Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.2% → 96.6%

Time: 19.9s

Alternatives: 8

Speedup: 1.9×

Specification

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 76.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\\ \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - {t_0}^{4} \cdot {t_0}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (cbrt (- (* (+ m n) 0.5) M))))
   (* (cos M) (exp (- (- (fabs (- m n)) l) (* (pow t_0 4.0) (pow t_0 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cbrt((((m + n) * 0.5) - M));
	return cos(M) * exp(((fabs((m - n)) - l) - (pow(t_0, 4.0) * pow(t_0, 2.0))));
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cbrt((((m + n) * 0.5) - M));
	return Math.cos(M) * Math.exp(((Math.abs((m - n)) - l) - (Math.pow(t_0, 4.0) * Math.pow(t_0, 2.0))));
}

Julia

function code(K, m, n, M, l)
	t_0 = cbrt(Float64(Float64(Float64(m + n) * 0.5) - M))
	return Float64(cos(M) * exp(Float64(Float64(abs(Float64(m - n)) - l) - Float64((t_0 ^ 4.0) * (t_0 ^ 2.0)))))
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(N[Power[t$95$0, 4.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 69.6%

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

   1. associate-/l* 69.6%

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

   2. +-commutative 69.6%

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

   3. fabs-sub 69.6%

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

   4. +-commutative 69.6%

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

3. Simplified 69.6%

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

4. Taylor expanded in K around 0 95.3%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
5. Step-by-step derivation

   1. cos-neg 95.3%

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

6. Simplified 95.3%

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
7. Step-by-step derivation

   1. add-cube-cbrt 95.3%

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

   2. cbrt-unprod 93.7%

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

   3. pow-prod-up 93.7%

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

   4. div-inv 93.7%

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

   5. fma-neg 93.7%

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

   6. metadata-eval 93.7%

      \[\leadsto \cos M \cdot e^{\left(-\sqrt[3]{{\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{\left(2 + 2\right)}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}\right) - \left(\ell - \left|n - m\right|\right)} \]

   7. metadata-eval 93.7%

      \[\leadsto \cos M \cdot e^{\left(-\sqrt[3]{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{\color{blue}{4}}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}\right) - \left(\ell - \left|n - m\right|\right)} \]

   8. unpow2 93.7%

      \[\leadsto \cos M \cdot e^{\left(-\sqrt[3]{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{4}} \cdot \sqrt[3]{\color{blue}{\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right)}}\right) - \left(\ell - \left|n - m\right|\right)} \]

   9. cbrt-prod 93.7%

      \[\leadsto \cos M \cdot e^{\left(-\sqrt[3]{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{4}} \cdot \color{blue}{\left(\sqrt[3]{\frac{m + n}{2} - M} \cdot \sqrt[3]{\frac{m + n}{2} - M}\right)}\right) - \left(\ell - \left|n - m\right|\right)} \]

   10. pow2 93.7%

       \[\leadsto \cos M \cdot e^{\left(-\sqrt[3]{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{4}} \cdot \color{blue}{{\left(\sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}}\right) - \left(\ell - \left|n - m\right|\right)} \]

   11. div-inv 93.7%

       \[\leadsto \cos M \cdot e^{\left(-\sqrt[3]{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{4}} \cdot {\left(\sqrt[3]{\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

   12. fma-neg 93.7%

       \[\leadsto \cos M \cdot e^{\left(-\sqrt[3]{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{4}} \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(m + n, \frac{1}{2}, -M\right)}}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

   13. metadata-eval 93.7%

       \[\leadsto \cos M \cdot e^{\left(-\sqrt[3]{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{4}} \cdot {\left(\sqrt[3]{\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

8. Applied egg-rr 93.7%

   \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{4}} \cdot {\left(\sqrt[3]{\mathsf{fma}\left(m + n, 0.5, -M\right)}\right)}^{2}}\right) - \left(\ell - \left|n - m\right|\right)} \]
9. Step-by-step derivation

