Average Error: 0.4 → 0.4
Time: 8.6s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1}{\frac{\frac{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)}{1 - 5 \cdot \left(v \cdot v\right)}}\]
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\frac{1}{\frac{\frac{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)}{1 - 5 \cdot \left(v \cdot v\right)}}
double f(double v, double t) {
        double r166565 = 1.0;
        double r166566 = 5.0;
        double r166567 = v;
        double r166568 = r166567 * r166567;
        double r166569 = r166566 * r166568;
        double r166570 = r166565 - r166569;
        double r166571 = atan2(1.0, 0.0);
        double r166572 = t;
        double r166573 = r166571 * r166572;
        double r166574 = 2.0;
        double r166575 = 3.0;
        double r166576 = r166575 * r166568;
        double r166577 = r166565 - r166576;
        double r166578 = r166574 * r166577;
        double r166579 = sqrt(r166578);
        double r166580 = r166573 * r166579;
        double r166581 = r166565 - r166568;
        double r166582 = r166580 * r166581;
        double r166583 = r166570 / r166582;
        return r166583;
}
↓
double f(double v, double t) {
        double r166584 = 1.0;
        double r166585 = atan2(1.0, 0.0);
        double r166586 = t;
        double r166587 = r166585 * r166586;
        double r166588 = 2.0;
        double r166589 = 1.0;
        double r166590 = r166589 * r166589;
        double r166591 = 3.0;
        double r166592 = v;
        double r166593 = r166592 * r166592;
        double r166594 = r166591 * r166593;
        double r166595 = r166594 * r166594;
        double r166596 = r166590 - r166595;
        double r166597 = r166588 * r166596;
        double r166598 = sqrt(r166597);
        double r166599 = r166587 * r166598;
        double r166600 = r166589 + r166594;
        double r166601 = sqrt(r166600);
        double r166602 = r166599 / r166601;
        double r166603 = r166589 - r166593;
        double r166604 = r166602 * r166603;
        double r166605 = 5.0;
        double r166606 = r166605 * r166593;
        double r166607 = r166589 - r166606;
        double r166608 = r166604 / r166607;
        double r166609 = r166584 / r166608;
        return r166609;
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
  2. Using strategy rm

  3. Applied clear-num0.4

    \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}{1 - 5 \cdot \left(v \cdot v\right)}}}\]
  4. Using strategy rm

  5. Applied flip--0.4

    \[\leadsto \frac{1}{\frac{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \color{blue}{\frac{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}{1 + 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \left(1 - v \cdot v\right)}{1 - 5 \cdot \left(v \cdot v\right)}}\]

  6. Applied associate-*r/0.4

    \[\leadsto \frac{1}{\frac{\left(\left(\pi \cdot t\right) \cdot \sqrt{\color{blue}{\frac{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}{1 + 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \left(1 - v \cdot v\right)}{1 - 5 \cdot \left(v \cdot v\right)}}\]

  7. Applied sqrt-div0.4

    \[\leadsto \frac{1}{\frac{\left(\left(\pi \cdot t\right) \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \left(1 - v \cdot v\right)}{1 - 5 \cdot \left(v \cdot v\right)}}\]

  8. Applied associate-*r/0.4

    \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}}} \cdot \left(1 - v \cdot v\right)}{1 - 5 \cdot \left(v \cdot v\right)}}\]

  9. Final simplification0.4

    \[\leadsto \frac{1}{\frac{\frac{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)}{1 - 5 \cdot \left(v \cdot v\right)}}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))