Average Error: 0.4 → 0.4
Time: 1.5m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\frac{\left(v \cdot v\right) \cdot \frac{v \cdot v}{\sqrt{2} \cdot \pi}}{t} \cdot \frac{-53}{8} + \frac{\frac{1}{\sqrt{2} \cdot \pi}}{t}\right) + \frac{-5}{2} \cdot \frac{v \cdot v}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\]
\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(\frac{\left(v \cdot v\right) \cdot \frac{v \cdot v}{\sqrt{2} \cdot \pi}}{t} \cdot \frac{-53}{8} + \frac{\frac{1}{\sqrt{2} \cdot \pi}}{t}\right) + \frac{-5}{2} \cdot \frac{v \cdot v}{\left(\sqrt{2} \cdot \pi\right) \cdot t}
double f(double v, double t) {
        double r44462108 = 1.0;
        double r44462109 = 5.0;
        double r44462110 = v;
        double r44462111 = r44462110 * r44462110;
        double r44462112 = r44462109 * r44462111;
        double r44462113 = r44462108 - r44462112;
        double r44462114 = atan2(1.0, 0.0);
        double r44462115 = t;
        double r44462116 = r44462114 * r44462115;
        double r44462117 = 2.0;
        double r44462118 = 3.0;
        double r44462119 = r44462118 * r44462111;
        double r44462120 = r44462108 - r44462119;
        double r44462121 = r44462117 * r44462120;
        double r44462122 = sqrt(r44462121);
        double r44462123 = r44462116 * r44462122;
        double r44462124 = r44462108 - r44462111;
        double r44462125 = r44462123 * r44462124;
        double r44462126 = r44462113 / r44462125;
        return r44462126;
}
↓
double f(double v, double t) {
        double r44462127 = v;
        double r44462128 = r44462127 * r44462127;
        double r44462129 = 2.0;
        double r44462130 = sqrt(r44462129);
        double r44462131 = atan2(1.0, 0.0);
        double r44462132 = r44462130 * r44462131;
        double r44462133 = r44462128 / r44462132;
        double r44462134 = r44462128 * r44462133;
        double r44462135 = t;
        double r44462136 = r44462134 / r44462135;
        double r44462137 = -6.625;
        double r44462138 = r44462136 * r44462137;
        double r44462139 = 1.0;
        double r44462140 = r44462139 / r44462132;
        double r44462141 = r44462140 / r44462135;
        double r44462142 = r44462138 + r44462141;
        double r44462143 = -2.5;
        double r44462144 = r44462132 * r44462135;
        double r44462145 = r44462128 / r44462144;
        double r44462146 = r44462143 * r44462145;
        double r44462147 = r44462142 + r44462146;
        return r44462147;
}

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. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\frac{53}{8} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)}\]

  3. Simplified0.6

    \[\leadsto \color{blue}{\frac{v \cdot v}{\left(\sqrt{2} \cdot \pi\right) \cdot t} \cdot \frac{-5}{2} + \left(\frac{1}{\left(\sqrt{2} \cdot \pi\right) \cdot t} + \frac{\frac{v \cdot v}{\sqrt{2} \cdot \pi} \cdot \left(v \cdot v\right)}{t} \cdot \frac{-53}{8}\right)}\]
  4. Using strategy rm

  5. Applied associate-/r*0.4

    \[\leadsto \frac{v \cdot v}{\left(\sqrt{2} \cdot \pi\right) \cdot t} \cdot \frac{-5}{2} + \left(\color{blue}{\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t}} + \frac{\frac{v \cdot v}{\sqrt{2} \cdot \pi} \cdot \left(v \cdot v\right)}{t} \cdot \frac{-53}{8}\right)\]

  6. Final simplification0.4

    \[\leadsto \left(\frac{\left(v \cdot v\right) \cdot \frac{v \cdot v}{\sqrt{2} \cdot \pi}}{t} \cdot \frac{-53}{8} + \frac{\frac{1}{\sqrt{2} \cdot \pi}}{t}\right) + \frac{-5}{2} \cdot \frac{v \cdot v}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\]

Reproduce

herbie shell --seed 2019104 
(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)))))