Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.1% → 96.4%

Time: 28.1s

Alternatives: 8

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

3. Simplified 76.0%

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

4. Taylor expanded in K around 0 95.5%

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

   1. cos-neg 95.5%

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

6. Simplified 95.5%

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

   1. add-cube-cbrt 95.5%

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

   2. exp-prod 95.5%

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

8. Applied egg-rr 95.5%

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

9. Final simplification 95.5%

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

Alternative 2: 96.5% 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 76.0%

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

3. Simplified 76.0%

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

4. Taylor expanded in K around 0 95.5%

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

   1. cos-neg 95.5%

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

6. Simplified 95.5%

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

7. Final simplification 95.5%

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

Alternative 3: 91.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\cos M}{e^{m + {\left(m \cdot 0.5 - M\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{{\left(n \cdot 0.5 - M\right)}^{2} + \left(\ell - n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1.5e+41)
   (/ (cos M) (exp (+ m (pow (- (* m 0.5) M) 2.0))))
   (/ (cos M) (exp (+ (pow (- (* n 0.5) M) 2.0) (- l n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.5e+41) {
		tmp = cos(M) / exp((m + pow(((m * 0.5) - M), 2.0)));
	} else {
		tmp = cos(M) / exp((pow(((n * 0.5) - M), 2.0) + (l - 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 <= (-1.5d+41)) then
        tmp = cos(m_1) / exp((m + (((m * 0.5d0) - m_1) ** 2.0d0)))
    else
        tmp = cos(m_1) / exp(((((n * 0.5d0) - m_1) ** 2.0d0) + (l - 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 <= -1.5e+41) {
		tmp = Math.cos(M) / Math.exp((m + Math.pow(((m * 0.5) - M), 2.0)));
	} else {
		tmp = Math.cos(M) / Math.exp((Math.pow(((n * 0.5) - M), 2.0) + (l - n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1.5e+41:
		tmp = math.cos(M) / math.exp((m + math.pow(((m * 0.5) - M), 2.0)))
	else:
		tmp = math.cos(M) / math.exp((math.pow(((n * 0.5) - M), 2.0) + (l - n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1.5e+41)
		tmp = Float64(cos(M) / exp(Float64(m + (Float64(Float64(m * 0.5) - M) ^ 2.0))));
	else
		tmp = Float64(cos(M) / exp(Float64((Float64(Float64(n * 0.5) - M) ^ 2.0) + Float64(l - n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1.5e+41)
		tmp = cos(M) / exp((m + (((m * 0.5) - M) ^ 2.0)));
	else
		tmp = cos(M) / exp(((((n * 0.5) - M) ^ 2.0) + (l - n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1.5e+41], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(m + N[Power[N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[Power[N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision] + N[(l - n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.4999999999999999e41

   1. Initial program 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

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

      2. exp-prod 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in n around 0 100.0%

      \[\leadsto \frac{\cos M}{\color{blue}{e^{{1}^{0.3333333333333333} \cdot \left({\left(0.5 \cdot m - M\right)}^{2} + \left(\ell + m\right)\right)}}} \]
   10. Step-by-step derivation

       1. pow-base-1 100.0%

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

       2. *-lft-identity 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in l around 0 100.0%

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

   if -1.4999999999999999e41 < m

   1. Initial program 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in K around 0 94.1%

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

      1. cos-neg 94.1%

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

   6. Simplified 94.1%

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

      1. add-cube-cbrt 94.1%

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

      2. exp-prod 94.1%

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

   8. Applied egg-rr 94.1%

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

   9. Taylor expanded in m around 0 88.1%

      \[\leadsto \frac{\cos M}{\color{blue}{e^{\left(\left({\left(0.5 \cdot n - M\right)}^{2} + \ell\right) - n\right) \cdot {1}^{0.3333333333333333}}}} \]
   10. Step-by-step derivation

       1. pow-base-1 88.1%

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

       2. *-rgt-identity 88.1%

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

       3. associate--l+ 88.1%

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

   11. Simplified 88.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\cos M}{e^{m + {\left(m \cdot 0.5 - M\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{{\left(n \cdot 0.5 - M\right)}^{2} + \left(\ell - n\right)}}\\ \end{array} \]