   1. fma-neg 93.7%

      \[\leadsto \cos M \cdot e^{\left(-\sqrt[3]{{\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)}}^{4}} \cdot {\left(\sqrt[3]{\mathsf{fma}\left(m + n, 0.5, -M\right)}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

   2. fma-neg 93.7%

      \[\leadsto \cos M \cdot e^{\left(-\sqrt[3]{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}} \cdot {\left(\sqrt[3]{\color{blue}{\left(m + n\right) \cdot 0.5 - M}}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

10. Simplified 93.7%

    \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\sqrt[3]{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}} \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}}\right) - \left(\ell - \left|n - m\right|\right)} \]
11. Step-by-step derivation

    1. pow1/3 93.7%

       \[\leadsto \cos M \cdot e^{\left(-\color{blue}{{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{4}\right)}^{0.3333333333333333}} \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    2. sqr-pow 93.7%

       \[\leadsto \cos M \cdot e^{\left(-{\color{blue}{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{\left(\frac{4}{2}\right)} \cdot {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{\left(\frac{4}{2}\right)}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    3. unpow-prod-down 95.3%

       \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left({\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{\left(\frac{4}{2}\right)}\right)}^{0.3333333333333333} \cdot {\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{\left(\frac{4}{2}\right)}\right)}^{0.3333333333333333}\right)} \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    4. metadata-eval 95.3%

       \[\leadsto \cos M \cdot e^{\left(-\left({\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{\color{blue}{2}}\right)}^{0.3333333333333333} \cdot {\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{\left(\frac{4}{2}\right)}\right)}^{0.3333333333333333}\right) \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    5. pow2 95.3%

       \[\leadsto \cos M \cdot e^{\left(-\left({\color{blue}{\left(\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right)\right)}}^{0.3333333333333333} \cdot {\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{\left(\frac{4}{2}\right)}\right)}^{0.3333333333333333}\right) \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    6. pow-prod-down 48.7%

       \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{0.3333333333333333} \cdot {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{0.3333333333333333}\right)} \cdot {\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{\left(\frac{4}{2}\right)}\right)}^{0.3333333333333333}\right) \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    7. pow1/3 48.7%

       \[\leadsto \cos M \cdot e^{\left(-\left(\left(\color{blue}{\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}} \cdot {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{0.3333333333333333}\right) \cdot {\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{\left(\frac{4}{2}\right)}\right)}^{0.3333333333333333}\right) \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    8. pow1/3 95.3%

       \[\leadsto \cos M \cdot e^{\left(-\left(\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M} \cdot \color{blue}{\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}}\right) \cdot {\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{\left(\frac{4}{2}\right)}\right)}^{0.3333333333333333}\right) \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    9. unpow2 95.3%

       \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{{\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}} \cdot {\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{\left(\frac{4}{2}\right)}\right)}^{0.3333333333333333}\right) \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    10. metadata-eval 95.3%

        \[\leadsto \cos M \cdot e^{\left(-\left({\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2} \cdot {\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{\color{blue}{2}}\right)}^{0.3333333333333333}\right) \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    11. pow2 95.3%

        \[\leadsto \cos M \cdot e^{\left(-\left({\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2} \cdot {\color{blue}{\left(\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right)\right)}}^{0.3333333333333333}\right) \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    12. pow-prod-down 48.7%

        \[\leadsto \cos M \cdot e^{\left(-\left({\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2} \cdot \color{blue}{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{0.3333333333333333} \cdot {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{0.3333333333333333}\right)}\right) \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    13. pow1/3 48.7%

        \[\leadsto \cos M \cdot e^{\left(-\left({\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2} \cdot \left(\color{blue}{\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}} \cdot {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{0.3333333333333333}\right)\right) \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    14. pow1/3 95.3%

        \[\leadsto \cos M \cdot e^{\left(-\left({\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2} \cdot \left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M} \cdot \color{blue}{\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}}\right)\right) \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

    15. unpow2 95.3%

        \[\leadsto \cos M \cdot e^{\left(-\left({\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2} \cdot \color{blue}{{\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}}\right) \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

