Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 97.0%

Time: 29.5s

Alternatives: 11

Speedup: 1.4×

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: 75.9% 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: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) / exp((pow((((m + n) / 2.0) - M), 2.0) + (l - fabs((n - 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) / exp((((((m + n) / 2.0d0) - m_1) ** 2.0d0) + (l - abs((n - m)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

3. Simplified 76.6%

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

4. Taylor expanded in K around 0 96.4%

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

   1. cos-neg 96.4%

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

6. Simplified 96.4%

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

7. Final simplification 96.4%

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

Alternative 2: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) / exp(fabs((((m + l) + pow(((0.5 * (m + n)) - M), 2.0)) - 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(abs((((m + l) + (((0.5d0 * (m + n)) - m_1) ** 2.0d0)) - n)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

3. Simplified 76.6%

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

   1. add-sqr-sqrt 68.0%

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

   2. sqrt-unprod 72.2%

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

   3. pow2 72.2%

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

5. Applied egg-rr 72.2%

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

   1. unpow2 72.2%

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

   2. rem-sqrt-square 72.2%

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

   3. associate-+r- 72.2%

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

   4. sub-neg 72.2%

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

   5. +-commutative 72.2%

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

   6. associate-+l+ 72.2%

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

   7. +-commutative 72.2%

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

   8. sub-neg 72.2%

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

   9. mul-1-neg 72.2%

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

   10. +-commutative 72.2%

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

   11. distribute-neg-in 72.2%

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

   12. mul-1-neg 72.2%

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

   13. remove-double-neg 72.2%

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

   14. sub-neg 72.2%

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

7. Simplified 72.2%

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

8. Taylor expanded in K around 0 91.4%

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

   1. cos-neg 91.4%

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

   2. fabs-sub 91.4%

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

   3. associate--r+ 91.4%

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

   4. unsub-neg 91.4%

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

   5. mul-1-neg 91.4%

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

   6. +-commutative 91.4%

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

   7. fabs-sub 91.4%

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

   8. associate--r+ 91.4%

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

   9. fabs-sub 91.4%

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

   10. cancel-sign-sub-inv 91.4%

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

   11. metadata-eval 91.4%

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

   12. *-lft-identity 91.4%

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

10. Simplified 91.4%

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

11. Final simplification 91.4%

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

Alternative 3: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;\frac{\cos M}{e^{\left|n - 0.25 \cdot \left(m \cdot m\right)\right|}}\\ \mathbf{elif}\;m \leq 8.5 \cdot 10^{-253}:\\ \;\;\;\;\frac{\cos M}{e^{\left|M \cdot M - n\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - \left|n - m\right|\right) + 0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -54.0)
   (/ (cos M) (exp (fabs (- n (* 0.25 (* m m))))))
   (if (<= m 8.5e-253)
     (/ (cos M) (exp (fabs (- (* M M) n))))
     (/ (cos M) (exp (+ (- l (fabs (- n m))) (* 0.25 (* n n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -54.0) {
		tmp = cos(M) / exp(fabs((n - (0.25 * (m * m)))));
	} else if (m <= 8.5e-253) {
		tmp = cos(M) / exp(fabs(((M * M) - n)));
	} else {
		tmp = cos(M) / exp(((l - fabs((n - m))) + (0.25 * (n * 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 <= (-54.0d0)) then
        tmp = cos(m_1) / exp(abs((n - (0.25d0 * (m * m)))))
    else if (m <= 8.5d-253) then
        tmp = cos(m_1) / exp(abs(((m_1 * m_1) - n)))
    else
        tmp = cos(m_1) / exp(((l - abs((n - m))) + (0.25d0 * (n * 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 <= -54.0) {
		tmp = Math.cos(M) / Math.exp(Math.abs((n - (0.25 * (m * m)))));
	} else if (m <= 8.5e-253) {
		tmp = Math.cos(M) / Math.exp(Math.abs(((M * M) - n)));
	} else {
		tmp = Math.cos(M) / Math.exp(((l - Math.abs((n - m))) + (0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -54.0:
		tmp = math.cos(M) / math.exp(math.fabs((n - (0.25 * (m * m)))))
	elif m <= 8.5e-253:
		tmp = math.cos(M) / math.exp(math.fabs(((M * M) - n)))
	else:
		tmp = math.cos(M) / math.exp(((l - math.fabs((n - m))) + (0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -54.0)
		tmp = Float64(cos(M) / exp(abs(Float64(n - Float64(0.25 * Float64(m * m))))));
	elseif (m <= 8.5e-253)
		tmp = Float64(cos(M) / exp(abs(Float64(Float64(M * M) - n))));
	else
		tmp = Float64(cos(M) / exp(Float64(Float64(l - abs(Float64(n - m))) + Float64(0.25 * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -54.0)
		tmp = cos(M) / exp(abs((n - (0.25 * (m * m)))));
	elseif (m <= 8.5e-253)
		tmp = cos(M) / exp(abs(((M * M) - n)));
	else
		tmp = cos(M) / exp(((l - abs((n - m))) + (0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -54.0], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[Abs[N[(n - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 8.5e-253], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[Abs[N[(N[(M * M), $MachinePrecision] - n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[(l - N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -54

