Average Error: 0.4 → 0.5
Time: 22.4s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot \left(\left(\sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot t\right)\right) \cdot \sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)}\]
\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)}
↓
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot \left(\left(\sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot t\right)\right) \cdot \sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)}
double f(double v, double t) {
        double r176151 = 1.0;
        double r176152 = 5.0;
        double r176153 = v;
        double r176154 = r176153 * r176153;
        double r176155 = r176152 * r176154;
        double r176156 = r176151 - r176155;
        double r176157 = atan2(1.0, 0.0);
        double r176158 = t;
        double r176159 = r176157 * r176158;
        double r176160 = 2.0;
        double r176161 = 3.0;
        double r176162 = r176161 * r176154;
        double r176163 = r176151 - r176162;
        double r176164 = r176160 * r176163;
        double r176165 = sqrt(r176164);
        double r176166 = r176159 * r176165;
        double r176167 = r176151 - r176154;
        double r176168 = r176166 * r176167;
        double r176169 = r176156 / r176168;
        return r176169;
}
↓
double f(double v, double t) {
        double r176170 = 1.0;
        double r176171 = 5.0;
        double r176172 = v;
        double r176173 = r176172 * r176172;
        double r176174 = r176171 * r176173;
        double r176175 = r176170 - r176174;
        double r176176 = atan2(1.0, 0.0);
        double r176177 = 2.0;
        double r176178 = 3.0;
        double r176179 = r176178 * r176173;
        double r176180 = r176170 - r176179;
        double r176181 = r176177 * r176180;
        double r176182 = sqrt(r176181);
        double r176183 = cbrt(r176182);
        double r176184 = r176183 * r176183;
        double r176185 = t;
        double r176186 = r176184 * r176185;
        double r176187 = r176176 * r176186;
        double r176188 = r176187 * r176183;
        double r176189 = r176170 - r176173;
        double r176190 = r176188 * r176189;
        double r176191 = r176175 / r176190;
        return r176191;
}

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 add-cube-cbrt0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied associate-*l*0.5

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

  7. Simplified0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019347 
(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)))))