Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.9% → 96.8%

Time: 26.2s

Alternatives: 7

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: 76.9% 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: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

2. Taylor expanded in K around 0 96.4%

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

   1. cos-neg 96.4%

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

   2. associate--r+ 96.4%

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

   3. *-commutative 96.4%

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

4. Simplified 96.4%

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

5. Final simplification 96.4%

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

Alternative 2: 87.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 1.06 \cdot 10^{-237}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.18 \cdot 10^{-5}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(t_0 - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 1.06e-237)
     (exp (- t_0 (+ l (* 0.25 (* (+ m n) (+ m n))))))
     (if (<= n 1.18e-5)
       (*
        (cos (- (/ (* (+ m n) K) 2.0) M))
        (exp (+ (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) (- t_0 l))))
       (exp (- t_0 (* 0.25 (pow (+ m n) 2.0))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (n <= 1.06e-237) {
		tmp = exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))));
	} else if (n <= 1.18e-5) {
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (t_0 - l)));
	} else {
		tmp = exp((t_0 - (0.25 * pow((m + n), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (n <= 1.06d-237) then
        tmp = exp((t_0 - (l + (0.25d0 * ((m + n) * (m + n))))))
    else if (n <= 1.18d-5) then
        tmp = cos(((((m + n) * k) / 2.0d0) - m_1)) * exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) + (t_0 - l)))
    else
        tmp = exp((t_0 - (0.25d0 * ((m + n) ** 2.0d0))))
    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.abs((n - m));
	double tmp;
	if (n <= 1.06e-237) {
		tmp = Math.exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))));
	} else if (n <= 1.18e-5) {
		tmp = Math.cos(((((m + n) * K) / 2.0) - M)) * Math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (t_0 - l)));
	} else {
		tmp = Math.exp((t_0 - (0.25 * Math.pow((m + n), 2.0))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 1.06e-237:
		tmp = math.exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))))
	elif n <= 1.18e-5:
		tmp = math.cos(((((m + n) * K) / 2.0) - M)) * math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (t_0 - l)))
	else:
		tmp = math.exp((t_0 - (0.25 * math.pow((m + n), 2.0))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 1.06e-237)
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * Float64(Float64(m + n) * Float64(m + n))))));
	elseif (n <= 1.18e-5)
		tmp = Float64(cos(Float64(Float64(Float64(Float64(m + n) * K) / 2.0) - M)) * exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) + Float64(t_0 - l))));
	else
		tmp = exp(Float64(t_0 - Float64(0.25 * (Float64(m + n) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (n <= 1.06e-237)
		tmp = exp((t_0 - (l + (0.25 * ((m + n) * (m + n))))));
	elseif (n <= 1.18e-5)
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (t_0 - l)));
	else
		tmp = exp((t_0 - (0.25 * ((m + n) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 1.06e-237], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.18e-5], N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$0 - N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.05999999999999994e-237

   1. Initial program 77.1%

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

   2. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

      2. associate--r+ 94.7%

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

      3. *-commutative 94.7%

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

   4. Simplified 94.7%

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

   5. Taylor expanded in M around 0 87.0%

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

      1. +-commutative 87.0%

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

      2. +-commutative 87.0%

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

   7. Simplified 87.0%

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

      1. +-commutative 87.0%

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

      2. pow2 87.0%

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

   9. Applied egg-rr 87.0%

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

   if 1.05999999999999994e-237 < n < 1.18000000000000005e-5

   1. Initial program 76.2%

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

   2. Taylor expanded in n around 0 76.2%

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

      1. +-commutative 76.2%

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

      2. unpow2 76.2%

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

      3. distribute-rgt-out 76.2%

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

      4. *-commutative 76.2%

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

      5. *-commutative 76.2%

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

   4. Simplified 76.2%

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

   if 1.18000000000000005e-5 < n

   1. Initial program 73.5%

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

   2. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. associate--r+ 100.0%

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

      3. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in M around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. +-commutative 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in l around 0 100.0%

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

      1. fabs-sub 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{e^{\left|n - m\right| - 0.25 \cdot {\left(m + n\right)}^{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.06 \cdot 10^{-237}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.18 \cdot 10^{-5}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \]

