), 3.0)))))) / ((double) (((double) fma(1.0, 1.0, ((double) fma(((double) (v * v)), ((double) (v * v)), ((double) (1.0 * ((double) (v * v)))))))) * ((double) sqrt(4.0))))));
}

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 add-sqr-sqrt0.0

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

  5. Applied associate-/r*0.0

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

  6. Applied associate-*l/0.0

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

  7. Applied frac-times0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))