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

      1. add-sqr-sqrt 67.2%

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

      2. sqrt-unprod 67.3%

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

      3. pow2 67.3%

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

   5. Applied egg-rr 67.3%

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

      1. unpow2 67.3%

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

      2. rem-sqrt-square 67.3%

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

      3. associate-+r- 67.3%

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

      4. sub-neg 67.3%

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

      5. +-commutative 67.3%

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

      6. associate-+l+ 67.3%

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

      7. +-commutative 67.3%

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

      8. sub-neg 67.3%

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

      9. mul-1-neg 67.3%

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

      10. +-commutative 67.3%

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

      11. distribute-neg-in 67.3%

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

      12. mul-1-neg 67.3%

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

      13. remove-double-neg 67.3%

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

      14. sub-neg 67.3%

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

   7. Simplified 67.3%

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

   8. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

      2. fabs-sub 96.8%

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

      3. associate--r+ 96.8%

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

      4. unsub-neg 96.8%

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

      5. mul-1-neg 96.8%

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

      6. +-commutative 96.8%

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

      7. fabs-sub 96.8%

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

      8. associate--r+ 96.8%

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

      9. fabs-sub 96.8%

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

      10. cancel-sign-sub-inv 96.8%

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

      11. metadata-eval 96.8%

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

      12. *-lft-identity 96.8%

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

   10. Simplified 96.8%

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

   11. Taylor expanded in m around inf 96.8%

       \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{0.25 \cdot {m}^{2}}\right|}} \]
   12. Step-by-step derivation

       1. unpow2 96.8%

          \[\leadsto \frac{\cos M}{e^{\left|n - 0.25 \cdot \color{blue}{\left(m \cdot m\right)}\right|}} \]

   13. Simplified 96.8%

       \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{0.25 \cdot \left(m \cdot m\right)}\right|}} \]

   if -54 < m < 8.4999999999999999e-253

   1. Initial program 81.8%

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

   3. Simplified 81.8%

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

      1. add-sqr-sqrt 64.8%

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

      2. sqrt-unprod 74.9%

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

      3. pow2 74.9%

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

   5. Applied egg-rr 74.9%

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

      1. unpow2 74.9%

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

      2. rem-sqrt-square 74.9%

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

      3. associate-+r- 74.9%

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

      4. sub-neg 74.9%

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

      5. +-commutative 74.9%

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

      6. associate-+l+ 74.9%

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

      7. +-commutative 74.9%

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

      8. sub-neg 74.9%

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

      9. mul-1-neg 74.9%

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

      10. +-commutative 74.9%

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

      11. distribute-neg-in 74.9%

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

      12. mul-1-neg 74.9%

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

      13. remove-double-neg 74.9%

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

      14. sub-neg 74.9%

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

   7. Simplified 74.9%

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

   8. Taylor expanded in K around 0 85.0%

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

      1. cos-neg 85.0%

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

      2. fabs-sub 85.0%

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

      3. associate--r+ 85.0%

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

      4. unsub-neg 85.0%

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

      5. mul-1-neg 85.0%

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

      6. +-commutative 85.0%

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

      7. fabs-sub 85.0%

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

      8. associate--r+ 85.0%

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

      9. fabs-sub 85.0%

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

      10. cancel-sign-sub-inv 85.0%

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

      11. metadata-eval 85.0%

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

      12. *-lft-identity 85.0%

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

   10. Simplified 85.0%

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

   11. Taylor expanded in M around inf 83.8%

       \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{{M}^{2}}\right|}} \]
   12. Step-by-step derivation

       1. unpow2 83.8%

          \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{M \cdot M}\right|}} \]

   13. Simplified 83.8%

       \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{M \cdot M}\right|}} \]

   if 8.4999999999999999e-253 < m

   1. Initial program 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in K around 0 97.1%

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

      1. cos-neg 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in n around inf 59.9%

      \[\leadsto \frac{\cos M}{e^{\color{blue}{0.25 \cdot {n}^{2}} + \left(\ell - \left|n - m\right|\right)}} \]
   8. Step-by-step derivation

      1. *-commutative 48.7%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{{n}^{2} \cdot 0.25} + \left(\ell - \left|n - m\right|\right)}} \]

      2. unpow2 48.7%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(n \cdot n\right)} \cdot 0.25 + \left(\ell - \left|n - m\right|\right)}} \]

