Average Error: 0.0 → 0.0
Time: 12.3s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\frac{\left(\sqrt{2} \cdot \left(\left(v \cdot v + 1\right) \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \sqrt{\left(1 + \left(v \cdot v\right) \cdot 3\right) \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}}{4}}{\left(v \cdot v + 1\right) \cdot \sqrt{1 + \left(v \cdot v\right) \cdot 3}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\frac{\left(\sqrt{2} \cdot \left(\left(v \cdot v + 1\right) \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \sqrt{\left(1 + \left(v \cdot v\right) \cdot 3\right) \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}}{4}}{\left(v \cdot v + 1\right) \cdot \sqrt{1 + \left(v \cdot v\right) \cdot 3}}
double f(double v) {
        double r7186316 = 2.0;
        double r7186317 = sqrt(r7186316);
        double r7186318 = 4.0;
        double r7186319 = r7186317 / r7186318;
        double r7186320 = 1.0;
        double r7186321 = 3.0;
        double r7186322 = v;
        double r7186323 = r7186322 * r7186322;
        double r7186324 = r7186321 * r7186323;
        double r7186325 = r7186320 - r7186324;
        double r7186326 = sqrt(r7186325);
        double r7186327 = r7186319 * r7186326;
        double r7186328 = r7186320 - r7186323;
        double r7186329 = r7186327 * r7186328;
        return r7186329;
}
↓
double f(double v) {
        double r7186330 = 2.0;
        double r7186331 = sqrt(r7186330);
        double r7186332 = v;
        double r7186333 = r7186332 * r7186332;
        double r7186334 = 1.0;
        double r7186335 = r7186333 + r7186334;
        double r7186336 = r7186334 - r7186333;
        double r7186337 = r7186335 * r7186336;
        double r7186338 = r7186331 * r7186337;
        double r7186339 = 3.0;
        double r7186340 = r7186333 * r7186339;
        double r7186341 = r7186334 + r7186340;
        double r7186342 = r7186334 - r7186340;
        double r7186343 = r7186341 * r7186342;
        double r7186344 = sqrt(r7186343);
        double r7186345 = r7186338 * r7186344;
        double r7186346 = 4.0;
        double r7186347 = r7186345 / r7186346;
        double r7186348 = sqrt(r7186341);
        double r7186349 = r7186335 * r7186348;
        double r7186350 = r7186347 / r7186349;
        return r7186350;
}

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

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

  4. Applied flip--0.0

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

  5. Applied sqrt-div0.0

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

  6. Applied associate-*r/0.0

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

  7. Applied frac-times0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019169 
(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))))