Alternative 4: 86.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.0058:\\ \;\;\;\;\frac{\cos M}{e^{m + {\left(m \cdot 0.5 - M\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\left(m + \ell\right) + M \cdot M}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -0.0058)
   (/ (cos M) (exp (+ m (pow (- (* m 0.5) M) 2.0))))
   (/ (cos M) (exp (+ (+ m l) (* M M))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.0058) {
		tmp = cos(M) / exp((m + pow(((m * 0.5) - M), 2.0)));
	} else {
		tmp = cos(M) / exp(((m + l) + (M * M)));
	}
	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 <= (-0.0058d0)) then
        tmp = cos(m_1) / exp((m + (((m * 0.5d0) - m_1) ** 2.0d0)))
    else
        tmp = cos(m_1) / exp(((m + l) + (m_1 * m_1)))
    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 <= -0.0058) {
		tmp = Math.cos(M) / Math.exp((m + Math.pow(((m * 0.5) - M), 2.0)));
	} else {
		tmp = Math.cos(M) / Math.exp(((m + l) + (M * M)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -0.0058:
		tmp = math.cos(M) / math.exp((m + math.pow(((m * 0.5) - M), 2.0)))
	else:
		tmp = math.cos(M) / math.exp(((m + l) + (M * M)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -0.0058)
		tmp = Float64(cos(M) / exp(Float64(m + (Float64(Float64(m * 0.5) - M) ^ 2.0))));
	else
		tmp = Float64(cos(M) / exp(Float64(Float64(m + l) + Float64(M * M))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -0.0058)
		tmp = cos(M) / exp((m + (((m * 0.5) - M) ^ 2.0)));
	else
		tmp = cos(M) / exp(((m + l) + (M * M)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -0.0058], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(m + N[Power[N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[(m + l), $MachinePrecision] + N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.0058

   1. Initial program 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

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

      2. exp-prod 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in n around 0 100.0%

      \[\leadsto \frac{\cos M}{\color{blue}{e^{{1}^{0.3333333333333333} \cdot \left({\left(0.5 \cdot m - M\right)}^{2} + \left(\ell + m\right)\right)}}} \]
   10. Step-by-step derivation

       1. pow-base-1 100.0%

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

       2. *-lft-identity 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in l around 0 100.0%

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

   if -0.0058 < m

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in K around 0 93.6%

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

      1. cos-neg 93.6%

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

   6. Simplified 93.6%

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

      1. add-cube-cbrt 93.6%

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

      2. exp-prod 93.6%

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

   8. Applied egg-rr 93.6%

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

   9. Taylor expanded in n around 0 85.5%

      \[\leadsto \frac{\cos M}{\color{blue}{e^{{1}^{0.3333333333333333} \cdot \left({\left(0.5 \cdot m - M\right)}^{2} + \left(\ell + m\right)\right)}}} \]
   10. Step-by-step derivation

       1. pow-base-1 85.5%

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

       2. *-lft-identity 85.5%

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

   11. Simplified 85.5%

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

   12. Taylor expanded in m around 0 85.0%

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

       1. unpow2 85.0%

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

   14. Simplified 85.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0058:\\ \;\;\;\;\frac{\cos M}{e^{m + {\left(m \cdot 0.5 - M\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\left(m + \ell\right) + M \cdot M}}\\ \end{array} \]

Alternative 5: 87.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) / exp((pow(((m * 0.5) - M), 2.0) + (m + 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(((((m * 0.5d0) - m_1) ** 2.0d0) + (m + 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.pow(((m * 0.5) - M), 2.0) + (m + l)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

3. Simplified 76.0%

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

4. Taylor expanded in K around 0 95.5%

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

   1. cos-neg 95.5%

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

6. Simplified 95.5%

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

   1. add-cube-cbrt 95.5%

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

   2. exp-prod 95.5%

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

8. Applied egg-rr 95.5%

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

9. Taylor expanded in n around 0 89.8%

   \[\leadsto \frac{\cos M}{\color{blue}{e^{{1}^{0.3333333333333333} \cdot \left({\left(0.5 \cdot m - M\right)}^{2} + \left(\ell + m\right)\right)}}} \]
10. Step-by-step derivation