   9. Simplified 59.9%

      \[\leadsto \frac{\cos M}{e^{\color{blue}{\left(n \cdot n\right) \cdot 0.25} + \left(\ell - \left|n - m\right|\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;\frac{\cos M}{e^{\left|n - 0.25 \cdot \left(m \cdot m\right)\right|}}\\ \mathbf{elif}\;m \leq 8.5 \cdot 10^{-253}:\\ \;\;\;\;\frac{\cos M}{e^{\left|M \cdot M - n\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - \left|n - m\right|\right) + 0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \]

Alternative 4: 85.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.35 \cdot 10^{-14} \lor \neg \left(M \leq 26.5\right):\\ \;\;\;\;\frac{\cos M}{e^{\left|M \cdot M - n\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\left|n - \ell\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1.35e-14) (not (<= M 26.5)))
   (/ (cos M) (exp (fabs (- (* M M) n))))
   (/ (cos M) (exp (fabs (- n l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.35e-14) || !(M <= 26.5)) {
		tmp = cos(M) / exp(fabs(((M * M) - n)));
	} else {
		tmp = cos(M) / exp(fabs((n - l)));
	}
	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 <= (-1.35d-14)) .or. (.not. (m_1 <= 26.5d0))) then
        tmp = cos(m_1) / exp(abs(((m_1 * m_1) - n)))
    else
        tmp = cos(m_1) / exp(abs((n - l)))
    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 <= -1.35e-14) || !(M <= 26.5)) {
		tmp = Math.cos(M) / Math.exp(Math.abs(((M * M) - n)));
	} else {
		tmp = Math.cos(M) / Math.exp(Math.abs((n - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1.35e-14) or not (M <= 26.5):
		tmp = math.cos(M) / math.exp(math.fabs(((M * M) - n)))
	else:
		tmp = math.cos(M) / math.exp(math.fabs((n - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1.35e-14) || !(M <= 26.5))
		tmp = Float64(cos(M) / exp(abs(Float64(Float64(M * M) - n))));
	else
		tmp = Float64(cos(M) / exp(abs(Float64(n - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -1.35e-14) || ~((M <= 26.5)))
		tmp = cos(M) / exp(abs(((M * M) - n)));
	else
		tmp = cos(M) / exp(abs((n - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1.35e-14], N[Not[LessEqual[M, 26.5]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[Abs[N[(N[(M * M), $MachinePrecision] - n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[Abs[N[(n - l), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1.3499999999999999e-14 or 26.5 < M

   1. Initial program 78.4%

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

   3. Simplified 78.4%

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

      1. add-sqr-sqrt 77.6%

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

      2. sqrt-unprod 77.7%

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

      3. pow2 77.7%

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

   5. Applied egg-rr 77.7%

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

      1. unpow2 77.7%

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

      2. rem-sqrt-square 77.7%

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

      3. associate-+r- 77.7%

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

      4. sub-neg 77.7%

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

      5. +-commutative 77.7%

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

      6. associate-+l+ 77.7%

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

      7. +-commutative 77.7%

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

      8. sub-neg 77.7%

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

      9. mul-1-neg 77.7%

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

      10. +-commutative 77.7%

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

      11. distribute-neg-in 77.7%

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

      12. mul-1-neg 77.7%

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

      13. remove-double-neg 77.7%

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

      14. sub-neg 77.7%

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

   7. Simplified 77.7%

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

   8. Taylor expanded in K around 0 96.3%

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

      1. cos-neg 96.3%

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

      2. fabs-sub 96.3%

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

      3. associate--r+ 96.3%

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

      4. unsub-neg 96.3%

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

      5. mul-1-neg 96.3%

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

      6. +-commutative 96.3%

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

      7. fabs-sub 96.3%

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

      8. associate--r+ 96.3%

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

      9. fabs-sub 96.3%

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

      10. cancel-sign-sub-inv 96.3%

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

      11. metadata-eval 96.3%

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

      12. *-lft-identity 96.3%

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

   10. Simplified 96.3%

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

   11. Taylor expanded in M around inf 96.3%

       \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{{M}^{2}}\right|}} \]
   12. Step-by-step derivation

       1. unpow2 96.3%

          \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{M \cdot M}\right|}} \]

   13. Simplified 96.3%

       \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{M \cdot M}\right|}} \]

   if -1.3499999999999999e-14 < M < 26.5

   1. Initial program 74.7%

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

   3. Simplified 74.7%

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

      1. add-sqr-sqrt 57.4%

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

      2. sqrt-unprod 66.2%

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

      3. pow2 66.2%

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

   5. Applied egg-rr 66.2%

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

      1. unpow2 66.2%

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

      2. rem-sqrt-square 66.2%

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

      3. associate-+r- 66.2%

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

      4. sub-neg 66.2%

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

      5. +-commutative 66.2%

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

      6. associate-+l+ 66.2%

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

      7. +-commutative 66.2%

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

      8. sub-neg 66.2%

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

      9. mul-1-neg 66.2%

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

      10. +-commutative 66.2%

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

      11. distribute-neg-in 66.2%

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

      12. mul-1-neg 66.2%

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

      13. remove-double-neg 66.2%

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

      14. sub-neg 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in K around 0 85.9%

