Average Error: 0.4 → 0.3
Time: 2.6m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\frac{1}{\pi}}{t} - \frac{\frac{\left(v \cdot v\right) \cdot 5}{\pi}}{t}}{\sqrt{\left(1 - \left(v \cdot v\right) \cdot 3\right) \cdot 2}}}{1 - v \cdot v}\]
\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{\frac{\frac{1}{\pi}}{t} - \frac{\frac{\left(v \cdot v\right) \cdot 5}{\pi}}{t}}{\sqrt{\left(1 - \left(v \cdot v\right) \cdot 3\right) \cdot 2}}}{1 - v \cdot v}
double f(double v, double t) {
        double r77701005 = 1.0;
        double r77701006 = 5.0;
        double r77701007 = v;
        double r77701008 = r77701007 * r77701007;
        double r77701009 = r77701006 * r77701008;
        double r77701010 = r77701005 - r77701009;
        double r77701011 = atan2(1.0, 0.0);
        double r77701012 = t;
        double r77701013 = r77701011 * r77701012;
        double r77701014 = 2.0;
        double r77701015 = 3.0;
        double r77701016 = r77701015 * r77701008;
        double r77701017 = r77701005 - r77701016;
        double r77701018 = r77701014 * r77701017;
        double r77701019 = sqrt(r77701018);
        double r77701020 = r77701013 * r77701019;
        double r77701021 = r77701005 - r77701008;
        double r77701022 = r77701020 * r77701021;
        double r77701023 = r77701010 / r77701022;
        return r77701023;
}
↓
double f(double v, double t) {
        double r77701024 = 1.0;
        double r77701025 = atan2(1.0, 0.0);
        double r77701026 = r77701024 / r77701025;
        double r77701027 = t;
        double r77701028 = r77701026 / r77701027;
        double r77701029 = v;
        double r77701030 = r77701029 * r77701029;
        double r77701031 = 5.0;
        double r77701032 = r77701030 * r77701031;
        double r77701033 = r77701032 / r77701025;
        double r77701034 = r77701033 / r77701027;
        double r77701035 = r77701028 - r77701034;
        double r77701036 = 3.0;
        double r77701037 = r77701030 * r77701036;
        double r77701038 = r77701024 - r77701037;
        double r77701039 = 2.0;
        double r77701040 = r77701038 * r77701039;
        double r77701041 = sqrt(r77701040);
        double r77701042 = r77701035 / r77701041;
        double r77701043 = r77701024 - r77701030;
        double r77701044 = r77701042 / r77701043;
        return r77701044;
}

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 associate-/r*0.4

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

  5. Applied associate-/r*0.4

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

  7. Applied associate-/r*0.3

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

  9. Applied div-sub0.3

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

  10. Applied div-sub0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019107 
(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)))))