Average Error: 0.5 → 0.3
Time: 8.9s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1}{t} \cdot \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{\pi}}{1 - v \cdot v}\]
\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)}
↓
\frac{\frac{1}{t} \cdot \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{\pi}}{1 - v \cdot v}
double f(double v, double t) {
        double r341996 = 1.0;
        double r341997 = 5.0;
        double r341998 = v;
        double r341999 = r341998 * r341998;
        double r342000 = r341997 * r341999;
        double r342001 = r341996 - r342000;
        double r342002 = atan2(1.0, 0.0);
        double r342003 = t;
        double r342004 = r342002 * r342003;
        double r342005 = 2.0;
        double r342006 = 3.0;
        double r342007 = r342006 * r341999;
        double r342008 = r341996 - r342007;
        double r342009 = r342005 * r342008;
        double r342010 = sqrt(r342009);
        double r342011 = r342004 * r342010;
        double r342012 = r341996 - r341999;
        double r342013 = r342011 * r342012;
        double r342014 = r342001 / r342013;
        return r342014;
}
↓
double f(double v, double t) {
        double r342015 = 1.0;
        double r342016 = t;
        double r342017 = r342015 / r342016;
        double r342018 = 1.0;
        double r342019 = 5.0;
        double r342020 = v;
        double r342021 = r342020 * r342020;
        double r342022 = r342019 * r342021;
        double r342023 = r342018 - r342022;
        double r342024 = 2.0;
        double r342025 = 3.0;
        double r342026 = r342025 * r342021;
        double r342027 = r342018 - r342026;
        double r342028 = r342024 * r342027;
        double r342029 = sqrt(r342028);
        double r342030 = r342023 / r342029;
        double r342031 = atan2(1.0, 0.0);
        double r342032 = r342030 / r342031;
        double r342033 = r342017 * r342032;
        double r342034 = r342018 - r342021;
        double r342035 = r342033 / r342034;
        return r342035;
}

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 associate-/r*0.5

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

  6. Simplified0.3

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

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

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

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

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

  10. Applied times-frac0.3

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

  11. Applied times-frac0.3

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

  12. Simplified0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020035 +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)))))