Average Error: 0.4 → 0.1
Time: 28.2s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\sqrt{1 + \left(3 \cdot \left(v \cdot v\right) + \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{\sqrt{2 \cdot \left(1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)\right)}}}{t}}{1 - v \cdot v}\]
\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)}
↓
\sqrt{1 + \left(3 \cdot \left(v \cdot v\right) + \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{\sqrt{2 \cdot \left(1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)\right)}}}{t}}{1 - v \cdot v}
double f(double v, double t) {
        double r3731025 = 1.0;
        double r3731026 = 5.0;
        double r3731027 = v;
        double r3731028 = r3731027 * r3731027;
        double r3731029 = r3731026 * r3731028;
        double r3731030 = r3731025 - r3731029;
        double r3731031 = atan2(1.0, 0.0);
        double r3731032 = t;
        double r3731033 = r3731031 * r3731032;
        double r3731034 = 2.0;
        double r3731035 = 3.0;
        double r3731036 = r3731035 * r3731028;
        double r3731037 = r3731025 - r3731036;
        double r3731038 = r3731034 * r3731037;
        double r3731039 = sqrt(r3731038);
        double r3731040 = r3731033 * r3731039;
        double r3731041 = r3731025 - r3731028;
        double r3731042 = r3731040 * r3731041;
        double r3731043 = r3731030 / r3731042;
        return r3731043;
}
↓
double f(double v, double t) {
        double r3731044 = 1.0;
        double r3731045 = 3.0;
        double r3731046 = v;
        double r3731047 = r3731046 * r3731046;
        double r3731048 = r3731045 * r3731047;
        double r3731049 = r3731048 * r3731048;
        double r3731050 = r3731048 + r3731049;
        double r3731051 = r3731044 + r3731050;
        double r3731052 = sqrt(r3731051);
        double r3731053 = 5.0;
        double r3731054 = r3731047 * r3731053;
        double r3731055 = r3731044 - r3731054;
        double r3731056 = atan2(1.0, 0.0);
        double r3731057 = r3731055 / r3731056;
        double r3731058 = 2.0;
        double r3731059 = r3731048 * r3731049;
        double r3731060 = r3731044 - r3731059;
        double r3731061 = r3731058 * r3731060;
        double r3731062 = sqrt(r3731061);
        double r3731063 = r3731057 / r3731062;
        double r3731064 = t;
        double r3731065 = r3731063 / r3731064;
        double r3731066 = r3731044 - r3731047;
        double r3731067 = r3731065 / r3731066;
        double r3731068 = r3731052 * r3731067;
        return r3731068;
}

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 flip3--0.4

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

  4. Applied associate-*r/0.4

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

  5. Applied sqrt-div0.5

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

  6. Applied associate-*r/0.5

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

  7. Applied associate-*l/0.5

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

  8. Applied associate-/r/0.5

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

  9. Simplified0.5

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

  11. Applied associate-/r*0.3

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

  13. Applied *-un-lft-identity0.3

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

  14. Applied add-sqr-sqrt0.3

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

  15. Applied times-frac0.3

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

  16. Applied times-frac0.3

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

  18. Applied associate-*l/0.1

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

  19. Simplified0.1

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

  20. Final simplification0.1

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

Reproduce

herbie shell --seed 2019153 
(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)))))