Average Error: 0.5 → 0.3
Time: 23.4s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\left(\frac{1}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}} \cdot \frac{\frac{\sqrt{5 \cdot \left(v \cdot v\right)} + \sqrt{1}}{\pi}}{t}\right) \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt{1} - \sqrt{5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
\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(\left(\frac{1}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}} \cdot \frac{\frac{\sqrt{5 \cdot \left(v \cdot v\right)} + \sqrt{1}}{\pi}}{t}\right) \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt{1} - \sqrt{5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}
double f(double v, double t) {
        double r195060 = 1.0;
        double r195061 = 5.0;
        double r195062 = v;
        double r195063 = r195062 * r195062;
        double r195064 = r195061 * r195063;
        double r195065 = r195060 - r195064;
        double r195066 = atan2(1.0, 0.0);
        double r195067 = t;
        double r195068 = r195066 * r195067;
        double r195069 = 2.0;
        double r195070 = 3.0;
        double r195071 = r195070 * r195063;
        double r195072 = r195060 - r195071;
        double r195073 = r195069 * r195072;
        double r195074 = sqrt(r195073);
        double r195075 = r195068 * r195074;
        double r195076 = r195060 - r195063;
        double r195077 = r195075 * r195076;
        double r195078 = r195065 / r195077;
        return r195078;
}
↓
double f(double v, double t) {
        double r195079 = 1.0;
        double r195080 = 2.0;
        double r195081 = 1.0;
        double r195082 = r195081 * r195081;
        double r195083 = 3.0;
        double r195084 = r195083 * r195083;
        double r195085 = v;
        double r195086 = 4.0;
        double r195087 = pow(r195085, r195086);
        double r195088 = r195084 * r195087;
        double r195089 = r195082 - r195088;
        double r195090 = r195080 * r195089;
        double r195091 = sqrt(r195090);
        double r195092 = r195079 / r195091;
        double r195093 = 5.0;
        double r195094 = r195085 * r195085;
        double r195095 = r195093 * r195094;
        double r195096 = sqrt(r195095);
        double r195097 = sqrt(r195081);
        double r195098 = r195096 + r195097;
        double r195099 = atan2(1.0, 0.0);
        double r195100 = r195098 / r195099;
        double r195101 = t;
        double r195102 = r195100 / r195101;
        double r195103 = r195092 * r195102;
        double r195104 = r195083 * r195094;
        double r195105 = r195081 + r195104;
        double r195106 = sqrt(r195105);
        double r195107 = r195103 * r195106;
        double r195108 = r195097 - r195096;
        double r195109 = r195081 - r195094;
        double r195110 = r195108 / r195109;
        double r195111 = r195107 * r195110;
        return r195111;
}

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-log-exp0.5

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied difference-of-squares0.5

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

  8. Applied times-frac0.5

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

  9. Simplified1.1

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

  10. Simplified0.5

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

  12. Applied flip--0.5

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

  13. Applied associate-*r/0.5

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

  14. Applied sqrt-div0.5

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

  15. Applied associate-*r/0.5

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

  16. Applied associate-/r/0.5

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

  17. Simplified0.3

    \[\leadsto \left(\color{blue}{\frac{\frac{\sqrt{5 \cdot \left(v \cdot v\right)} + \sqrt{1}}{\pi}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)} \cdot t}} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt{1} - \sqrt{5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
  18. Using strategy rm

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

    \[\leadsto \left(\frac{\frac{\sqrt{5 \cdot \left(v \cdot v\right)} + \sqrt{1}}{\color{blue}{1 \cdot \pi}}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)} \cdot t} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt{1} - \sqrt{5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]

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

    \[\leadsto \left(\frac{\frac{\color{blue}{1 \cdot \left(\sqrt{5 \cdot \left(v \cdot v\right)} + \sqrt{1}\right)}}{1 \cdot \pi}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)} \cdot t} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt{1} - \sqrt{5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]

  21. Applied times-frac0.3

    \[\leadsto \left(\frac{\color{blue}{\frac{1}{1} \cdot \frac{\sqrt{5 \cdot \left(v \cdot v\right)} + \sqrt{1}}{\pi}}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)} \cdot t} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt{1} - \sqrt{5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]

  22. Applied times-frac0.3

    \[\leadsto \left(\color{blue}{\left(\frac{\frac{1}{1}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}} \cdot \frac{\frac{\sqrt{5 \cdot \left(v \cdot v\right)} + \sqrt{1}}{\pi}}{t}\right)} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt{1} - \sqrt{5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]

  23. Simplified0.3

    \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}}} \cdot \frac{\frac{\sqrt{5 \cdot \left(v \cdot v\right)} + \sqrt{1}}{\pi}}{t}\right) \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt{1} - \sqrt{5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]

  24. Final simplification0.3

    \[\leadsto \left(\left(\frac{1}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}} \cdot \frac{\frac{\sqrt{5 \cdot \left(v \cdot v\right)} + \sqrt{1}}{\pi}}{t}\right) \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt{1} - \sqrt{5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]

Reproduce

herbie shell --seed 2019326 +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)))))