Average Error: 0.4 → 0.5
Time: 13.7s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\frac{9}{2} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1 \cdot \left(\frac{\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}}{t} \cdot \frac{\sqrt[3]{\sqrt{1}}}{\sqrt{2} \cdot \pi}\right) - 4 \cdot \left(\frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right) \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]
\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)}
↓
\left(\frac{9}{2} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1 \cdot \left(\frac{\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}}{t} \cdot \frac{\sqrt[3]{\sqrt{1}}}{\sqrt{2} \cdot \pi}\right) - 4 \cdot \left(\frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right) \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}
double f(double v, double t) {
        double r282182 = 1.0;
        double r282183 = 5.0;
        double r282184 = v;
        double r282185 = r282184 * r282184;
        double r282186 = r282183 * r282185;
        double r282187 = r282182 - r282186;
        double r282188 = atan2(1.0, 0.0);
        double r282189 = t;
        double r282190 = r282188 * r282189;
        double r282191 = 2.0;
        double r282192 = 3.0;
        double r282193 = r282192 * r282185;
        double r282194 = r282182 - r282193;
        double r282195 = r282191 * r282194;
        double r282196 = sqrt(r282195);
        double r282197 = r282190 * r282196;
        double r282198 = r282182 - r282185;
        double r282199 = r282197 * r282198;
        double r282200 = r282187 / r282199;
        return r282200;
}
↓
double f(double v, double t) {
        double r282201 = 9.0;
        double r282202 = 2.0;
        double r282203 = r282201 / r282202;
        double r282204 = v;
        double r282205 = 4.0;
        double r282206 = pow(r282204, r282205);
        double r282207 = t;
        double r282208 = sqrt(r282202);
        double r282209 = 1.0;
        double r282210 = sqrt(r282209);
        double r282211 = atan2(1.0, 0.0);
        double r282212 = r282210 * r282211;
        double r282213 = r282208 * r282212;
        double r282214 = r282207 * r282213;
        double r282215 = r282206 / r282214;
        double r282216 = r282203 * r282215;
        double r282217 = cbrt(r282210);
        double r282218 = r282217 * r282217;
        double r282219 = r282218 / r282207;
        double r282220 = r282208 * r282211;
        double r282221 = r282217 / r282220;
        double r282222 = r282219 * r282221;
        double r282223 = r282209 * r282222;
        double r282224 = 4.0;
        double r282225 = 2.0;
        double r282226 = pow(r282204, r282225);
        double r282227 = r282226 * r282210;
        double r282228 = r282207 * r282220;
        double r282229 = r282227 / r282228;
        double r282230 = r282206 * r282210;
        double r282231 = r282230 / r282228;
        double r282232 = r282229 + r282231;
        double r282233 = r282224 * r282232;
        double r282234 = r282223 - r282233;
        double r282235 = r282216 + r282234;
        double r282236 = 3.0;
        double r282237 = r282204 * r282204;
        double r282238 = r282236 * r282237;
        double r282239 = r282209 + r282238;
        double r282240 = sqrt(r282239);
        double r282241 = r282235 * r282240;
        return r282241;
}

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. Using strategy rm

  3. Applied flip--0.4

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

  4. Applied associate-*r/0.4

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

  5. Applied sqrt-div0.4

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

  6. Applied associate-*r/0.4

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

  7. Applied associate-*l/0.4

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

  8. Applied associate-/r/0.4

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

  9. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\left(\left(4.5 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + 1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right) - \left(4 \cdot \frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + 4 \cdot \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

  10. Simplified0.6

    \[\leadsto \color{blue}{\left(\frac{9}{2} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - 4 \cdot \left(\frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt0.6

    \[\leadsto \left(\frac{9}{2} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1 \cdot \frac{\color{blue}{\left(\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}\right) \cdot \sqrt[3]{\sqrt{1}}}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - 4 \cdot \left(\frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right) \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

  13. Applied times-frac0.5

    \[\leadsto \left(\frac{9}{2} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1 \cdot \color{blue}{\left(\frac{\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}}{t} \cdot \frac{\sqrt[3]{\sqrt{1}}}{\sqrt{2} \cdot \pi}\right)} - 4 \cdot \left(\frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right) \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

  14. Final simplification0.5

    \[\leadsto \left(\frac{9}{2} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1 \cdot \left(\frac{\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}}{t} \cdot \frac{\sqrt[3]{\sqrt{1}}}{\sqrt{2} \cdot \pi}\right) - 4 \cdot \left(\frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right) \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

Reproduce

herbie shell --seed 2019304 
(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)))))