Average Error: 0.5 → 0.1
Time: 26.4s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{\sqrt{\left(1 - v \cdot \left(v \cdot 3\right)\right) \cdot 2}}}{t}}{1 \cdot \left(1 \cdot 1\right) - \left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)}\]
\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(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{\sqrt{\left(1 - v \cdot \left(v \cdot 3\right)\right) \cdot 2}}}{t}}{1 \cdot \left(1 \cdot 1\right) - \left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)}
double f(double v, double t) {
        double r6007128 = 1.0;
        double r6007129 = 5.0;
        double r6007130 = v;
        double r6007131 = r6007130 * r6007130;
        double r6007132 = r6007129 * r6007131;
        double r6007133 = r6007128 - r6007132;
        double r6007134 = atan2(1.0, 0.0);
        double r6007135 = t;
        double r6007136 = r6007134 * r6007135;
        double r6007137 = 2.0;
        double r6007138 = 3.0;
        double r6007139 = r6007138 * r6007131;
        double r6007140 = r6007128 - r6007139;
        double r6007141 = r6007137 * r6007140;
        double r6007142 = sqrt(r6007141);
        double r6007143 = r6007136 * r6007142;
        double r6007144 = r6007128 - r6007131;
        double r6007145 = r6007143 * r6007144;
        double r6007146 = r6007133 / r6007145;
        return r6007146;
}
↓
double f(double v, double t) {
        double r6007147 = 1.0;
        double r6007148 = r6007147 * r6007147;
        double r6007149 = v;
        double r6007150 = r6007149 * r6007149;
        double r6007151 = r6007150 * r6007147;
        double r6007152 = r6007150 * r6007150;
        double r6007153 = r6007151 + r6007152;
        double r6007154 = r6007148 + r6007153;
        double r6007155 = 5.0;
        double r6007156 = r6007150 * r6007155;
        double r6007157 = r6007147 - r6007156;
        double r6007158 = atan2(1.0, 0.0);
        double r6007159 = r6007157 / r6007158;
        double r6007160 = 3.0;
        double r6007161 = r6007149 * r6007160;
        double r6007162 = r6007149 * r6007161;
        double r6007163 = r6007147 - r6007162;
        double r6007164 = 2.0;
        double r6007165 = r6007163 * r6007164;
        double r6007166 = sqrt(r6007165);
        double r6007167 = r6007159 / r6007166;
        double r6007168 = t;
        double r6007169 = r6007167 / r6007168;
        double r6007170 = r6007147 * r6007148;
        double r6007171 = r6007149 * r6007150;
        double r6007172 = r6007171 * r6007171;
        double r6007173 = r6007170 - r6007172;
        double r6007174 = r6007169 / r6007173;
        double r6007175 = r6007154 * r6007174;
        return r6007175;
}

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 associate-*l*0.5

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

  5. Applied flip3--0.5

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

  6. Applied associate-*r/0.5

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

  7. Applied associate-/r/0.5

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

  8. Simplified0.3

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

  10. Applied associate-/r*0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))