Average Error: 0.5 → 0.4
Time: 12.2s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(t \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
\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(t \cdot \left(\sqrt{2} \cdot \pi\right)\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 r224359 = 1.0;
        double r224360 = 5.0;
        double r224361 = v;
        double r224362 = r224361 * r224361;
        double r224363 = r224360 * r224362;
        double r224364 = r224359 - r224363;
        double r224365 = atan2(1.0, 0.0);
        double r224366 = t;
        double r224367 = r224365 * r224366;
        double r224368 = 2.0;
        double r224369 = 3.0;
        double r224370 = r224369 * r224362;
        double r224371 = r224359 - r224370;
        double r224372 = r224368 * r224371;
        double r224373 = sqrt(r224372);
        double r224374 = r224367 * r224373;
        double r224375 = r224359 - r224362;
        double r224376 = r224374 * r224375;
        double r224377 = r224364 / r224376;
        return r224377;
}
↓
double f(double v, double t) {
        double r224378 = 1.0;
        double r224379 = 5.0;
        double r224380 = v;
        double r224381 = r224380 * r224380;
        double r224382 = r224379 * r224381;
        double r224383 = r224378 - r224382;
        double r224384 = t;
        double r224385 = 2.0;
        double r224386 = sqrt(r224385);
        double r224387 = atan2(1.0, 0.0);
        double r224388 = r224386 * r224387;
        double r224389 = r224384 * r224388;
        double r224390 = 3.0;
        double r224391 = r224390 * r224381;
        double r224392 = r224378 - r224391;
        double r224393 = sqrt(r224392);
        double r224394 = r224389 * r224393;
        double r224395 = r224378 - r224381;
        double r224396 = r224394 * r224395;
        double r224397 = r224383 / r224396;
        return r224397;
}

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 sqrt-prod0.5

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

  4. Applied associate-*r*0.5

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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)))))