12. Applied egg-rr 95.3%

    \[\leadsto \cos M \cdot e^{\left(-\color{blue}{{\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{4}} \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]

13. Final simplification 95.3%

    \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{4} \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5 - M}\right)}^{2}} \]

Alternative 2: 96.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \cos M \cdot e^{\left(\left(n - m\right) - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (- (- n m) l) (pow (- (* (+ m n) 0.5) M) 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp((((n - m) - l) - pow((((m + n) * 0.5) - M), 2.0)));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp((((n - m) - l) - ((((m + n) * 0.5d0) - m_1) ** 2.0d0)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.exp((((n - m) - l) - Math.pow((((m + n) * 0.5) - M), 2.0)));
}

Python

def code(K, m, n, M, l):
	return math.cos(M) * math.exp((((n - m) - l) - math.pow((((m + n) * 0.5) - M), 2.0)))

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(Float64(Float64(n - m) - l) - (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp((((n - m) - l) - ((((m + n) * 0.5) - M) ^ 2.0)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(n - m), $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.6%

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

   1. associate-/l* 69.6%

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

   2. +-commutative 69.6%

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

   3. fabs-sub 69.6%

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

   4. +-commutative 69.6%

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

3. Simplified 69.6%

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

4. Taylor expanded in K around 0 95.3%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
5. Step-by-step derivation

   1. cos-neg 95.3%

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

6. Simplified 95.3%

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
7. Step-by-step derivation

   1. sub-neg 95.3%

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

   2. distribute-neg-out 95.3%

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

   3. div-inv 95.3%

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

   4. fma-neg 95.3%

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

   5. metadata-eval 95.3%

      \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2} + \left(\ell - \left|n - m\right|\right)\right)} \]

   6. add-sqr-sqrt 50.8%

      \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right|\right)\right)} \]

   7. fabs-sqr 50.8%

      \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right)\right)} \]

   8. add-sqr-sqrt 95.2%

      \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \color{blue}{\left(n - m\right)}\right)\right)} \]

8. Applied egg-rr 95.2%

   \[\leadsto \cos M \cdot e^{\color{blue}{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left(n - m\right)\right)\right)}} \]
9. Step-by-step derivation

   1. neg-sub0 95.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{0 - \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left(n - m\right)\right)\right)}} \]

   2. +-commutative 95.2%

      \[\leadsto \cos M \cdot e^{0 - \color{blue}{\left(\left(\ell - \left(n - m\right)\right) + {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}\right)}} \]

   3. associate--r+ 95.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{\left(0 - \left(\ell - \left(n - m\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}} \]

   4. sub-neg 95.2%

      \[\leadsto \cos M \cdot e^{\left(0 - \color{blue}{\left(\ell + \left(-\left(n - m\right)\right)\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   5. associate--r+ 95.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left(0 - \ell\right) - \left(-\left(n - m\right)\right)\right)} - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   6. neg-sub0 95.2%

      \[\leadsto \cos M \cdot e^{\left(\color{blue}{\left(-\ell\right)} - \left(-\left(n - m\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   7. sub-neg 95.2%

      \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\color{blue}{\left(n + \left(-m\right)\right)}\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   8. mul-1-neg 95.2%

      \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\left(n + \color{blue}{-1 \cdot m}\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   9. +-commutative 95.2%

      \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\color{blue}{\left(-1 \cdot m + n\right)}\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   10. distribute-neg-in 95.2%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \color{blue}{\left(\left(--1 \cdot m\right) + \left(-n\right)\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   11. mul-1-neg 95.2%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(\left(-\color{blue}{\left(-m\right)}\right) + \left(-n\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   12. remove-double-neg 95.2%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(\color{blue}{m} + \left(-n\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   13. sub-neg 95.2%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \color{blue}{\left(m - n\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   14. fma-neg 95.2%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - {\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)}}^{2}} \]

10. Simplified 95.2%

    \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left(-\ell\right) - \left(m - n\right)\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}} \]

