Average Error: 0.4 → 0.3
Time: 25.9s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \frac{\frac{v \cdot v}{t}}{\sqrt{2} \cdot \pi} \cdot \frac{5}{2}\right) - \frac{\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \frac{53}{8}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\]
\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)}
↓
\left(\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \frac{\frac{v \cdot v}{t}}{\sqrt{2} \cdot \pi} \cdot \frac{5}{2}\right) - \frac{\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \frac{53}{8}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}
double f(double v, double t) {
        double r6816362 = 1.0;
        double r6816363 = 5.0;
        double r6816364 = v;
        double r6816365 = r6816364 * r6816364;
        double r6816366 = r6816363 * r6816365;
        double r6816367 = r6816362 - r6816366;
        double r6816368 = atan2(1.0, 0.0);
        double r6816369 = t;
        double r6816370 = r6816368 * r6816369;
        double r6816371 = 2.0;
        double r6816372 = 3.0;
        double r6816373 = r6816372 * r6816365;
        double r6816374 = r6816362 - r6816373;
        double r6816375 = r6816371 * r6816374;
        double r6816376 = sqrt(r6816375);
        double r6816377 = r6816370 * r6816376;
        double r6816378 = r6816362 - r6816365;
        double r6816379 = r6816377 * r6816378;
        double r6816380 = r6816367 / r6816379;
        return r6816380;
}
↓
double f(double v, double t) {
        double r6816381 = 1.0;
        double r6816382 = 2.0;
        double r6816383 = sqrt(r6816382);
        double r6816384 = atan2(1.0, 0.0);
        double r6816385 = r6816383 * r6816384;
        double r6816386 = r6816381 / r6816385;
        double r6816387 = t;
        double r6816388 = r6816386 / r6816387;
        double r6816389 = v;
        double r6816390 = r6816389 * r6816389;
        double r6816391 = r6816390 / r6816387;
        double r6816392 = r6816391 / r6816385;
        double r6816393 = 2.5;
        double r6816394 = r6816392 * r6816393;
        double r6816395 = r6816388 - r6816394;
        double r6816396 = r6816390 * r6816390;
        double r6816397 = 6.625;
        double r6816398 = r6816396 * r6816397;
        double r6816399 = r6816387 * r6816385;
        double r6816400 = r6816398 / r6816399;
        double r6816401 = r6816395 - r6816400;
        return r6816401;
}

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. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\frac{53}{8} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)}\]

  3. Simplified0.5

    \[\leadsto \color{blue}{\left(\frac{\frac{1}{t}}{\sqrt{2} \cdot \pi} - \frac{\frac{v \cdot v}{t}}{\sqrt{2} \cdot \pi} \cdot \frac{5}{2}\right) - \frac{\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \frac{53}{8}}{\left(\sqrt{2} \cdot \pi\right) \cdot t}}\]
  4. Using strategy rm

  5. Applied associate-/l/0.6

    \[\leadsto \left(\color{blue}{\frac{1}{\left(\sqrt{2} \cdot \pi\right) \cdot t}} - \frac{\frac{v \cdot v}{t}}{\sqrt{2} \cdot \pi} \cdot \frac{5}{2}\right) - \frac{\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \frac{53}{8}}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\]
  6. Using strategy rm

  7. Applied associate-/r*0.3

    \[\leadsto \left(\color{blue}{\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t}} - \frac{\frac{v \cdot v}{t}}{\sqrt{2} \cdot \pi} \cdot \frac{5}{2}\right) - \frac{\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \frac{53}{8}}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\]

  8. Final simplification0.3

    \[\leadsto \left(\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \frac{\frac{v \cdot v}{t}}{\sqrt{2} \cdot \pi} \cdot \frac{5}{2}\right) - \frac{\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \frac{53}{8}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\]

Reproduce

herbie shell --seed 2019158 
(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)))))