Average Error: 0.4 → 0.1
Time: 2.0m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\sqrt{\left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right) + \left(4 - \left(\left(v \cdot v\right) \cdot -6\right) \cdot 2\right)} \cdot \frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{1 - v \cdot v}}{\sqrt{8 + \left(\left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}}}{t}\]
\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)}
↓
\sqrt{\left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right) + \left(4 - \left(\left(v \cdot v\right) \cdot -6\right) \cdot 2\right)} \cdot \frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{1 - v \cdot v}}{\sqrt{8 + \left(\left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}}}{t}
double f(double v, double t) {
        double r5948389 = 1.0;
        double r5948390 = 5.0;
        double r5948391 = v;
        double r5948392 = r5948391 * r5948391;
        double r5948393 = r5948390 * r5948392;
        double r5948394 = r5948389 - r5948393;
        double r5948395 = atan2(1.0, 0.0);
        double r5948396 = t;
        double r5948397 = r5948395 * r5948396;
        double r5948398 = 2.0;
        double r5948399 = 3.0;
        double r5948400 = r5948399 * r5948392;
        double r5948401 = r5948389 - r5948400;
        double r5948402 = r5948398 * r5948401;
        double r5948403 = sqrt(r5948402);
        double r5948404 = r5948397 * r5948403;
        double r5948405 = r5948389 - r5948392;
        double r5948406 = r5948404 * r5948405;
        double r5948407 = r5948394 / r5948406;
        return r5948407;
}
↓
double f(double v, double t) {
        double r5948408 = v;
        double r5948409 = r5948408 * r5948408;
        double r5948410 = -6.0;
        double r5948411 = r5948409 * r5948410;
        double r5948412 = r5948411 * r5948411;
        double r5948413 = 4.0;
        double r5948414 = 2.0;
        double r5948415 = r5948411 * r5948414;
        double r5948416 = r5948413 - r5948415;
        double r5948417 = r5948412 + r5948416;
        double r5948418 = sqrt(r5948417);
        double r5948419 = 1.0;
        double r5948420 = 5.0;
        double r5948421 = r5948409 * r5948420;
        double r5948422 = r5948419 - r5948421;
        double r5948423 = atan2(1.0, 0.0);
        double r5948424 = r5948422 / r5948423;
        double r5948425 = r5948419 - r5948409;
        double r5948426 = r5948424 / r5948425;
        double r5948427 = 8.0;
        double r5948428 = r5948412 * r5948411;
        double r5948429 = r5948427 + r5948428;
        double r5948430 = sqrt(r5948429);
        double r5948431 = r5948426 / r5948430;
        double r5948432 = t;
        double r5948433 = r5948431 / r5948432;
        double r5948434 = r5948418 * r5948433;
        return r5948434;
}

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. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}}{\pi}}{t \cdot \sqrt{-6 \cdot \left(v \cdot v\right) + 2}}}\]
  3. Using strategy rm

  4. Applied flip3-+0.3

    \[\leadsto \frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}}{\pi}}{t \cdot \sqrt{\color{blue}{\frac{{\left(-6 \cdot \left(v \cdot v\right)\right)}^{3} + {2}^{3}}{\left(-6 \cdot \left(v \cdot v\right)\right) \cdot \left(-6 \cdot \left(v \cdot v\right)\right) + \left(2 \cdot 2 - \left(-6 \cdot \left(v \cdot v\right)\right) \cdot 2\right)}}}}\]

  5. Applied sqrt-div0.3

    \[\leadsto \frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}}{\pi}}{t \cdot \color{blue}{\frac{\sqrt{{\left(-6 \cdot \left(v \cdot v\right)\right)}^{3} + {2}^{3}}}{\sqrt{\left(-6 \cdot \left(v \cdot v\right)\right) \cdot \left(-6 \cdot \left(v \cdot v\right)\right) + \left(2 \cdot 2 - \left(-6 \cdot \left(v \cdot v\right)\right) \cdot 2\right)}}}}\]

  6. Applied associate-*r/0.3

    \[\leadsto \frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}}{\pi}}{\color{blue}{\frac{t \cdot \sqrt{{\left(-6 \cdot \left(v \cdot v\right)\right)}^{3} + {2}^{3}}}{\sqrt{\left(-6 \cdot \left(v \cdot v\right)\right) \cdot \left(-6 \cdot \left(v \cdot v\right)\right) + \left(2 \cdot 2 - \left(-6 \cdot \left(v \cdot v\right)\right) \cdot 2\right)}}}}\]

  7. Applied associate-/r/0.3

    \[\leadsto \color{blue}{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}}{\pi}}{t \cdot \sqrt{{\left(-6 \cdot \left(v \cdot v\right)\right)}^{3} + {2}^{3}}} \cdot \sqrt{\left(-6 \cdot \left(v \cdot v\right)\right) \cdot \left(-6 \cdot \left(v \cdot v\right)\right) + \left(2 \cdot 2 - \left(-6 \cdot \left(v \cdot v\right)\right) \cdot 2\right)}}\]

  8. Simplified0.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{1 - v \cdot v}}{\sqrt{8 + \left(\left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}}}{t}} \cdot \sqrt{\left(-6 \cdot \left(v \cdot v\right)\right) \cdot \left(-6 \cdot \left(v \cdot v\right)\right) + \left(2 \cdot 2 - \left(-6 \cdot \left(v \cdot v\right)\right) \cdot 2\right)}\]

  9. Final simplification0.1

    \[\leadsto \sqrt{\left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right) + \left(4 - \left(\left(v \cdot v\right) \cdot -6\right) \cdot 2\right)} \cdot \frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{1 - v \cdot v}}{\sqrt{8 + \left(\left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}}}{t}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))