11. Final simplification 95.2%

    \[\leadsto \cos M \cdot e^{\left(\left(n - m\right) - \ell\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}} \]

Alternative 3: 84.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+108}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left(n - m\right) - \ell\right) + \left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -5.2e+108)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (*
    (cos M)
    (exp (+ (- (- n m) l) (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -5.2e+108) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-5.2d+108)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp((((n - m) - l) + (((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -5.2e+108) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -5.2e+108:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -5.2e+108)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(n - m) - l) + Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -5.2e+108)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -5.2e+108], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(n - m), $MachinePrecision] - l), $MachinePrecision] + N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -5.2000000000000005e108

   1. Initial program 58.1%

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

      1. associate-/l* 58.1%

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

      2. +-commutative 58.1%

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

      3. fabs-sub 58.1%

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

      4. +-commutative 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   5. Step-by-step derivation

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around inf 100.0%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \cos M \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

   9. Simplified 100.0%

      \[\leadsto \cos M \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

   if -5.2000000000000005e108 < m

   1. Initial program 71.9%

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

      1. associate-/l* 71.9%

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

      2. +-commutative 71.9%

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

      3. fabs-sub 71.9%

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

      4. +-commutative 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in K around 0 94.3%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   5. Step-by-step derivation

      1. cos-neg 94.3%

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

   6. Simplified 94.3%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   7. Step-by-step derivation

      1. sub-neg 94.3%

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

      2. distribute-neg-out 94.3%

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

      3. div-inv 94.3%

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

      4. fma-neg 94.3%

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

      5. metadata-eval 94.3%

         \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2} + \left(\ell - \left|n - m\right|\right)\right)} \]

      6. add-sqr-sqrt 42.3%

         \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right|\right)\right)} \]

      7. fabs-sqr 42.3%

         \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right)\right)} \]

      8. add-sqr-sqrt 94.3%

         \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \color{blue}{\left(n - m\right)}\right)\right)} \]

   8. Applied egg-rr 94.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left(n - m\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. neg-sub0 94.3%

         \[\leadsto \cos M \cdot e^{\color{blue}{0 - \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left(n - m\right)\right)\right)}} \]

      2. +-commutative 94.3%

         \[\leadsto \cos M \cdot e^{0 - \color{blue}{\left(\left(\ell - \left(n - m\right)\right) + {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}\right)}} \]

      3. associate--r+ 94.3%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(0 - \left(\ell - \left(n - m\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}} \]

      4. sub-neg 94.3%

         \[\leadsto \cos M \cdot e^{\left(0 - \color{blue}{\left(\ell + \left(-\left(n - m\right)\right)\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      5. associate--r+ 94.3%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left(0 - \ell\right) - \left(-\left(n - m\right)\right)\right)} - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      6. neg-sub0 94.3%

         \[\leadsto \cos M \cdot e^{\left(\color{blue}{\left(-\ell\right)} - \left(-\left(n - m\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      7. sub-neg 94.3%

         \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\color{blue}{\left(n + \left(-m\right)\right)}\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      8. mul-1-neg 94.3%

         \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\left(n + \color{blue}{-1 \cdot m}\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      9. +-commutative 94.3%

         \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\color{blue}{\left(-1 \cdot m + n\right)}\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      10. distribute-neg-in 94.3%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \color{blue}{\left(\left(--1 \cdot m\right) + \left(-n\right)\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      11. mul-1-neg 94.3%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(\left(-\color{blue}{\left(-m\right)}\right) + \left(-n\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      12. remove-double-neg 94.3%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(\color{blue}{m} + \left(-n\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      13. sub-neg 94.3%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \color{blue}{\left(m - n\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      14. fma-neg 94.3%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - {\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)}}^{2}} \]

   10. Simplified 94.3%

       \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left(-\ell\right) - \left(m - n\right)\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}} \]

   11. Taylor expanded in n around 0 76.6%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}} \]
   12. Step-by-step derivation

       1. +-commutative 76.6%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}} \]

       2. unpow2 76.6%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)} \]

