Average Error: 0.4 → 0.3
Time: 7.2s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
\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{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}
double f(double v, double t) {
        double r246207 = 1.0;
        double r246208 = 5.0;
        double r246209 = v;
        double r246210 = r246209 * r246209;
        double r246211 = r246208 * r246210;
        double r246212 = r246207 - r246211;
        double r246213 = atan2(1.0, 0.0);
        double r246214 = t;
        double r246215 = r246213 * r246214;
        double r246216 = 2.0;
        double r246217 = 3.0;
        double r246218 = r246217 * r246210;
        double r246219 = r246207 - r246218;
        double r246220 = r246216 * r246219;
        double r246221 = sqrt(r246220);
        double r246222 = r246215 * r246221;
        double r246223 = r246207 - r246210;
        double r246224 = r246222 * r246223;
        double r246225 = r246212 / r246224;
        return r246225;
}
↓
double f(double v, double t) {
        double r246226 = 1.0;
        double r246227 = 5.0;
        double r246228 = v;
        double r246229 = r246228 * r246228;
        double r246230 = r246227 * r246229;
        double r246231 = r246226 - r246230;
        double r246232 = sqrt(r246231);
        double r246233 = atan2(1.0, 0.0);
        double r246234 = r246232 / r246233;
        double r246235 = t;
        double r246236 = 2.0;
        double r246237 = 3.0;
        double r246238 = r246237 * r246229;
        double r246239 = r246226 - r246238;
        double r246240 = r246236 * r246239;
        double r246241 = sqrt(r246240);
        double r246242 = r246235 * r246241;
        double r246243 = r246234 / r246242;
        double r246244 = r246226 - r246229;
        double r246245 = r246232 / r246244;
        double r246246 = r246243 * r246245;
        return r246246;
}

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-*l*0.4

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied times-frac0.4

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

  8. Applied associate-/r*0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020024 +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)))))