Average Error: 0.0 → 0.0
Time: 13.0s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\frac{\sqrt{2}}{4} \cdot \left(\left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right| \cdot \sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right) \cdot \left(1 - v \cdot v\right)\]
\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{2}}{4} \cdot \left(\left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right| \cdot \sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right) \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r167194 = 2.0;
        double r167195 = sqrt(r167194);
        double r167196 = 4.0;
        double r167197 = r167195 / r167196;
        double r167198 = 1.0;
        double r167199 = 3.0;
        double r167200 = v;
        double r167201 = r167200 * r167200;
        double r167202 = r167199 * r167201;
        double r167203 = r167198 - r167202;
        double r167204 = sqrt(r167203);
        double r167205 = r167197 * r167204;
        double r167206 = r167198 - r167201;
        double r167207 = r167205 * r167206;
        return r167207;
}
↓
double f(double v) {
        double r167208 = 2.0;
        double r167209 = sqrt(r167208);
        double r167210 = 4.0;
        double r167211 = r167209 / r167210;
        double r167212 = 1.0;
        double r167213 = 3.0;
        double r167214 = v;
        double r167215 = r167214 * r167214;
        double r167216 = r167213 * r167215;
        double r167217 = r167212 - r167216;
        double r167218 = cbrt(r167217);
        double r167219 = fabs(r167218);
        double r167220 = sqrt(r167218);
        double r167221 = r167219 * r167220;
        double r167222 = r167211 * r167221;
        double r167223 = r167212 - r167215;
        double r167224 = r167222 * r167223;
        return r167224;
}

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-cube-cbrt0.0

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

  4. Applied sqrt-prod0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))