Average Error: 1.7 → 0.7
Time: 1.3m
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \left(\frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om} + \frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om}\right), 1\right)}} + \frac{1}{2}}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}
↓
\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \left(\frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om} + \frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om}\right), 1\right)}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r7072198 = 1.0;
        double r7072199 = 2.0;
        double r7072200 = r7072198 / r7072199;
        double r7072201 = l;
        double r7072202 = r7072199 * r7072201;
        double r7072203 = Om;
        double r7072204 = r7072202 / r7072203;
        double r7072205 = pow(r7072204, r7072199);
        double r7072206 = kx;
        double r7072207 = sin(r7072206);
        double r7072208 = pow(r7072207, r7072199);
        double r7072209 = ky;
        double r7072210 = sin(r7072209);
        double r7072211 = pow(r7072210, r7072199);
        double r7072212 = r7072208 + r7072211;
        double r7072213 = r7072205 * r7072212;
        double r7072214 = r7072198 + r7072213;
        double r7072215 = sqrt(r7072214);
        double r7072216 = r7072198 / r7072215;
        double r7072217 = r7072198 + r7072216;
        double r7072218 = r7072200 * r7072217;
        double r7072219 = sqrt(r7072218);
        return r7072219;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r7072220 = 0.5;
        double r7072221 = 4.0;
        double r7072222 = l;
        double r7072223 = kx;
        double r7072224 = sin(r7072223);
        double r7072225 = r7072222 * r7072224;
        double r7072226 = Om;
        double r7072227 = r7072225 / r7072226;
        double r7072228 = r7072227 * r7072227;
        double r7072229 = ky;
        double r7072230 = sin(r7072229);
        double r7072231 = r7072230 * r7072222;
        double r7072232 = r7072231 / r7072226;
        double r7072233 = r7072232 * r7072232;
        double r7072234 = r7072228 + r7072233;
        double r7072235 = 1.0;
        double r7072236 = fma(r7072221, r7072234, r7072235);
        double r7072237 = sqrt(r7072236);
        double r7072238 = r7072220 / r7072237;
        double r7072239 = r7072238 + r7072220;
        double r7072240 = sqrt(r7072239);
        return r7072240;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Initial program 1.7

    \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]

  2. Simplified1.7

    \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(\left(\frac{\ell \cdot 2}{Om} \cdot \frac{\ell \cdot 2}{Om}\right), \left(\mathsf{fma}\left(\left(\sin ky\right), \left(\sin ky\right), \left(\sin kx \cdot \sin kx\right)\right)\right), 1\right)}} + \frac{1}{2}}}\]

  3. Taylor expanded around inf 16.7

    \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\color{blue}{4 \cdot \frac{{\ell}^{2} \cdot {\left(\sin ky\right)}^{2}}{{Om}^{2}} + \left(4 \cdot \frac{{\left(\sin kx\right)}^{2} \cdot {\ell}^{2}}{{Om}^{2}} + 1\right)}}} + \frac{1}{2}}\]

  4. Simplified0.7

    \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(4, \left(\frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om} + \frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om}\right), 1\right)}}} + \frac{1}{2}}\]

  5. Taylor expanded around -inf 0.7

    \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \left(\frac{\sin ky \cdot \ell}{Om} \cdot \color{blue}{\frac{\ell \cdot \sin ky}{Om}} + \frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om}\right), 1\right)}} + \frac{1}{2}}\]

  6. Final simplification0.7

    \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \left(\frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om} + \frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om}\right), 1\right)}} + \frac{1}{2}}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))