    1. pow-base-1 89.8%

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

    2. *-lft-identity 89.8%

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

11. Simplified 89.8%

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

12. Final simplification 89.8%

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

Alternative 6: 85.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+26}:\\ \;\;\;\;\frac{\cos M}{e^{\left(m + \ell\right) + 0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\left(m + \ell\right) + M \cdot M}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -5e+26)
   (/ (cos M) (exp (+ (+ m l) (* 0.25 (* m m)))))
   (/ (cos M) (exp (+ (+ m l) (* M M))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -5e+26) {
		tmp = cos(M) / exp(((m + l) + (0.25 * (m * m))));
	} else {
		tmp = cos(M) / exp(((m + l) + (M * M)));
	}
	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 <= (-5d+26)) then
        tmp = cos(m_1) / exp(((m + l) + (0.25d0 * (m * m))))
    else
        tmp = cos(m_1) / exp(((m + l) + (m_1 * m_1)))
    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 <= -5e+26) {
		tmp = Math.cos(M) / Math.exp(((m + l) + (0.25 * (m * m))));
	} else {
		tmp = Math.cos(M) / Math.exp(((m + l) + (M * M)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -5e+26:
		tmp = math.cos(M) / math.exp(((m + l) + (0.25 * (m * m))))
	else:
		tmp = math.cos(M) / math.exp(((m + l) + (M * M)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -5e+26)
		tmp = Float64(cos(M) / exp(Float64(Float64(m + l) + Float64(0.25 * Float64(m * m)))));
	else
		tmp = Float64(cos(M) / exp(Float64(Float64(m + l) + Float64(M * M))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -5e+26)
		tmp = cos(M) / exp(((m + l) + (0.25 * (m * m))));
	else
		tmp = cos(M) / exp(((m + l) + (M * M)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -5e+26], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[(m + l), $MachinePrecision] + N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[(m + l), $MachinePrecision] + N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -5.0000000000000001e26

   1. Initial program 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

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

      2. exp-prod 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in n around 0 100.0%

      \[\leadsto \frac{\cos M}{\color{blue}{e^{{1}^{0.3333333333333333} \cdot \left({\left(0.5 \cdot m - M\right)}^{2} + \left(\ell + m\right)\right)}}} \]
   10. Step-by-step derivation

       1. pow-base-1 100.0%

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

       2. *-lft-identity 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in M around 0 100.0%

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

       1. associate-+r+ 100.0%

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

       2. +-commutative 100.0%

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

       3. unpow2 100.0%

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

   14. Simplified 100.0%

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

   if -5.0000000000000001e26 < m

   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 K around 0 93.9%

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

      1. cos-neg 93.9%

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

   6. Simplified 93.9%

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

      1. add-cube-cbrt 93.9%

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

      2. exp-prod 93.9%

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

   8. Applied egg-rr 93.9%

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

   9. Taylor expanded in n around 0 86.2%

      \[\leadsto \frac{\cos M}{\color{blue}{e^{{1}^{0.3333333333333333} \cdot \left({\left(0.5 \cdot m - M\right)}^{2} + \left(\ell + m\right)\right)}}} \]
   10. Step-by-step derivation

       1. pow-base-1 86.2%

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

       2. *-lft-identity 86.2%

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

   11. Simplified 86.2%

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

   12. Taylor expanded in m around 0 84.7%

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

       1. unpow2 84.7%

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

   14. Simplified 84.7%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+26}:\\ \;\;\;\;\frac{\cos M}{e^{\left(m + \ell\right) + 0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\left(m + \ell\right) + M \cdot M}}\\ \end{array} \]

Alternative 7: 69.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) / exp((l + (M * 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((l + (m_1 * m_1)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

3. Simplified 76.0%

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

4. Taylor expanded in K around 0 95.5%

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

   1. cos-neg 95.5%

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

6. Simplified 95.5%

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

   1. add-cube-cbrt 95.5%

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

   2. exp-prod 95.5%

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

8. Applied egg-rr 95.5%

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

9. Taylor expanded in n around 0 89.8%

   \[\leadsto \frac{\cos M}{\color{blue}{e^{{1}^{0.3333333333333333} \cdot \left({\left(0.5 \cdot m - M\right)}^{2} + \left(\ell + m\right)\right)}}} \]
10. Step-by-step derivation

    1. pow-base-1 89.8%

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

    2. *-lft-identity 89.8%

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

11. Simplified 89.8%

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

12. Taylor expanded in m around 0 74.5%

    \[\leadsto \frac{\cos M}{\color{blue}{e^{\ell + {M}^{2}}}} \]
13. Step-by-step derivation

    1. +-commutative 74.5%

       \[\leadsto \frac{\cos M}{e^{\color{blue}{{M}^{2} + \ell}}} \]

    2. unpow2 74.5%

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

14. Simplified 74.5%

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

15. Final simplification 74.5%

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

Alternative 8: 7.0% 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 76.0%

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

3. Simplified 76.0%

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

4. Taylor expanded in K around 0 95.5%

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

   1. cos-neg 95.5%

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

6. Simplified 95.5%

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

   1. add-cube-cbrt 95.5%

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

   2. exp-prod 95.5%

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

8. Applied egg-rr 95.5%

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

9. Taylor expanded in m around inf 5.5%

   \[\leadsto \frac{\cos M}{\color{blue}{1}} \]

10. Final simplification 5.5%

    \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023217 
(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)))))))