Average Error: 1.0 → 0.0
Time: 23.3s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\left({\left(1 \cdot 1\right)}^{3} + {\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right)}^{3}\right) \cdot \frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right) - 1 \cdot 1\right) + 1 \cdot {1}^{3}\right)}\]
\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\frac{\left({\left(1 \cdot 1\right)}^{3} + {\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right)}^{3}\right) \cdot \frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right) - 1 \cdot 1\right) + 1 \cdot {1}^{3}\right)}
double f(double v) {
        double r132194 = 4.0;
        double r132195 = 3.0;
        double r132196 = atan2(1.0, 0.0);
        double r132197 = r132195 * r132196;
        double r132198 = 1.0;
        double r132199 = v;
        double r132200 = r132199 * r132199;
        double r132201 = r132198 - r132200;
        double r132202 = r132197 * r132201;
        double r132203 = 2.0;
        double r132204 = 6.0;
        double r132205 = r132204 * r132200;
        double r132206 = r132203 - r132205;
        double r132207 = sqrt(r132206);
        double r132208 = r132202 * r132207;
        double r132209 = r132194 / r132208;
        return r132209;
}
↓
double f(double v) {
        double r132210 = 1.0;
        double r132211 = r132210 * r132210;
        double r132212 = 3.0;
        double r132213 = pow(r132211, r132212);
        double r132214 = v;
        double r132215 = r132214 * r132214;
        double r132216 = r132215 + r132210;
        double r132217 = r132215 * r132216;
        double r132218 = pow(r132217, r132212);
        double r132219 = r132213 + r132218;
        double r132220 = 4.0;
        double r132221 = 3.0;
        double r132222 = atan2(1.0, 0.0);
        double r132223 = r132221 * r132222;
        double r132224 = r132220 / r132223;
        double r132225 = pow(r132210, r132212);
        double r132226 = 6.0;
        double r132227 = pow(r132214, r132226);
        double r132228 = r132225 - r132227;
        double r132229 = r132224 / r132228;
        double r132230 = r132219 * r132229;
        double r132231 = 2.0;
        double r132232 = 6.0;
        double r132233 = r132232 * r132215;
        double r132234 = r132231 - r132233;
        double r132235 = sqrt(r132234);
        double r132236 = r132217 - r132211;
        double r132237 = r132217 * r132236;
        double r132238 = r132210 * r132225;
        double r132239 = r132237 + r132238;
        double r132240 = r132235 * r132239;
        double r132241 = r132230 / r132240;
        return r132241;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
  2. Using strategy rm

  3. Applied flip3--1.0

    \[\leadsto \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \color{blue}{\frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  4. Applied associate-*r/1.0

    \[\leadsto \frac{4}{\color{blue}{\frac{\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  5. Applied associate-*l/1.0

    \[\leadsto \frac{4}{\color{blue}{\frac{\left(\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}}\]

  6. Applied associate-/r/1.0

    \[\leadsto \color{blue}{\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}\]

  7. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]
  8. Using strategy rm

  9. Applied flip3-+0.0

    \[\leadsto \frac{\frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \color{blue}{\frac{{\left(1 \cdot 1\right)}^{3} + {\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}^{3}}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right) - \left(1 \cdot 1\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}}\]

  10. Applied frac-times0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}} \cdot \left({\left(1 \cdot 1\right)}^{3} + {\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}^{3}\right)}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right) - \left(1 \cdot 1\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)}}\]

  11. Simplified0.0

    \[\leadsto \frac{\color{blue}{\left({\left(1 \cdot 1\right)}^{3} + {\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right)}^{3}\right) \cdot \frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right) - \left(1 \cdot 1\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)}\]

  12. Simplified0.0

    \[\leadsto \frac{\left({\left(1 \cdot 1\right)}^{3} + {\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right)}^{3}\right) \cdot \frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}{\color{blue}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right) - 1 \cdot 1\right) + 1 \cdot {1}^{3}\right)}}\]

  13. Final simplification0.0

    \[\leadsto \frac{\left({\left(1 \cdot 1\right)}^{3} + {\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right)}^{3}\right) \cdot \frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right) - 1 \cdot 1\right) + 1 \cdot {1}^{3}\right)}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))