       3. distribute-rgt-out 81.8%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   13. Simplified 81.8%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+108}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left(n - m\right) - \ell\right) + \left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right)}\\ \end{array} \]

Alternative 4: 90.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 6.3 \cdot 10^{+94}:\\ \;\;\;\;\cos M \cdot e^{\left(\left(n - m\right) - \ell\right) + \left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{{n}^{2} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 6.3e+94)
   (*
    (cos M)
    (exp (+ (- (- n m) l) (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)))))
   (* (cos M) (exp (* (pow n 2.0) -0.25)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 6.3e+94) {
		tmp = cos(M) * exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))));
	} else {
		tmp = cos(M) * exp((pow(n, 2.0) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 6.3d+94) then
        tmp = cos(m_1) * exp((((n - m) - l) + (((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n))))
    else
        tmp = cos(m_1) * exp(((n ** 2.0d0) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 6.3e+94) {
		tmp = Math.cos(M) * Math.exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))));
	} else {
		tmp = Math.cos(M) * Math.exp((Math.pow(n, 2.0) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 6.3e+94:
		tmp = math.cos(M) * math.exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))))
	else:
		tmp = math.cos(M) * math.exp((math.pow(n, 2.0) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 6.3e+94)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(n - m) - l) + Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)))));
	else
		tmp = Float64(cos(M) * exp(Float64((n ^ 2.0) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 6.3e+94)
		tmp = cos(M) * exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))));
	else
		tmp = cos(M) * exp(((n ^ 2.0) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 6.3e+94], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(n - m), $MachinePrecision] - l), $MachinePrecision] + N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 6.3000000000000001e94

   1. Initial program 70.9%

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

      1. associate-/l* 70.9%

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

      2. +-commutative 70.9%

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

      3. fabs-sub 70.9%

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

      4. +-commutative 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in K around 0 94.4%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   5. Step-by-step derivation

      1. cos-neg 94.4%

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

   6. Simplified 94.4%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   7. Step-by-step derivation

      1. sub-neg 94.4%

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

      2. distribute-neg-out 94.4%

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

      3. div-inv 94.4%

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

      4. fma-neg 94.4%

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

      5. metadata-eval 94.4%

         \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2} + \left(\ell - \left|n - m\right|\right)\right)} \]

      6. add-sqr-sqrt 42.2%

         \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right|\right)\right)} \]

      7. fabs-sqr 42.2%

         \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right)\right)} \]

      8. add-sqr-sqrt 94.3%

         \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \color{blue}{\left(n - m\right)}\right)\right)} \]

   8. Applied egg-rr 94.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left(n - m\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. neg-sub0 94.3%

         \[\leadsto \cos M \cdot e^{\color{blue}{0 - \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left(n - m\right)\right)\right)}} \]

      2. +-commutative 94.3%

         \[\leadsto \cos M \cdot e^{0 - \color{blue}{\left(\left(\ell - \left(n - m\right)\right) + {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}\right)}} \]

      3. associate--r+ 94.3%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(0 - \left(\ell - \left(n - m\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}} \]

      4. sub-neg 94.3%

         \[\leadsto \cos M \cdot e^{\left(0 - \color{blue}{\left(\ell + \left(-\left(n - m\right)\right)\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      5. associate--r+ 94.3%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left(0 - \ell\right) - \left(-\left(n - m\right)\right)\right)} - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      6. neg-sub0 94.3%

         \[\leadsto \cos M \cdot e^{\left(\color{blue}{\left(-\ell\right)} - \left(-\left(n - m\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      7. sub-neg 94.3%

         \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\color{blue}{\left(n + \left(-m\right)\right)}\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      8. mul-1-neg 94.3%

         \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\left(n + \color{blue}{-1 \cdot m}\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      9. +-commutative 94.3%

         \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\color{blue}{\left(-1 \cdot m + n\right)}\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      10. distribute-neg-in 94.3%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \color{blue}{\left(\left(--1 \cdot m\right) + \left(-n\right)\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      11. mul-1-neg 94.3%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(\left(-\color{blue}{\left(-m\right)}\right) + \left(-n\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      12. remove-double-neg 94.3%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(\color{blue}{m} + \left(-n\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      13. sub-neg 94.3%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \color{blue}{\left(m - n\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      14. fma-neg 94.3%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - {\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)}}^{2}} \]

