Average Error: 14.7 → 1.2
Time: 50.0s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}, -\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}, -\left(\ell - \left|m - n\right|\right)\right)}} \cdot \sqrt[3]{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}} \cdot \sqrt[3]{\sqrt[3]{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}}\right) \cdot \left(\sqrt[3]{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}} \cdot \sqrt[3]{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}\right)\]
\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)}
↓
\left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}, -\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}, -\left(\ell - \left|m - n\right|\right)\right)}} \cdot \sqrt[3]{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}} \cdot \sqrt[3]{\sqrt[3]{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}}\right) \cdot \left(\sqrt[3]{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}} \cdot \sqrt[3]{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}\right)
double f(double K, double m, double n, double M, double l) {
        double r77943 = K;
        double r77944 = m;
        double r77945 = n;
        double r77946 = r77944 + r77945;
        double r77947 = r77943 * r77946;
        double r77948 = 2.0;
        double r77949 = r77947 / r77948;
        double r77950 = M;
        double r77951 = r77949 - r77950;
        double r77952 = cos(r77951);
        double r77953 = r77946 / r77948;
        double r77954 = r77953 - r77950;
        double r77955 = pow(r77954, r77948);
        double r77956 = -r77955;
        double r77957 = l;
        double r77958 = r77944 - r77945;
        double r77959 = fabs(r77958);
        double r77960 = r77957 - r77959;
        double r77961 = r77956 - r77960;
        double r77962 = exp(r77961);
        double r77963 = r77952 * r77962;
        return r77963;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r77964 = m;
        double r77965 = n;
        double r77966 = r77964 + r77965;
        double r77967 = 2.0;
        double r77968 = r77966 / r77967;
        double r77969 = M;
        double r77970 = r77968 - r77969;
        double r77971 = pow(r77970, r77967);
        double r77972 = cbrt(r77971);
        double r77973 = r77972 * r77972;
        double r77974 = -r77972;
        double r77975 = l;
        double r77976 = r77964 - r77965;
        double r77977 = fabs(r77976);
        double r77978 = r77975 - r77977;
        double r77979 = -r77978;
        double r77980 = fma(r77973, r77974, r77979);
        double r77981 = exp(r77980);
        double r77982 = cbrt(r77981);
        double r77983 = -r77971;
        double r77984 = r77983 - r77978;
        double r77985 = exp(r77984);
        double r77986 = cbrt(r77985);
        double r77987 = r77982 * r77986;
        double r77988 = cbrt(r77987);
        double r77989 = cbrt(r77986);
        double r77990 = r77988 * r77989;
        double r77991 = r77986 * r77986;
        double r77992 = r77990 * r77991;
        return r77992;
}

Error

Bits error versus K

Bits error versus m

Bits error versus n

Bits error versus M

Bits error versus l

Derivation

  1. Initial program 14.7

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

  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

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

  6. Applied add-cube-cbrt1.2

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

  7. Applied cbrt-prod1.2

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

  9. Applied add-cube-cbrt1.2

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

  10. Applied add-cube-cbrt1.2

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

  11. Applied distribute-lft-neg-in1.2

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

  12. Applied prod-diff1.2

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

  13. Applied exp-sum28.2

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

  14. Simplified28.2

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

  15. Simplified1.2

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

  16. Final simplification1.2

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))