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

      1. cos-neg 85.9%

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

      2. fabs-sub 85.9%

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

      3. associate--r+ 85.9%

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

      4. unsub-neg 85.9%

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

      5. mul-1-neg 85.9%

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

      6. +-commutative 85.9%

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

      7. fabs-sub 85.9%

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

      8. associate--r+ 85.9%

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

      9. fabs-sub 85.9%

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

      10. cancel-sign-sub-inv 85.9%

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

      11. metadata-eval 85.9%

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

      12. *-lft-identity 85.9%

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

   10. Simplified 85.9%

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

   11. Taylor expanded in l around inf 70.8%

       \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{\ell}\right|}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.35 \cdot 10^{-14} \lor \neg \left(M \leq 26.5\right):\\ \;\;\;\;\frac{\cos M}{e^{\left|M \cdot M - n\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\left|n - \ell\right|}}\\ \end{array} \]

Alternative 5: 82.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;\frac{\cos M}{e^{\left|n - 0.25 \cdot \left(m \cdot m\right)\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\left|M \cdot M - n\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -54.0)
   (/ (cos M) (exp (fabs (- n (* 0.25 (* m m))))))
   (/ (cos M) (exp (fabs (- (* M M) n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -54.0) {
		tmp = cos(M) / exp(fabs((n - (0.25 * (m * m)))));
	} else {
		tmp = cos(M) / exp(fabs(((M * M) - 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 <= (-54.0d0)) then
        tmp = cos(m_1) / exp(abs((n - (0.25d0 * (m * m)))))
    else
        tmp = cos(m_1) / exp(abs(((m_1 * m_1) - 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 <= -54.0) {
		tmp = Math.cos(M) / Math.exp(Math.abs((n - (0.25 * (m * m)))));
	} else {
		tmp = Math.cos(M) / Math.exp(Math.abs(((M * M) - n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -54.0:
		tmp = math.cos(M) / math.exp(math.fabs((n - (0.25 * (m * m)))))
	else:
		tmp = math.cos(M) / math.exp(math.fabs(((M * M) - n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -54.0)
		tmp = Float64(cos(M) / exp(abs(Float64(n - Float64(0.25 * Float64(m * m))))));
	else
		tmp = Float64(cos(M) / exp(abs(Float64(Float64(M * M) - n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -54.0)
		tmp = cos(M) / exp(abs((n - (0.25 * (m * m)))));
	else
		tmp = cos(M) / exp(abs(((M * M) - n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -54.0], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[Abs[N[(n - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[Abs[N[(N[(M * M), $MachinePrecision] - n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -54

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

      1. add-sqr-sqrt 67.2%

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

      2. sqrt-unprod 67.3%

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

      3. pow2 67.3%

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

   5. Applied egg-rr 67.3%

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

      1. unpow2 67.3%

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

      2. rem-sqrt-square 67.3%

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

      3. associate-+r- 67.3%

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

      4. sub-neg 67.3%

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

      5. +-commutative 67.3%

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

      6. associate-+l+ 67.3%

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

      7. +-commutative 67.3%

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

      8. sub-neg 67.3%

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

      9. mul-1-neg 67.3%

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

      10. +-commutative 67.3%

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

      11. distribute-neg-in 67.3%

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

      12. mul-1-neg 67.3%

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

      13. remove-double-neg 67.3%

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

      14. sub-neg 67.3%

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

   7. Simplified 67.3%

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

   8. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

      2. fabs-sub 96.8%

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

      3. associate--r+ 96.8%

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

      4. unsub-neg 96.8%

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

      5. mul-1-neg 96.8%

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

      6. +-commutative 96.8%

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

      7. fabs-sub 96.8%

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

      8. associate--r+ 96.8%

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

      9. fabs-sub 96.8%

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

      10. cancel-sign-sub-inv 96.8%

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

      11. metadata-eval 96.8%

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

      12. *-lft-identity 96.8%

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

   10. Simplified 96.8%

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

   11. Taylor expanded in m around inf 96.8%

       \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{0.25 \cdot {m}^{2}}\right|}} \]
   12. Step-by-step derivation

       1. unpow2 96.8%

          \[\leadsto \frac{\cos M}{e^{\left|n - 0.25 \cdot \color{blue}{\left(m \cdot m\right)}\right|}} \]

