Average Error: 17.3 → 0.4
Time: 1.7m
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(\mathsf{fma}\left(\frac{1}{60}, \left({\ell}^{5}\right), \left(\ell \cdot 2 + \left(\frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right) \cdot \ell\right)\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\left(\mathsf{fma}\left(\frac{1}{60}, \left({\ell}^{5}\right), \left(\ell \cdot 2 + \left(\frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right) \cdot \ell\right)\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U
double f(double J, double l, double K, double U) {
        double r16125108 = J;
        double r16125109 = l;
        double r16125110 = exp(r16125109);
        double r16125111 = -r16125109;
        double r16125112 = exp(r16125111);
        double r16125113 = r16125110 - r16125112;
        double r16125114 = r16125108 * r16125113;
        double r16125115 = K;
        double r16125116 = 2.0;
        double r16125117 = r16125115 / r16125116;
        double r16125118 = cos(r16125117);
        double r16125119 = r16125114 * r16125118;
        double r16125120 = U;
        double r16125121 = r16125119 + r16125120;
        return r16125121;
}
↓
double f(double J, double l, double K, double U) {
        double r16125122 = 0.016666666666666666;
        double r16125123 = l;
        double r16125124 = 5.0;
        double r16125125 = pow(r16125123, r16125124);
        double r16125126 = 2.0;
        double r16125127 = r16125123 * r16125126;
        double r16125128 = 0.3333333333333333;
        double r16125129 = r16125123 * r16125123;
        double r16125130 = r16125128 * r16125129;
        double r16125131 = r16125130 * r16125123;
        double r16125132 = r16125127 + r16125131;
        double r16125133 = fma(r16125122, r16125125, r16125132);
        double r16125134 = J;
        double r16125135 = r16125133 * r16125134;
        double r16125136 = K;
        double r16125137 = r16125136 / r16125126;
        double r16125138 = cos(r16125137);
        double r16125139 = r16125135 * r16125138;
        double r16125140 = U;
        double r16125141 = r16125139 + r16125140;
        return r16125141;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.3

    \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell + \left(\frac{1}{3} \cdot {\ell}^{3} + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  3. Simplified0.4

    \[\leadsto \left(J \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{60}, \left({\ell}^{5}\right), \left(\ell \cdot \mathsf{fma}\left(\frac{1}{3}, \left(\ell \cdot \ell\right), 2\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
  4. Using strategy rm

  5. Applied fma-udef0.4

    \[\leadsto \left(J \cdot \mathsf{fma}\left(\frac{1}{60}, \left({\ell}^{5}\right), \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  6. Applied distribute-lft-in0.4

    \[\leadsto \left(J \cdot \mathsf{fma}\left(\frac{1}{60}, \left({\ell}^{5}\right), \color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right) + \ell \cdot 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  7. Final simplification0.4

    \[\leadsto \left(\mathsf{fma}\left(\frac{1}{60}, \left({\ell}^{5}\right), \left(\ell \cdot 2 + \left(\frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right) \cdot \ell\right)\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))