Average Error: 0.0 → 0.0
Time: 4.7s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(\sqrt{2} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}{4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left(\sqrt{2} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}{4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r233132 = 2.0;
        double r233133 = sqrt(r233132);
        double r233134 = 4.0;
        double r233135 = r233133 / r233134;
        double r233136 = 1.0;
        double r233137 = 3.0;
        double r233138 = v;
        double r233139 = r233138 * r233138;
        double r233140 = r233137 * r233139;
        double r233141 = r233136 - r233140;
        double r233142 = sqrt(r233141);
        double r233143 = r233135 * r233142;
        double r233144 = r233136 - r233139;
        double r233145 = r233143 * r233144;
        return r233145;
}
↓
double f(double v) {
        double r233146 = 2.0;
        double r233147 = sqrt(r233146);
        double r233148 = 1.0;
        double r233149 = r233148 * r233148;
        double r233150 = 3.0;
        double r233151 = v;
        double r233152 = r233151 * r233151;
        double r233153 = r233150 * r233152;
        double r233154 = r233153 * r233153;
        double r233155 = r233149 - r233154;
        double r233156 = sqrt(r233155);
        double r233157 = r233147 * r233156;
        double r233158 = r233148 - r233152;
        double r233159 = r233157 * r233158;
        double r233160 = 4.0;
        double r233161 = r233148 + r233153;
        double r233162 = sqrt(r233161);
        double r233163 = r233160 * r233162;
        double r233164 = r233159 / r233163;
        return r233164;
}

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 flip--0.0

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

  4. Applied sqrt-div0.0

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

  5. Applied frac-times0.0

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

  6. Applied associate-*l/0.0

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

  7. Final simplification0.0

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

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))))