   10. Simplified 94.3%

       \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left(-\ell\right) - \left(m - n\right)\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}} \]

   11. Taylor expanded in n around 0 82.9%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}} \]
   12. Step-by-step derivation

       1. +-commutative 82.9%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}} \]

       2. unpow2 82.9%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)} \]

       3. distribute-rgt-out 87.1%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   13. Simplified 87.1%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   if 6.3000000000000001e94 < n

   1. Initial program 62.5%

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

      1. associate-/l* 62.5%

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

      2. +-commutative 62.5%

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

      3. fabs-sub 62.5%

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

      4. +-commutative 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   5. Step-by-step derivation

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 100.0%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

   9. Simplified 100.0%

      \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 6.3 \cdot 10^{+94}:\\ \;\;\;\;\cos M \cdot e^{\left(\left(n - m\right) - \ell\right) + \left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{{n}^{2} \cdot -0.25}\\ \end{array} \]

Alternative 5: 85.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\cos M \cdot e^{\left(\left(n - m\right) - \ell\right) + \left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M 2e+14)
   (*
    (cos M)
    (exp (+ (- (- n m) l) (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)))))
   (* (cos M) (exp (- (pow M 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= 2e+14) {
		tmp = cos(M) * exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))));
	} else {
		tmp = cos(M) * exp(-pow(M, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m_1 <= 2d+14) then
        tmp = cos(m_1) * exp((((n - m) - l) + (((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n))))
    else
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= 2e+14) {
		tmp = Math.cos(M) * Math.exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))));
	} else {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= 2e+14:
		tmp = math.cos(M) * math.exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))))
	else:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= 2e+14)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(n - m) - l) + Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)))));
	else
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= 2e+14)
		tmp = cos(M) * exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))));
	else
		tmp = cos(M) * exp(-(M ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, 2e+14], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(n - m), $MachinePrecision] - l), $MachinePrecision] + N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2e14

   1. Initial program 71.3%

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

      1. associate-/l* 71.3%

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

      2. +-commutative 71.3%

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

      3. fabs-sub 71.3%

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

      4. +-commutative 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in K around 0 94.3%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   5. Step-by-step derivation

      1. cos-neg 94.3%

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

   6. Simplified 94.3%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   7. Step-by-step derivation

      1. sub-neg 94.3%

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

      2. distribute-neg-out 94.3%

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

      3. div-inv 94.3%

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

      4. fma-neg 94.3%

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

      5. metadata-eval 94.3%

         \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2} + \left(\ell - \left|n - m\right|\right)\right)} \]

      6. add-sqr-sqrt 52.4%

         \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right|\right)\right)} \]

      7. fabs-sqr 52.4%

         \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right)\right)} \]

      8. add-sqr-sqrt 94.2%

         \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \color{blue}{\left(n - m\right)}\right)\right)} \]

   8. Applied egg-rr 94.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left(n - m\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. neg-sub0 94.2%

         \[\leadsto \cos M \cdot e^{\color{blue}{0 - \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left(n - m\right)\right)\right)}} \]

      2. +-commutative 94.2%

         \[\leadsto \cos M \cdot e^{0 - \color{blue}{\left(\left(\ell - \left(n - m\right)\right) + {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}\right)}} \]

      3. associate--r+ 94.2%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(0 - \left(\ell - \left(n - m\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}} \]

      4. sub-neg 94.2%

         \[\leadsto \cos M \cdot e^{\left(0 - \color{blue}{\left(\ell + \left(-\left(n - m\right)\right)\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      5. associate--r+ 94.2%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left(0 - \ell\right) - \left(-\left(n - m\right)\right)\right)} - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      6. neg-sub0 94.2%

         \[\leadsto \cos M \cdot e^{\left(\color{blue}{\left(-\ell\right)} - \left(-\left(n - m\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      7. sub-neg 94.2%