   13. Simplified 96.8%

       \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{0.25 \cdot \left(m \cdot m\right)}\right|}} \]

   if -54 < m

   1. Initial program 78.6%

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

   3. Simplified 78.6%

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

      1. add-sqr-sqrt 68.2%

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

      2. sqrt-unprod 73.7%

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

      3. pow2 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. unpow2 73.7%

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

      2. rem-sqrt-square 73.7%

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

      3. associate-+r- 73.7%

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

      4. sub-neg 73.7%

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

      5. +-commutative 73.7%

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

      6. associate-+l+ 73.7%

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

      7. +-commutative 73.7%

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

      8. sub-neg 73.7%

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

      9. mul-1-neg 73.7%

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

      10. +-commutative 73.7%

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

      11. distribute-neg-in 73.7%

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

      12. mul-1-neg 73.7%

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

      13. remove-double-neg 73.7%

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

      14. sub-neg 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in K around 0 89.7%

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

      1. cos-neg 89.7%

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

      2. fabs-sub 89.7%

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

      3. associate--r+ 89.7%

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

      4. unsub-neg 89.7%

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

      5. mul-1-neg 89.7%

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

      6. +-commutative 89.7%

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

      7. fabs-sub 89.7%

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

      8. associate--r+ 89.7%

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

      9. fabs-sub 89.7%

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

      10. cancel-sign-sub-inv 89.7%

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

      11. metadata-eval 89.7%

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

      12. *-lft-identity 89.7%

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

   10. Simplified 89.7%

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

   11. Taylor expanded in M around inf 80.8%

       \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{{M}^{2}}\right|}} \]
   12. Step-by-step derivation

       1. unpow2 80.8%

          \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{M \cdot M}\right|}} \]

   13. Simplified 80.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;\frac{\cos M}{e^{\left|n - 0.25 \cdot \left(m \cdot m\right)\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\left|M \cdot M - n\right|}}\\ \end{array} \]

Alternative 6: 72.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\cos M}{e^{\left|n - \ell\right|}} \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) / exp(fabs((n - 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(abs((n - l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

3. Simplified 76.6%

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

   1. add-sqr-sqrt 68.0%

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

   2. sqrt-unprod 72.2%

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

   3. pow2 72.2%

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

5. Applied egg-rr 72.2%

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

   1. unpow2 72.2%

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

   2. rem-sqrt-square 72.2%

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

   3. associate-+r- 72.2%

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

   4. sub-neg 72.2%

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

   5. +-commutative 72.2%

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

   6. associate-+l+ 72.2%

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

   7. +-commutative 72.2%

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

   8. sub-neg 72.2%

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

   9. mul-1-neg 72.2%

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

   10. +-commutative 72.2%

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

   11. distribute-neg-in 72.2%

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

   12. mul-1-neg 72.2%

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

   13. remove-double-neg 72.2%

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

   14. sub-neg 72.2%

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

7. Simplified 72.2%

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

8. Taylor expanded in K around 0 91.4%

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

   1. cos-neg 91.4%

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

   2. fabs-sub 91.4%

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

   3. associate--r+ 91.4%

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

   4. unsub-neg 91.4%

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

   5. mul-1-neg 91.4%

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

   6. +-commutative 91.4%

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

   7. fabs-sub 91.4%

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

   8. associate--r+ 91.4%

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

   9. fabs-sub 91.4%

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

   10. cancel-sign-sub-inv 91.4%

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

   11. metadata-eval 91.4%

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

   12. *-lft-identity 91.4%

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

10. Simplified 91.4%

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

11. Taylor expanded in l around inf 73.6%

    \[\leadsto \frac{\cos M}{e^{\left|n - \color{blue}{\ell}\right|}} \]

12. Final simplification 73.6%

    \[\leadsto \frac{\cos M}{e^{\left|n - \ell\right|}} \]

