Average Error: 0.0 → 0.0
Time: 4.7s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(\sqrt{2} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right) \cdot \left(1 - v \cdot v\right)}{4 \cdot \sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left(\sqrt{2} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right) \cdot \left(1 - v \cdot v\right)}{4 \cdot \sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}
double f(double v) {
        double r226153 = 2.0;
        double r226154 = sqrt(r226153);
        double r226155 = 4.0;
        double r226156 = r226154 / r226155;
        double r226157 = 1.0;
        double r226158 = 3.0;
        double r226159 = v;
        double r226160 = r226159 * r226159;
        double r226161 = r226158 * r226160;
        double r226162 = r226157 - r226161;
        double r226163 = sqrt(r226162);
        double r226164 = r226156 * r226163;
        double r226165 = r226157 - r226160;
        double r226166 = r226164 * r226165;
        return r226166;
}
↓
double f(double v) {
        double r226167 = 2.0;
        double r226168 = sqrt(r226167);
        double r226169 = 1.0;
        double r226170 = 3.0;
        double r226171 = pow(r226169, r226170);
        double r226172 = 3.0;
        double r226173 = v;
        double r226174 = r226173 * r226173;
        double r226175 = r226172 * r226174;
        double r226176 = pow(r226175, r226170);
        double r226177 = r226171 - r226176;
        double r226178 = sqrt(r226177);
        double r226179 = r226168 * r226178;
        double r226180 = r226169 - r226174;
        double r226181 = r226179 * r226180;
        double r226182 = 4.0;
        double r226183 = r226169 * r226169;
        double r226184 = r226175 * r226175;
        double r226185 = r226169 * r226175;
        double r226186 = r226184 + r226185;
        double r226187 = r226183 + r226186;
        double r226188 = sqrt(r226187);
        double r226189 = r226182 * r226188;
        double r226190 = r226181 / r226189;
        return r226190;
}

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

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

  4. Applied sqrt-div0.0

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

  5. Applied frac-times0.0

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

  6. Applied associate-*l/0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019353 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))