Average Error: 0.0 → 0.0
Time: 16.1s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left({1}^{3} - {v}^{6}\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right)}{\left(1 \cdot 1 + \left({v}^{4} + 1 \cdot \left(v \cdot v\right)\right)\right) \cdot \sqrt{1 \cdot 1 + \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right) + 1\right)}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left({1}^{3} - {v}^{6}\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right)}{\left(1 \cdot 1 + \left({v}^{4} + 1 \cdot \left(v \cdot v\right)\right)\right) \cdot \sqrt{1 \cdot 1 + \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right) + 1\right)}}
double f(double v) {
        double r128101 = 2.0;
        double r128102 = sqrt(r128101);
        double r128103 = 4.0;
        double r128104 = r128102 / r128103;
        double r128105 = 1.0;
        double r128106 = 3.0;
        double r128107 = v;
        double r128108 = r128107 * r128107;
        double r128109 = r128106 * r128108;
        double r128110 = r128105 - r128109;
        double r128111 = sqrt(r128110);
        double r128112 = r128104 * r128111;
        double r128113 = r128105 - r128108;
        double r128114 = r128112 * r128113;
        return r128114;
}
↓
double f(double v) {
        double r128115 = 1.0;
        double r128116 = 3.0;
        double r128117 = pow(r128115, r128116);
        double r128118 = v;
        double r128119 = 6.0;
        double r128120 = pow(r128118, r128119);
        double r128121 = r128117 - r128120;
        double r128122 = 2.0;
        double r128123 = sqrt(r128122);
        double r128124 = 4.0;
        double r128125 = r128123 / r128124;
        double r128126 = 3.0;
        double r128127 = r128118 * r128118;
        double r128128 = r128126 * r128127;
        double r128129 = pow(r128128, r128116);
        double r128130 = r128117 - r128129;
        double r128131 = sqrt(r128130);
        double r128132 = r128125 * r128131;
        double r128133 = r128121 * r128132;
        double r128134 = r128115 * r128115;
        double r128135 = 4.0;
        double r128136 = pow(r128118, r128135);
        double r128137 = r128115 * r128127;
        double r128138 = r128136 + r128137;
        double r128139 = r128134 + r128138;
        double r128140 = r128128 + r128115;
        double r128141 = r128128 * r128140;
        double r128142 = r128134 + r128141;
        double r128143 = sqrt(r128142);
        double r128144 = r128139 * r128143;
        double r128145 = r128133 / r128144;
        return r128145;
}

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 flip3--0.0

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

  5. Applied sqrt-div0.0

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

  6. Applied associate-*r/0.0

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

  7. Applied frac-times0.0

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

  8. Simplified0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019212 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))