Average Error: 0.0 → 0.0
Time: 18.7s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}}{4 \cdot \sqrt{1 \cdot 1 + \left(3 \cdot {v}^{2}\right) \cdot \left(1 + 3 \cdot \left(v \cdot v\right)\right)}} \cdot \left(1 - v \cdot v\right)\]
\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 \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}}{4 \cdot \sqrt{1 \cdot 1 + \left(3 \cdot {v}^{2}\right) \cdot \left(1 + 3 \cdot \left(v \cdot v\right)\right)}} \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r120008 = 2.0;
        double r120009 = sqrt(r120008);
        double r120010 = 4.0;
        double r120011 = r120009 / r120010;
        double r120012 = 1.0;
        double r120013 = 3.0;
        double r120014 = v;
        double r120015 = r120014 * r120014;
        double r120016 = r120013 * r120015;
        double r120017 = r120012 - r120016;
        double r120018 = sqrt(r120017);
        double r120019 = r120011 * r120018;
        double r120020 = r120012 - r120015;
        double r120021 = r120019 * r120020;
        return r120021;
}
↓
double f(double v) {
        double r120022 = 2.0;
        double r120023 = sqrt(r120022);
        double r120024 = 1.0;
        double r120025 = 3.0;
        double r120026 = pow(r120024, r120025);
        double r120027 = 3.0;
        double r120028 = v;
        double r120029 = r120028 * r120028;
        double r120030 = r120027 * r120029;
        double r120031 = pow(r120030, r120025);
        double r120032 = r120026 - r120031;
        double r120033 = sqrt(r120032);
        double r120034 = r120023 * r120033;
        double r120035 = 4.0;
        double r120036 = r120024 * r120024;
        double r120037 = 2.0;
        double r120038 = pow(r120028, r120037);
        double r120039 = r120027 * r120038;
        double r120040 = r120024 + r120030;
        double r120041 = r120039 * r120040;
        double r120042 = r120036 + r120041;
        double r120043 = sqrt(r120042);
        double r120044 = r120035 * r120043;
        double r120045 = r120034 / r120044;
        double r120046 = r120024 - r120029;
        double r120047 = r120045 * r120046;
        return r120047;
}

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. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))