Average Error: 0.0 → 0.2
Time: 5.1s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1} - \left(1.5 \cdot \frac{{v}^{2}}{\sqrt{1}} + 1.125 \cdot \frac{{v}^{4}}{{\left(\sqrt{1}\right)}^{3}}\right)\right)\right) \cdot \left(1 - v \cdot v\right)\]
\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(\sqrt{1} - \left(1.5 \cdot \frac{{v}^{2}}{\sqrt{1}} + 1.125 \cdot \frac{{v}^{4}}{{\left(\sqrt{1}\right)}^{3}}\right)\right)\right) \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r242543 = 2.0;
        double r242544 = sqrt(r242543);
        double r242545 = 4.0;
        double r242546 = r242544 / r242545;
        double r242547 = 1.0;
        double r242548 = 3.0;
        double r242549 = v;
        double r242550 = r242549 * r242549;
        double r242551 = r242548 * r242550;
        double r242552 = r242547 - r242551;
        double r242553 = sqrt(r242552);
        double r242554 = r242546 * r242553;
        double r242555 = r242547 - r242550;
        double r242556 = r242554 * r242555;
        return r242556;
}
↓
double f(double v) {
        double r242557 = 2.0;
        double r242558 = sqrt(r242557);
        double r242559 = 4.0;
        double r242560 = r242558 / r242559;
        double r242561 = 1.0;
        double r242562 = sqrt(r242561);
        double r242563 = 1.5;
        double r242564 = v;
        double r242565 = 2.0;
        double r242566 = pow(r242564, r242565);
        double r242567 = r242566 / r242562;
        double r242568 = r242563 * r242567;
        double r242569 = 1.125;
        double r242570 = 4.0;
        double r242571 = pow(r242564, r242570);
        double r242572 = 3.0;
        double r242573 = pow(r242562, r242572);
        double r242574 = r242571 / r242573;
        double r242575 = r242569 * r242574;
        double r242576 = r242568 + r242575;
        double r242577 = r242562 - r242576;
        double r242578 = r242560 * r242577;
        double r242579 = r242564 * r242564;
        double r242580 = r242561 - r242579;
        double r242581 = r242578 * r242580;
        return r242581;
}

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. Taylor expanded around 0 0.2

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\left(\sqrt{1} - \left(1.5 \cdot \frac{{v}^{2}}{\sqrt{1}} + 1.125 \cdot \frac{{v}^{4}}{{\left(\sqrt{1}\right)}^{3}}\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\]

  3. Final simplification0.2

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1} - \left(1.5 \cdot \frac{{v}^{2}}{\sqrt{1}} + 1.125 \cdot \frac{{v}^{4}}{{\left(\sqrt{1}\right)}^{3}}\right)\right)\right) \cdot \left(1 - v \cdot v\right)\]

Reproduce

herbie shell --seed 2020018 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))