Average Error: 0.0 → 0.0
Time: 19.2s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(-v \cdot v\right) + \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot 1\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(-v \cdot v\right) + \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot 1
double f(double v) {
        double r6896209 = 2.0;
        double r6896210 = sqrt(r6896209);
        double r6896211 = 4.0;
        double r6896212 = r6896210 / r6896211;
        double r6896213 = 1.0;
        double r6896214 = 3.0;
        double r6896215 = v;
        double r6896216 = r6896215 * r6896215;
        double r6896217 = r6896214 * r6896216;
        double r6896218 = r6896213 - r6896217;
        double r6896219 = sqrt(r6896218);
        double r6896220 = r6896212 * r6896219;
        double r6896221 = r6896213 - r6896216;
        double r6896222 = r6896220 * r6896221;
        return r6896222;
}
↓
double f(double v) {
        double r6896223 = 1.0;
        double r6896224 = v;
        double r6896225 = r6896224 * r6896224;
        double r6896226 = 3.0;
        double r6896227 = r6896225 * r6896226;
        double r6896228 = r6896223 - r6896227;
        double r6896229 = sqrt(r6896228);
        double r6896230 = 2.0;
        double r6896231 = sqrt(r6896230);
        double r6896232 = 4.0;
        double r6896233 = r6896231 / r6896232;
        double r6896234 = r6896229 * r6896233;
        double r6896235 = -r6896225;
        double r6896236 = r6896234 * r6896235;
        double r6896237 = r6896234 * r6896223;
        double r6896238 = r6896236 + r6896237;
        return r6896238;
}

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 sub-neg0.0

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

  4. Applied distribute-lft-in0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))