Average Error: 0.5 → 0.4
Time: 13.7s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(t \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
\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(t \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)}
double f(double v, double t) {
        double r240182 = 1.0;
        double r240183 = 5.0;
        double r240184 = v;
        double r240185 = r240184 * r240184;
        double r240186 = r240183 * r240185;
        double r240187 = r240182 - r240186;
        double r240188 = atan2(1.0, 0.0);
        double r240189 = t;
        double r240190 = r240188 * r240189;
        double r240191 = 2.0;
        double r240192 = 3.0;
        double r240193 = r240192 * r240185;
        double r240194 = r240182 - r240193;
        double r240195 = r240191 * r240194;
        double r240196 = sqrt(r240195);
        double r240197 = r240190 * r240196;
        double r240198 = r240182 - r240185;
        double r240199 = r240197 * r240198;
        double r240200 = r240187 / r240199;
        return r240200;
}
↓
double f(double v, double t) {
        double r240201 = 1.0;
        double r240202 = 5.0;
        double r240203 = v;
        double r240204 = r240203 * r240203;
        double r240205 = r240202 * r240204;
        double r240206 = r240201 - r240205;
        double r240207 = t;
        double r240208 = 2.0;
        double r240209 = sqrt(r240208);
        double r240210 = atan2(1.0, 0.0);
        double r240211 = r240209 * r240210;
        double r240212 = r240207 * r240211;
        double r240213 = 3.0;
        double r240214 = r240213 * r240204;
        double r240215 = r240201 - r240214;
        double r240216 = sqrt(r240215);
        double r240217 = r240212 * r240216;
        double r240218 = r240201 - r240204;
        double r240219 = r240217 * r240218;
        double r240220 = r240206 / r240219;
        return r240220;
}

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 sqrt-prod0.5

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

  4. Applied associate-*r*0.5

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020042 +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)))))