         \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\color{blue}{\left(n + \left(-m\right)\right)}\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      8. mul-1-neg 94.2%

         \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\left(n + \color{blue}{-1 \cdot m}\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      9. +-commutative 94.2%

         \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\color{blue}{\left(-1 \cdot m + n\right)}\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      10. distribute-neg-in 94.2%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \color{blue}{\left(\left(--1 \cdot m\right) + \left(-n\right)\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      11. mul-1-neg 94.2%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(\left(-\color{blue}{\left(-m\right)}\right) + \left(-n\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      12. remove-double-neg 94.2%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(\color{blue}{m} + \left(-n\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      13. sub-neg 94.2%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \color{blue}{\left(m - n\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

      14. fma-neg 94.2%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - {\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)}}^{2}} \]

   10. Simplified 94.2%

       \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left(-\ell\right) - \left(m - n\right)\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}} \]

   11. Taylor expanded in n around 0 76.0%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}} \]
   12. Step-by-step derivation

       1. +-commutative 76.0%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}} \]

       2. unpow2 76.0%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)} \]

       3. distribute-rgt-out 80.3%

          \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   13. Simplified 80.3%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   if 2e14 < M

   1. Initial program 61.4%

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

      1. associate-/l* 61.4%

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

      2. +-commutative 61.4%

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

      3. fabs-sub 61.4%

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

      4. +-commutative 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
   5. Step-by-step derivation

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around inf 100.0%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   9. Simplified 100.0%

      \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\cos M \cdot e^{\left(\left(n - m\right) - \ell\right) + \left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \]

Alternative 6: 82.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \cos M \cdot e^{\left(\left(n - m\right) - \ell\right) + \left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (+ (- (- n m) l) (* (- (* m 0.5) M) (- (- M (* m 0.5)) n))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp((((n - m) - l) + (((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))));
}

Python

def code(K, m, n, M, l):
	return math.cos(M) * math.exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(Float64(Float64(n - m) - l) + Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp((((n - m) - l) + (((m * 0.5) - M) * ((M - (m * 0.5)) - n))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(n - m), $MachinePrecision] - l), $MachinePrecision] + N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.6%

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

   1. associate-/l* 69.6%

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

   2. +-commutative 69.6%

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

   3. fabs-sub 69.6%

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

   4. +-commutative 69.6%

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

3. Simplified 69.6%

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

4. Taylor expanded in K around 0 95.3%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
5. Step-by-step derivation

   1. cos-neg 95.3%

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

6. Simplified 95.3%

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \]
7. Step-by-step derivation

   1. sub-neg 95.3%

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

   2. distribute-neg-out 95.3%

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

   3. div-inv 95.3%

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

   4. fma-neg 95.3%

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

   5. metadata-eval 95.3%

      \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2} + \left(\ell - \left|n - m\right|\right)\right)} \]

   6. add-sqr-sqrt 50.8%

      \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right|\right)\right)} \]

   7. fabs-sqr 50.8%

      \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right)\right)} \]

   8. add-sqr-sqrt 95.2%

      \[\leadsto \cos M \cdot e^{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \color{blue}{\left(n - m\right)}\right)\right)} \]

8. Applied egg-rr 95.2%

   \[\leadsto \cos M \cdot e^{\color{blue}{-\left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left(n - m\right)\right)\right)}} \]
9. Step-by-step derivation

   1. neg-sub0 95.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{0 - \left({\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2} + \left(\ell - \left(n - m\right)\right)\right)}} \]

   2. +-commutative 95.2%

      \[\leadsto \cos M \cdot e^{0 - \color{blue}{\left(\left(\ell - \left(n - m\right)\right) + {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}\right)}} \]

   3. associate--r+ 95.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{\left(0 - \left(\ell - \left(n - m\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}} \]