Alternative 7: 54.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos \left(\left(n \cdot 0.5\right) \cdot K - M\right)}{e^{n \cdot \left(n \cdot 0.25\right)}}\\ \mathbf{if}\;n \leq -0.115:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-231}:\\ \;\;\;\;-0.5 \cdot \frac{n \cdot \left(K \cdot \sin \left(0.5 \cdot \left(m \cdot K\right) - M\right)\right)}{e^{\ell}}\\ \mathbf{elif}\;n \leq 0.00068:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (/ (cos (- (* (* n 0.5) K) M)) (exp (* n (* n 0.25))))))
   (if (<= n -0.115)
     t_0
     (if (<= n 3.4e-231)
       (* -0.5 (/ (* n (* K (sin (- (* 0.5 (* m K)) M)))) (exp l)))
       (if (<= n 0.00068) (/ (cos M) (exp l)) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos((((n * 0.5) * K) - M)) / exp((n * (n * 0.25)));
	double tmp;
	if (n <= -0.115) {
		tmp = t_0;
	} else if (n <= 3.4e-231) {
		tmp = -0.5 * ((n * (K * sin(((0.5 * (m * K)) - M)))) / exp(l));
	} else if (n <= 0.00068) {
		tmp = cos(M) / exp(l);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = cos((((n * 0.5d0) * k) - m_1)) / exp((n * (n * 0.25d0)))
    if (n <= (-0.115d0)) then
        tmp = t_0
    else if (n <= 3.4d-231) then
        tmp = (-0.5d0) * ((n * (k * sin(((0.5d0 * (m * k)) - m_1)))) / exp(l))
    else if (n <= 0.00068d0) then
        tmp = cos(m_1) / exp(l)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos((((n * 0.5) * K) - M)) / Math.exp((n * (n * 0.25)));
	double tmp;
	if (n <= -0.115) {
		tmp = t_0;
	} else if (n <= 3.4e-231) {
		tmp = -0.5 * ((n * (K * Math.sin(((0.5 * (m * K)) - M)))) / Math.exp(l));
	} else if (n <= 0.00068) {
		tmp = Math.cos(M) / Math.exp(l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos((((n * 0.5) * K) - M)) / math.exp((n * (n * 0.25)))
	tmp = 0
	if n <= -0.115:
		tmp = t_0
	elif n <= 3.4e-231:
		tmp = -0.5 * ((n * (K * math.sin(((0.5 * (m * K)) - M)))) / math.exp(l))
	elif n <= 0.00068:
		tmp = math.cos(M) / math.exp(l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(Float64(Float64(Float64(n * 0.5) * K) - M)) / exp(Float64(n * Float64(n * 0.25))))
	tmp = 0.0
	if (n <= -0.115)
		tmp = t_0;
	elseif (n <= 3.4e-231)
		tmp = Float64(-0.5 * Float64(Float64(n * Float64(K * sin(Float64(Float64(0.5 * Float64(m * K)) - M)))) / exp(l)));
	elseif (n <= 0.00068)
		tmp = Float64(cos(M) / exp(l));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos((((n * 0.5) * K) - M)) / exp((n * (n * 0.25)));
	tmp = 0.0;
	if (n <= -0.115)
		tmp = t_0;
	elseif (n <= 3.4e-231)
		tmp = -0.5 * ((n * (K * sin(((0.5 * (m * K)) - M)))) / exp(l));
	elseif (n <= 0.00068)
		tmp = cos(M) / exp(l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[N[(N[(N[(n * 0.5), $MachinePrecision] * K), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(n * N[(n * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -0.115], t$95$0, If[LessEqual[n, 3.4e-231], N[(-0.5 * N[(N[(n * N[(K * N[Sin[N[(N[(0.5 * N[(m * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.00068], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -0.115000000000000005 or 6.8e-4 < n

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in n around inf 64.2%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{0.25 \cdot {n}^{2}} + \left(\ell - \left|n - m\right|\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 64.2%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{{n}^{2} \cdot 0.25} + \left(\ell - \left|n - m\right|\right)}} \]

      2. unpow2 64.2%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(n \cdot n\right)} \cdot 0.25 + \left(\ell - \left|n - m\right|\right)}} \]

   6. Simplified 64.2%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(n \cdot n\right) \cdot 0.25} + \left(\ell - \left|n - m\right|\right)}} \]

   7. Taylor expanded in m around 0 66.0%

      \[\leadsto \frac{\cos \left(\color{blue}{0.5 \cdot \left(n \cdot K\right)} - M\right)}{e^{\left(n \cdot n\right) \cdot 0.25 + \left(\ell - \left|n - m\right|\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 66.0%

         \[\leadsto \frac{\cos \left(\color{blue}{\left(0.5 \cdot n\right) \cdot K} - M\right)}{e^{\left(n \cdot n\right) \cdot 0.25 + \left(\ell - \left|n - m\right|\right)}} \]

   9. Simplified 66.0%

      \[\leadsto \frac{\cos \left(\color{blue}{\left(0.5 \cdot n\right) \cdot K} - M\right)}{e^{\left(n \cdot n\right) \cdot 0.25 + \left(\ell - \left|n - m\right|\right)}} \]

   10. Taylor expanded in n around inf 76.1%

       \[\leadsto \frac{\cos \left(\left(0.5 \cdot n\right) \cdot K - M\right)}{e^{\color{blue}{0.25 \cdot {n}^{2}}}} \]
   11. Step-by-step derivation

       1. *-commutative 76.1%

          \[\leadsto \frac{\cos \left(\left(0.5 \cdot n\right) \cdot K - M\right)}{e^{\color{blue}{{n}^{2} \cdot 0.25}}} \]

       2. unpow2 76.1%

          \[\leadsto \frac{\cos \left(\left(0.5 \cdot n\right) \cdot K - M\right)}{e^{\color{blue}{\left(n \cdot n\right)} \cdot 0.25}} \]

