Average Error: 0.5 → 0.3
Time: 11.7s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\frac{\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{t} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{{\left(\sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right)}^{3}}\right) \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
\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{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{t} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{{\left(\sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right)}^{3}}\right) \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}
double f(double v, double t) {
        double r286228 = 1.0;
        double r286229 = 5.0;
        double r286230 = v;
        double r286231 = r286230 * r286230;
        double r286232 = r286229 * r286231;
        double r286233 = r286228 - r286232;
        double r286234 = atan2(1.0, 0.0);
        double r286235 = t;
        double r286236 = r286234 * r286235;
        double r286237 = 2.0;
        double r286238 = 3.0;
        double r286239 = r286238 * r286231;
        double r286240 = r286228 - r286239;
        double r286241 = r286237 * r286240;
        double r286242 = sqrt(r286241);
        double r286243 = r286236 * r286242;
        double r286244 = r286228 - r286231;
        double r286245 = r286243 * r286244;
        double r286246 = r286233 / r286245;
        return r286246;
}
↓
double f(double v, double t) {
        double r286247 = 1.0;
        double r286248 = 5.0;
        double r286249 = v;
        double r286250 = r286249 * r286249;
        double r286251 = r286248 * r286250;
        double r286252 = r286247 - r286251;
        double r286253 = cbrt(r286252);
        double r286254 = atan2(1.0, 0.0);
        double r286255 = r286253 / r286254;
        double r286256 = t;
        double r286257 = r286255 / r286256;
        double r286258 = 2.0;
        double r286259 = 3.0;
        double r286260 = r286259 * r286250;
        double r286261 = r286247 - r286260;
        double r286262 = r286258 * r286261;
        double r286263 = sqrt(r286262);
        double r286264 = cbrt(r286263);
        double r286265 = 3.0;
        double r286266 = pow(r286264, r286265);
        double r286267 = r286253 / r286266;
        double r286268 = r286257 * r286267;
        double r286269 = r286247 - r286250;
        double r286270 = r286253 / r286269;
        double r286271 = r286268 * r286270;
        return r286271;
}

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 add-cube-cbrt0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied add-cube-cbrt0.6

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

  7. Applied times-frac0.5

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

  8. Simplified0.5

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

  10. Applied associate-/r*0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020002 +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)))))