Average Error: 1.0 → 0.0
Time: 18.2s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3 \cdot \pi}}{1 - v \cdot v}}{\sqrt{\left(2 \cdot 2\right) \cdot 2 - \left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(v \cdot v\right) \cdot 6\right)\right)}} \cdot \sqrt{2 \cdot 2 + \left(\left(\left(v \cdot v\right) \cdot 6\right) \cdot 2 + \left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(v \cdot v\right) \cdot 6\right)\right)}\]
\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\frac{\frac{\frac{4}{3 \cdot \pi}}{1 - v \cdot v}}{\sqrt{\left(2 \cdot 2\right) \cdot 2 - \left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(v \cdot v\right) \cdot 6\right)\right)}} \cdot \sqrt{2 \cdot 2 + \left(\left(\left(v \cdot v\right) \cdot 6\right) \cdot 2 + \left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(v \cdot v\right) \cdot 6\right)\right)}
double f(double v) {
        double r6305487 = 4.0;
        double r6305488 = 3.0;
        double r6305489 = atan2(1.0, 0.0);
        double r6305490 = r6305488 * r6305489;
        double r6305491 = 1.0;
        double r6305492 = v;
        double r6305493 = r6305492 * r6305492;
        double r6305494 = r6305491 - r6305493;
        double r6305495 = r6305490 * r6305494;
        double r6305496 = 2.0;
        double r6305497 = 6.0;
        double r6305498 = r6305497 * r6305493;
        double r6305499 = r6305496 - r6305498;
        double r6305500 = sqrt(r6305499);
        double r6305501 = r6305495 * r6305500;
        double r6305502 = r6305487 / r6305501;
        return r6305502;
}
↓
double f(double v) {
        double r6305503 = 4.0;
        double r6305504 = 3.0;
        double r6305505 = atan2(1.0, 0.0);
        double r6305506 = r6305504 * r6305505;
        double r6305507 = r6305503 / r6305506;
        double r6305508 = 1.0;
        double r6305509 = v;
        double r6305510 = r6305509 * r6305509;
        double r6305511 = r6305508 - r6305510;
        double r6305512 = r6305507 / r6305511;
        double r6305513 = 2.0;
        double r6305514 = r6305513 * r6305513;
        double r6305515 = r6305514 * r6305513;
        double r6305516 = 6.0;
        double r6305517 = r6305510 * r6305516;
        double r6305518 = r6305517 * r6305517;
        double r6305519 = r6305517 * r6305518;
        double r6305520 = r6305515 - r6305519;
        double r6305521 = sqrt(r6305520);
        double r6305522 = r6305512 / r6305521;
        double r6305523 = r6305517 * r6305513;
        double r6305524 = r6305523 + r6305518;
        double r6305525 = r6305514 + r6305524;
        double r6305526 = sqrt(r6305525);
        double r6305527 = r6305522 * r6305526;
        return r6305527;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
  2. Using strategy rm

  3. Applied flip3--1.0

    \[\leadsto \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\frac{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}}}\]

  4. Applied sqrt-div1.0

    \[\leadsto \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \color{blue}{\frac{\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}}{\sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}}}\]

  5. Applied associate-*r/1.0

    \[\leadsto \frac{4}{\color{blue}{\frac{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}}{\sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}}}\]

  6. Applied associate-/r/1.0

    \[\leadsto \color{blue}{\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}} \cdot \sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}\]

  7. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{\sqrt{\left(2 \cdot 2\right) \cdot 2 - \left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(v \cdot v\right) \cdot 6\right)\right)}}} \cdot \sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}\]

  8. Final simplification0.0

    \[\leadsto \frac{\frac{\frac{4}{3 \cdot \pi}}{1 - v \cdot v}}{\sqrt{\left(2 \cdot 2\right) \cdot 2 - \left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(v \cdot v\right) \cdot 6\right)\right)}} \cdot \sqrt{2 \cdot 2 + \left(\left(\left(v \cdot v\right) \cdot 6\right) \cdot 2 + \left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(v \cdot v\right) \cdot 6\right)\right)}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))