Average Error: 0.4 → 0.3
Time: 20.7s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}\]
\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)}
↓
\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}
double f(double v, double t) {
        double r256116 = 1.0;
        double r256117 = 5.0;
        double r256118 = v;
        double r256119 = r256118 * r256118;
        double r256120 = r256117 * r256119;
        double r256121 = r256116 - r256120;
        double r256122 = atan2(1.0, 0.0);
        double r256123 = t;
        double r256124 = r256122 * r256123;
        double r256125 = 2.0;
        double r256126 = 3.0;
        double r256127 = r256126 * r256119;
        double r256128 = r256116 - r256127;
        double r256129 = r256125 * r256128;
        double r256130 = sqrt(r256129);
        double r256131 = r256124 * r256130;
        double r256132 = r256116 - r256119;
        double r256133 = r256131 * r256132;
        double r256134 = r256121 / r256133;
        return r256134;
}
↓
double f(double v, double t) {
        double r256135 = 1.0;
        double r256136 = 5.0;
        double r256137 = v;
        double r256138 = r256137 * r256137;
        double r256139 = r256136 * r256138;
        double r256140 = r256135 - r256139;
        double r256141 = atan2(1.0, 0.0);
        double r256142 = r256140 / r256141;
        double r256143 = t;
        double r256144 = 2.0;
        double r256145 = 3.0;
        double r256146 = r256145 * r256138;
        double r256147 = r256135 - r256146;
        double r256148 = r256144 * r256147;
        double r256149 = sqrt(r256148);
        double r256150 = r256143 * r256149;
        double r256151 = r256142 / r256150;
        double r256152 = r256135 - r256138;
        double r256153 = r256151 / r256152;
        return r256153;
}

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-*l*0.4

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

  7. Applied associate-/r*0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019325 +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)))))