Average Error: 15.6 → 1.2
Time: 8.0s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[e^{\left(-\left(\left(\sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\frac{m + n}{2} - M} \cdot \sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\]
\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}
↓
e^{\left(-\left(\left(\sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\frac{m + n}{2} - M} \cdot \sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}
double f(double K, double m, double n, double M, double l) {
        double r128444 = K;
        double r128445 = m;
        double r128446 = n;
        double r128447 = r128445 + r128446;
        double r128448 = r128444 * r128447;
        double r128449 = 2.0;
        double r128450 = r128448 / r128449;
        double r128451 = M;
        double r128452 = r128450 - r128451;
        double r128453 = cos(r128452);
        double r128454 = r128447 / r128449;
        double r128455 = r128454 - r128451;
        double r128456 = pow(r128455, r128449);
        double r128457 = -r128456;
        double r128458 = l;
        double r128459 = r128445 - r128446;
        double r128460 = fabs(r128459);
        double r128461 = r128458 - r128460;
        double r128462 = r128457 - r128461;
        double r128463 = exp(r128462);
        double r128464 = r128453 * r128463;
        return r128464;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r128465 = m;
        double r128466 = n;
        double r128467 = r128465 + r128466;
        double r128468 = 2.0;
        double r128469 = r128467 / r128468;
        double r128470 = M;
        double r128471 = r128469 - r128470;
        double r128472 = pow(r128471, r128468);
        double r128473 = cbrt(r128472);
        double r128474 = r128473 * r128473;
        double r128475 = cbrt(r128474);
        double r128476 = r128475 * r128475;
        double r128477 = r128476 * r128475;
        double r128478 = cbrt(r128471);
        double r128479 = r128478 * r128478;
        double r128480 = pow(r128479, r128468);
        double r128481 = cbrt(r128480);
        double r128482 = pow(r128478, r128468);
        double r128483 = cbrt(r128482);
        double r128484 = r128481 * r128483;
        double r128485 = r128477 * r128484;
        double r128486 = -r128485;
        double r128487 = l;
        double r128488 = r128465 - r128466;
        double r128489 = fabs(r128488);
        double r128490 = r128487 - r128489;
        double r128491 = r128486 - r128490;
        double r128492 = exp(r128491);
        return r128492;
}

Error

Bits error versus K

Bits error versus m

Bits error versus n

Bits error versus M

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.6

    \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]

  2. Taylor expanded around 0 1.2

    \[\leadsto \color{blue}{1} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt1.2

    \[\leadsto 1 \cdot e^{\left(-\color{blue}{\left(\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) - \left(\ell - \left|m - n\right|\right)}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt1.2

    \[\leadsto 1 \cdot e^{\left(-\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right)} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}\right) - \left(\ell - \left|m - n\right|\right)}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt1.2

    \[\leadsto 1 \cdot e^{\left(-\left(\left(\sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \sqrt[3]{{\color{blue}{\left(\left(\sqrt[3]{\frac{m + n}{2} - M} \cdot \sqrt[3]{\frac{m + n}{2} - M}\right) \cdot \sqrt[3]{\frac{m + n}{2} - M}\right)}}^{2}}\right) - \left(\ell - \left|m - n\right|\right)}\]

  9. Applied unpow-prod-down1.2

    \[\leadsto 1 \cdot e^{\left(-\left(\left(\sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\frac{m + n}{2} - M} \cdot \sqrt[3]{\frac{m + n}{2} - M}\right)}^{2} \cdot {\left(\sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}}}\right) - \left(\ell - \left|m - n\right|\right)}\]

  10. Applied cbrt-prod1.2

    \[\leadsto 1 \cdot e^{\left(-\left(\left(\sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{\frac{m + n}{2} - M} \cdot \sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}}\right)}\right) - \left(\ell - \left|m - n\right|\right)}\]

  11. Final simplification1.2

    \[\leadsto e^{\left(-\left(\left(\sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\frac{m + n}{2} - M} \cdot \sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2) M)) (exp (- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n)))))))