   4. sub-neg 95.2%

      \[\leadsto \cos M \cdot e^{\left(0 - \color{blue}{\left(\ell + \left(-\left(n - m\right)\right)\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   5. associate--r+ 95.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left(0 - \ell\right) - \left(-\left(n - m\right)\right)\right)} - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   6. neg-sub0 95.2%

      \[\leadsto \cos M \cdot e^{\left(\color{blue}{\left(-\ell\right)} - \left(-\left(n - m\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   7. sub-neg 95.2%

      \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\color{blue}{\left(n + \left(-m\right)\right)}\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   8. mul-1-neg 95.2%

      \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\left(n + \color{blue}{-1 \cdot m}\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   9. +-commutative 95.2%

      \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(-\color{blue}{\left(-1 \cdot m + n\right)}\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   10. distribute-neg-in 95.2%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \color{blue}{\left(\left(--1 \cdot m\right) + \left(-n\right)\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   11. mul-1-neg 95.2%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(\left(-\color{blue}{\left(-m\right)}\right) + \left(-n\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   12. remove-double-neg 95.2%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(\color{blue}{m} + \left(-n\right)\right)\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   13. sub-neg 95.2%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \color{blue}{\left(m - n\right)}\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}} \]

   14. fma-neg 95.2%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - {\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right)}}^{2}} \]

10. Simplified 95.2%

    \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left(-\ell\right) - \left(m - n\right)\right) - {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}}} \]

11. Taylor expanded in n around 0 77.0%

    \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}} \]
12. Step-by-step derivation

    1. +-commutative 77.0%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}} \]

    2. unpow2 77.0%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)} \]

    3. distribute-rgt-out 82.9%

       \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

13. Simplified 82.9%

    \[\leadsto \cos M \cdot e^{\left(\left(-\ell\right) - \left(m - n\right)\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

14. Final simplification 82.9%

    \[\leadsto \cos M \cdot e^{\left(\left(n - m\right) - \ell\right) + \left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right)} \]

Alternative 7: 35.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \cos M \cdot e^{-\ell} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (* (cos M) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp(-l);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp(-l)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.exp(-l);
}

Python

def code(K, m, n, M, l):
	return math.cos(M) * math.exp(-l)

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(-l)))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp(-l);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.6%

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

   1. associate-/l* 69.6%

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

   2. +-commutative 69.6%

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

   3. fabs-sub 69.6%

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

   4. +-commutative 69.6%

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

3. Simplified 69.6%

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

4. Taylor expanded in l around inf 28.6%

   \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
5. Step-by-step derivation

   1. mul-1-neg 28.6%

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

6. Simplified 28.6%

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

7. Taylor expanded in K around 0 35.4%

   \[\leadsto \color{blue}{\cos \left(-M\right) \cdot e^{-\ell}} \]
8. Step-by-step derivation

   1. cos-neg 35.4%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]

9. Simplified 35.4%

   \[\leadsto \color{blue}{\cos M \cdot e^{-\ell}} \]

10. Final simplification 35.4%

    \[\leadsto \cos M \cdot e^{-\ell} \]

Alternative 8: 6.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \cos M \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (cos M))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M);
}

Python

def code(K, m, n, M, l):
	return math.cos(M)

Julia

function code(K, m, n, M, l)
	return cos(M)
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Cos[M], $MachinePrecision]

Derivation

1. Initial program 69.6%

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

   1. associate-/l* 69.6%

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

   2. +-commutative 69.6%

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

   3. fabs-sub 69.6%

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

   4. +-commutative 69.6%

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

3. Simplified 69.6%

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

4. Taylor expanded in l around inf 28.6%

   \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
5. Step-by-step derivation

   1. mul-1-neg 28.6%

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

6. Simplified 28.6%

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

7. Taylor expanded in l around 0 6.1%

   \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right) - M\right)} \]

8. Taylor expanded in K around 0 6.9%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \]
9. Step-by-step derivation

   1. cos-neg 6.9%

      \[\leadsto \color{blue}{\cos M} \]

10. Simplified 6.9%

    \[\leadsto \color{blue}{\cos M} \]

11. Final simplification 6.9%

    \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023319 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))