Average Error: 0.5 → 0.5
Time: 13.1s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\left(\pi \cdot t\right) \cdot \sqrt{2}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
\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 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\left(\pi \cdot t\right) \cdot \sqrt{2}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)}
double f(double v, double t) {
        double r160496 = 1.0;
        double r160497 = 5.0;
        double r160498 = v;
        double r160499 = r160498 * r160498;
        double r160500 = r160497 * r160499;
        double r160501 = r160496 - r160500;
        double r160502 = atan2(1.0, 0.0);
        double r160503 = t;
        double r160504 = r160502 * r160503;
        double r160505 = 2.0;
        double r160506 = 3.0;
        double r160507 = r160506 * r160499;
        double r160508 = r160496 - r160507;
        double r160509 = r160505 * r160508;
        double r160510 = sqrt(r160509);
        double r160511 = r160504 * r160510;
        double r160512 = r160496 - r160499;
        double r160513 = r160511 * r160512;
        double r160514 = r160501 / r160513;
        return r160514;
}
↓
double f(double v, double t) {
        double r160515 = 1.0;
        double r160516 = 5.0;
        double r160517 = v;
        double r160518 = r160517 * r160517;
        double r160519 = r160516 * r160518;
        double r160520 = r160515 - r160519;
        double r160521 = atan2(1.0, 0.0);
        double r160522 = t;
        double r160523 = r160521 * r160522;
        double r160524 = 2.0;
        double r160525 = sqrt(r160524);
        double r160526 = r160523 * r160525;
        double r160527 = 3.0;
        double r160528 = r160527 * r160518;
        double r160529 = r160515 - r160528;
        double r160530 = sqrt(r160529);
        double r160531 = r160526 * r160530;
        double r160532 = r160515 - r160518;
        double r160533 = r160531 * r160532;
        double r160534 = r160520 / r160533;
        return r160534;
}

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 0.5

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

  3. Applied div-inv0.5

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

  5. Applied sqrt-prod0.5

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

  6. Applied associate-*r*0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019308 
(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)))))