       3. associate-*r* 76.1%

          \[\leadsto \frac{\cos \left(\left(0.5 \cdot n\right) \cdot K - M\right)}{e^{\color{blue}{n \cdot \left(n \cdot 0.25\right)}}} \]

   12. Simplified 76.1%

       \[\leadsto \frac{\cos \left(\left(0.5 \cdot n\right) \cdot K - M\right)}{e^{\color{blue}{n \cdot \left(n \cdot 0.25\right)}}} \]

   if -0.115000000000000005 < n < 3.4e-231

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in l around inf 32.1%

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

   5. Taylor expanded in n around 0 32.1%

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

   6. Taylor expanded in n around inf 38.4%

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

   if 3.4e-231 < n < 6.8e-4

   1. Initial program 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in l around inf 47.3%

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

   5. Taylor expanded in K around 0 55.4%

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

      1. cos-neg 55.4%

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

   7. Simplified 55.4%

      \[\leadsto \color{blue}{\frac{\cos M}{e^{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.115:\\ \;\;\;\;\frac{\cos \left(\left(n \cdot 0.5\right) \cdot K - M\right)}{e^{n \cdot \left(n \cdot 0.25\right)}}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-231}:\\ \;\;\;\;-0.5 \cdot \frac{n \cdot \left(K \cdot \sin \left(0.5 \cdot \left(m \cdot K\right) - M\right)\right)}{e^{\ell}}\\ \mathbf{elif}\;n \leq 0.00068:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left(n \cdot 0.5\right) \cdot K - M\right)}{e^{n \cdot \left(n \cdot 0.25\right)}}\\ \end{array} \]

Alternative 8: 57.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -53 \lor \neg \left(n \leq 0.00068\right):\\ \;\;\;\;\frac{\cos \left(\left(n \cdot 0.5\right) \cdot K - M\right)}{e^{n \cdot \left(n \cdot 0.25\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -53.0) (not (<= n 0.00068)))
   (/ (cos (- (* (* n 0.5) K) M)) (exp (* n (* n 0.25))))
   (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -53.0) || !(n <= 0.00068)) {
		tmp = cos((((n * 0.5) * K) - M)) / exp((n * (n * 0.25)));
	} else {
		tmp = cos(M) / exp(l);
	}
	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 <= (-53.0d0)) .or. (.not. (n <= 0.00068d0))) then
        tmp = cos((((n * 0.5d0) * k) - m_1)) / exp((n * (n * 0.25d0)))
    else
        tmp = cos(m_1) / exp(l)
    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 <= -53.0) || !(n <= 0.00068)) {
		tmp = Math.cos((((n * 0.5) * K) - M)) / Math.exp((n * (n * 0.25)));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -53.0) or not (n <= 0.00068):
		tmp = math.cos((((n * 0.5) * K) - M)) / math.exp((n * (n * 0.25)))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -53.0) || !(n <= 0.00068))
		tmp = Float64(cos(Float64(Float64(Float64(n * 0.5) * K) - M)) / exp(Float64(n * Float64(n * 0.25))));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -53.0) || ~((n <= 0.00068)))
		tmp = cos((((n * 0.5) * K) - M)) / exp((n * (n * 0.25)));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -53.0], N[Not[LessEqual[n, 0.00068]], $MachinePrecision]], N[(N[Cos[N[(N[(N[(n * 0.5), $MachinePrecision] * K), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(n * N[(n * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -53 or 6.8e-4 < n

   1. Initial program 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in n around inf 64.8%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{0.25 \cdot {n}^{2}} + \left(\ell - \left|n - m\right|\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 64.8%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{{n}^{2} \cdot 0.25} + \left(\ell - \left|n - m\right|\right)}} \]

      2. unpow2 64.8%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(n \cdot n\right)} \cdot 0.25 + \left(\ell - \left|n - m\right|\right)}} \]

   6. Simplified 64.8%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(n \cdot n\right) \cdot 0.25} + \left(\ell - \left|n - m\right|\right)}} \]

   7. Taylor expanded in m around 0 65.7%

      \[\leadsto \frac{\cos \left(\color{blue}{0.5 \cdot \left(n \cdot K\right)} - M\right)}{e^{\left(n \cdot n\right) \cdot 0.25 + \left(\ell - \left|n - m\right|\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 65.7%

         \[\leadsto \frac{\cos \left(\color{blue}{\left(0.5 \cdot n\right) \cdot K} - M\right)}{e^{\left(n \cdot n\right) \cdot 0.25 + \left(\ell - \left|n - m\right|\right)}} \]

   9. Simplified 65.7%

      \[\leadsto \frac{\cos \left(\color{blue}{\left(0.5 \cdot n\right) \cdot K} - M\right)}{e^{\left(n \cdot n\right) \cdot 0.25 + \left(\ell - \left|n - m\right|\right)}} \]

