Average Error: 0.0 → 0.0
Time: 10.1s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{4}} \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{\sqrt{4}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right) \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{4}} \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{\sqrt{4}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right) \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r130242 = 2.0;
        double r130243 = sqrt(r130242);
        double r130244 = 4.0;
        double r130245 = r130243 / r130244;
        double r130246 = 1.0;
        double r130247 = 3.0;
        double r130248 = v;
        double r130249 = r130248 * r130248;
        double r130250 = r130247 * r130249;
        double r130251 = r130246 - r130250;
        double r130252 = sqrt(r130251);
        double r130253 = r130245 * r130252;
        double r130254 = r130246 - r130249;
        double r130255 = r130253 * r130254;
        return r130255;
}
↓
double f(double v) {
        double r130256 = 2.0;
        double r130257 = sqrt(r130256);
        double r130258 = cbrt(r130257);
        double r130259 = r130258 * r130258;
        double r130260 = 4.0;
        double r130261 = sqrt(r130260);
        double r130262 = r130259 / r130261;
        double r130263 = r130258 / r130261;
        double r130264 = 1.0;
        double r130265 = 3.0;
        double r130266 = v;
        double r130267 = r130266 * r130266;
        double r130268 = r130265 * r130267;
        double r130269 = r130264 - r130268;
        double r130270 = sqrt(r130269);
        double r130271 = r130263 * r130270;
        double r130272 = r130262 * r130271;
        double r130273 = r130264 - r130267;
        double r130274 = r130272 * r130273;
        return r130274;
}

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 add-sqr-sqrt0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied times-frac0.0

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

  6. Applied associate-*l*0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +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))))