Average Error: 0.4 → 0.3
Time: 10.2s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\mathsf{fma}\left(5 \cdot \left(v \cdot v\right), 1 + 5 \cdot \left(v \cdot v\right), 1 \cdot 1\right) \cdot \pi}}{\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot 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{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\mathsf{fma}\left(5 \cdot \left(v \cdot v\right), 1 + 5 \cdot \left(v \cdot v\right), 1 \cdot 1\right) \cdot \pi}}{\left(t \cdot \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 r425153 = 1.0;
        double r425154 = 5.0;
        double r425155 = v;
        double r425156 = r425155 * r425155;
        double r425157 = r425154 * r425156;
        double r425158 = r425153 - r425157;
        double r425159 = atan2(1.0, 0.0);
        double r425160 = t;
        double r425161 = r425159 * r425160;
        double r425162 = 2.0;
        double r425163 = 3.0;
        double r425164 = r425163 * r425156;
        double r425165 = r425153 - r425164;
        double r425166 = r425162 * r425165;
        double r425167 = sqrt(r425166);
        double r425168 = r425161 * r425167;
        double r425169 = r425153 - r425156;
        double r425170 = r425168 * r425169;
        double r425171 = r425158 / r425170;
        return r425171;
}
↓
double f(double v, double t) {
        double r425172 = 1.0;
        double r425173 = 3.0;
        double r425174 = pow(r425172, r425173);
        double r425175 = 5.0;
        double r425176 = v;
        double r425177 = r425176 * r425176;
        double r425178 = r425175 * r425177;
        double r425179 = pow(r425178, r425173);
        double r425180 = r425174 - r425179;
        double r425181 = r425172 + r425178;
        double r425182 = r425172 * r425172;
        double r425183 = fma(r425178, r425181, r425182);
        double r425184 = atan2(1.0, 0.0);
        double r425185 = r425183 * r425184;
        double r425186 = r425180 / r425185;
        double r425187 = t;
        double r425188 = 2.0;
        double r425189 = 3.0;
        double r425190 = r425189 * r425177;
        double r425191 = r425172 - r425190;
        double r425192 = r425188 * r425191;
        double r425193 = sqrt(r425192);
        double r425194 = r425187 * r425193;
        double r425195 = r425172 - r425177;
        double r425196 = r425194 * r425195;
        double r425197 = r425186 / r425196;
        return r425197;
}

Error

Bits error versus v

Bits error versus t

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 flip3--0.4

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

  6. Applied associate-/l/0.4

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

  7. Simplified0.4

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

  9. Applied associate-/r*0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020036 +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)))))