   10. Taylor expanded in n around inf 76.8%

       \[\leadsto \frac{\cos \left(\left(0.5 \cdot n\right) \cdot K - M\right)}{e^{\color{blue}{0.25 \cdot {n}^{2}}}} \]
   11. Step-by-step derivation

       1. *-commutative 76.8%

          \[\leadsto \frac{\cos \left(\left(0.5 \cdot n\right) \cdot K - M\right)}{e^{\color{blue}{{n}^{2} \cdot 0.25}}} \]

       2. unpow2 76.8%

          \[\leadsto \frac{\cos \left(\left(0.5 \cdot n\right) \cdot K - M\right)}{e^{\color{blue}{\left(n \cdot n\right)} \cdot 0.25}} \]

       3. associate-*r* 76.8%

          \[\leadsto \frac{\cos \left(\left(0.5 \cdot n\right) \cdot K - M\right)}{e^{\color{blue}{n \cdot \left(n \cdot 0.25\right)}}} \]

   12. Simplified 76.8%

       \[\leadsto \frac{\cos \left(\left(0.5 \cdot n\right) \cdot K - M\right)}{e^{\color{blue}{n \cdot \left(n \cdot 0.25\right)}}} \]

   if -53 < n < 6.8e-4

   1. Initial program 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in l around inf 38.0%

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

   5. Taylor expanded in K around 0 41.8%

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

      1. cos-neg 41.8%

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

   7. Simplified 41.8%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -53 \lor \neg \left(n \leq 0.00068\right):\\ \;\;\;\;\frac{\cos \left(\left(n \cdot 0.5\right) \cdot K - M\right)}{e^{n \cdot \left(n \cdot 0.25\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]

Alternative 9: 35.4% accurate, 2.1× speedup

Math

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

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(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 76.6%

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

3. Simplified 76.6%

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

4. Taylor expanded in l around inf 29.2%

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

5. Taylor expanded in K around 0 33.2%

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

   1. cos-neg 33.2%

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

7. Simplified 33.2%

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

8. Final simplification 33.2%

   \[\leadsto \frac{\cos M}{e^{\ell}} \]

Alternative 10: 35.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{\ell}} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (/ 1.0 (exp l)))

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 / 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 = 1.0d0 / exp(l)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 / Math.exp(l);
}

Python

def code(K, m, n, M, l):
	return 1.0 / math.exp(l)

Julia

function code(K, m, n, M, l)
	return Float64(1.0 / exp(l))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0 / exp(l);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 / N[Exp[l], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

3. Simplified 76.6%

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

4. Taylor expanded in l around inf 29.2%

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

   1. div-inv 29.2%

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

   2. metadata-eval 29.2%

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

   3. flip-- 19.7%

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

6. Applied egg-rr 19.7%

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

7. Taylor expanded in n around inf 22.0%

   \[\leadsto \frac{\cos \left(\frac{\color{blue}{0.25 \cdot \left({n}^{2} \cdot {K}^{2}\right)}}{K \cdot \left(\left(m + n\right) \cdot 0.5\right) + M}\right)}{e^{\ell}} \]
8. Step-by-step derivation

   1. unpow2 22.0%

      \[\leadsto \frac{\cos \left(\frac{0.25 \cdot \left(\color{blue}{\left(n \cdot n\right)} \cdot {K}^{2}\right)}{K \cdot \left(\left(m + n\right) \cdot 0.5\right) + M}\right)}{e^{\ell}} \]

   2. unpow2 22.0%

      \[\leadsto \frac{\cos \left(\frac{0.25 \cdot \left(\left(n \cdot n\right) \cdot \color{blue}{\left(K \cdot K\right)}\right)}{K \cdot \left(\left(m + n\right) \cdot 0.5\right) + M}\right)}{e^{\ell}} \]

9. Simplified 22.0%

   \[\leadsto \frac{\cos \left(\frac{\color{blue}{0.25 \cdot \left(\left(n \cdot n\right) \cdot \left(K \cdot K\right)\right)}}{K \cdot \left(\left(m + n\right) \cdot 0.5\right) + M}\right)}{e^{\ell}} \]

10. Taylor expanded in n around 0 32.8%

    \[\leadsto \frac{\color{blue}{1}}{e^{\ell}} \]

11. Final simplification 32.8%

    \[\leadsto \frac{1}{e^{\ell}} \]

Alternative 11: 6.9% 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 76.6%

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

3. Simplified 76.6%

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

4. Taylor expanded in l around inf 29.2%

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

5. Taylor expanded in l around 0 7.5%

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

6. Taylor expanded in K around 0 8.2%

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

   1. cos-neg 8.2%

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

8. Simplified 8.2%

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

9. Final simplification 8.2%

   \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023189 
(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)))))))