Average Error: 0.0 → 0.0
Time: 51.7s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \sqrt[3]{\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot \left(\left(\frac{1}{32} \cdot \sqrt{1 + \left(v \cdot v\right) \cdot -3}\right) \cdot \sqrt{2}\right)}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(1 - v \cdot v\right) \cdot \sqrt[3]{\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot \left(\left(\frac{1}{32} \cdot \sqrt{1 + \left(v \cdot v\right) \cdot -3}\right) \cdot \sqrt{2}\right)}
double f(double v) {
        double r59901212 = 2.0;
        double r59901213 = sqrt(r59901212);
        double r59901214 = 4.0;
        double r59901215 = r59901213 / r59901214;
        double r59901216 = 1.0;
        double r59901217 = 3.0;
        double r59901218 = v;
        double r59901219 = r59901218 * r59901218;
        double r59901220 = r59901217 * r59901219;
        double r59901221 = r59901216 - r59901220;
        double r59901222 = sqrt(r59901221);
        double r59901223 = r59901215 * r59901222;
        double r59901224 = r59901216 - r59901219;
        double r59901225 = r59901223 * r59901224;
        return r59901225;
}
↓
double f(double v) {
        double r59901226 = 1.0;
        double r59901227 = v;
        double r59901228 = r59901227 * r59901227;
        double r59901229 = r59901226 - r59901228;
        double r59901230 = -3.0;
        double r59901231 = r59901228 * r59901230;
        double r59901232 = r59901226 + r59901231;
        double r59901233 = 0.03125;
        double r59901234 = sqrt(r59901232);
        double r59901235 = r59901233 * r59901234;
        double r59901236 = 2.0;
        double r59901237 = sqrt(r59901236);
        double r59901238 = r59901235 * r59901237;
        double r59901239 = r59901232 * r59901238;
        double r59901240 = cbrt(r59901239);
        double r59901241 = r59901229 * r59901240;
        return r59901241;
}

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-cbrt-cube0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube1.0

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

  6. Applied cbrt-undiv0.0

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

  7. Applied cbrt-unprod0.0

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

  8. Simplified0.0

    \[\leadsto \sqrt[3]{\color{blue}{\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot \left(\left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \frac{1}{32}\right) \cdot \sqrt{2}\right)}} \cdot \left(1 - v \cdot v\right)\]

  9. Final simplification0.0

    \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt[3]{\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot \left(\left(\frac{1}{32} \cdot \sqrt{1 + \left(v \cdot v\right) \cdot -3}\right) \cdot \sqrt{2}\right)}\]

Reproduce

herbie shell --seed 2019112 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))