Average Error: 0.0 → 0.0
Time: 14.6s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right) \cdot \left(1 - v \cdot v\right)}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)\right)}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right) \cdot \left(1 - v \cdot v\right)}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)\right)}}
double f(double v) {
        double r167433 = 2.0;
        double r167434 = sqrt(r167433);
        double r167435 = 4.0;
        double r167436 = r167434 / r167435;
        double r167437 = 1.0;
        double r167438 = 3.0;
        double r167439 = v;
        double r167440 = r167439 * r167439;
        double r167441 = r167438 * r167440;
        double r167442 = r167437 - r167441;
        double r167443 = sqrt(r167442);
        double r167444 = r167436 * r167443;
        double r167445 = r167437 - r167440;
        double r167446 = r167444 * r167445;
        return r167446;
}
↓
double f(double v) {
        double r167447 = 2.0;
        double r167448 = sqrt(r167447);
        double r167449 = 4.0;
        double r167450 = r167448 / r167449;
        double r167451 = 1.0;
        double r167452 = 3.0;
        double r167453 = pow(r167451, r167452);
        double r167454 = 3.0;
        double r167455 = v;
        double r167456 = r167455 * r167455;
        double r167457 = r167454 * r167456;
        double r167458 = pow(r167457, r167452);
        double r167459 = r167453 - r167458;
        double r167460 = sqrt(r167459);
        double r167461 = r167450 * r167460;
        double r167462 = r167451 - r167456;
        double r167463 = r167461 * r167462;
        double r167464 = r167451 * r167451;
        double r167465 = r167457 * r167457;
        double r167466 = exp(r167457);
        double r167467 = log(r167466);
        double r167468 = r167451 * r167467;
        double r167469 = r167465 + r167468;
        double r167470 = r167464 + r167469;
        double r167471 = sqrt(r167470);
        double r167472 = r167463 / r167471;
        return r167472;
}

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 flip3--0.0

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

  4. Applied sqrt-div0.0

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

  5. Applied associate-*r/0.0

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

  6. Applied associate-*l/0.0

    \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right) \cdot \left(1 - v \cdot v\right)}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\]
  7. Using strategy rm

  8. Applied add-log-exp0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019350 +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))))