Alternative 3: 45.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.7 \cdot 10^{+17}:\\ \;\;\;\;\cos \left(\left(\left(m + n\right) \cdot 0.5\right) \cdot K - M\right) \cdot e^{m - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{0.25 \cdot {m}^{2} - \left(n + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -2.7e+17)
   (* (cos (- (* (* (+ m n) 0.5) K) M)) (exp (- m l)))
   (exp (- (* 0.25 (pow m 2.0)) (+ n l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.7e+17) {
		tmp = cos(((((m + n) * 0.5) * K) - M)) * exp((m - l));
	} else {
		tmp = exp(((0.25 * pow(m, 2.0)) - (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 <= (-2.7d+17)) then
        tmp = cos(((((m + n) * 0.5d0) * k) - m_1)) * exp((m - l))
    else
        tmp = exp(((0.25d0 * (m ** 2.0d0)) - (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 <= -2.7e+17) {
		tmp = Math.cos(((((m + n) * 0.5) * K) - M)) * Math.exp((m - l));
	} else {
		tmp = Math.exp(((0.25 * Math.pow(m, 2.0)) - (n + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -2.7e+17:
		tmp = math.cos(((((m + n) * 0.5) * K) - M)) * math.exp((m - l))
	else:
		tmp = math.exp(((0.25 * math.pow(m, 2.0)) - (n + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -2.7e+17)
		tmp = Float64(cos(Float64(Float64(Float64(Float64(m + n) * 0.5) * K) - M)) * exp(Float64(m - l)));
	else
		tmp = exp(Float64(Float64(0.25 * (m ^ 2.0)) - Float64(n + l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -2.7e+17)
		tmp = cos(((((m + n) * 0.5) * K) - M)) * exp((m - l));
	else
		tmp = exp(((0.25 * (m ^ 2.0)) - (n + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -2.7e+17], N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] * K), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(m - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision] - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.7e17

   1. Initial program 72.7%

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

   3. Applied egg-rr 7.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(K \cdot \left(m + n\right), 0.5, -M\right)\right) \cdot e^{\left(m - n\right) + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right)}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 7.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(K \cdot \left(m + n\right), 0.5, -M\right)\right) \cdot e^{\left(m - n\right) + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right)}\right)\right)} \]

      2. expm1-log1p 8.3%

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

      3. fma-neg 8.3%

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

      4. associate-*r* 8.3%

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

      5. +-commutative 8.3%

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

      6. associate-+l- 8.3%

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

      7. associate--r- 8.3%

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

      8. +-commutative 8.3%

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

   5. Simplified 8.3%

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

   6. Taylor expanded in n around inf 42.1%

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

   7. Taylor expanded in n around 0 69.1%

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

   if -2.7e17 < m

   1. Initial program 76.8%

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

   3. Applied egg-rr 19.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(K \cdot \left(m + n\right), 0.5, -M\right)\right) \cdot e^{\left(m - n\right) + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right)}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 19.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(K \cdot \left(m + n\right), 0.5, -M\right)\right) \cdot e^{\left(m - n\right) + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right)}\right)\right)} \]

      2. expm1-log1p 20.7%

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

      3. fma-neg 20.7%

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

      4. associate-*r* 20.7%

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

      5. +-commutative 20.7%

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

      6. associate-+l- 20.7%

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

      7. associate--r- 20.7%

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

      8. +-commutative 20.7%

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

   5. Simplified 20.7%

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

   6. Taylor expanded in K around 0 20.6%

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

      1. cos-neg 20.6%

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

   8. Simplified 20.6%

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

   9. Taylor expanded in M around 0 26.1%

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

   10. Taylor expanded in m around inf 40.7%

       \[\leadsto e^{\color{blue}{0.25 \cdot {m}^{2}} - \left(\ell + n\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.7 \cdot 10^{+17}:\\ \;\;\;\;\cos \left(\left(\left(m + n\right) \cdot 0.5\right) \cdot K - M\right) \cdot e^{m - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{0.25 \cdot {m}^{2} - \left(n + \ell\right)}\\ \end{array} \]

Alternative 4: 86.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- (fabs (- n m)) (+ l (* 0.25 (* (+ m n) (+ m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) - (l + (0.25 * ((m + n) * (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 = exp((abs((n - m)) - (l + (0.25d0 * ((m + n) * (m + n))))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((Math.abs((n - m)) - (l + (0.25 * ((m + n) * (m + n))))));
}

Python

def code(K, m, n, M, l):
	return math.exp((math.fabs((n - m)) - (l + (0.25 * ((m + n) * (m + n))))))

Julia

function code(K, m, n, M, l)
	return exp(Float64(abs(Float64(n - m)) - Float64(l + Float64(0.25 * Float64(Float64(m + n) * Float64(m + n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp((abs((n - m)) - (l + (0.25 * ((m + n) * (m + n))))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(l + N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.0%

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

2. Taylor expanded in K around 0 96.4%

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

   1. cos-neg 96.4%

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

   2. associate--r+ 96.4%

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

   3. *-commutative 96.4%

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

4. Simplified 96.4%

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

5. Taylor expanded in M around 0 87.2%

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

   1. +-commutative 87.2%

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

   2. +-commutative 87.2%

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

7. Simplified 87.2%

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

   1. +-commutative 87.2%

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

   2. pow2 87.2%

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

9. Applied egg-rr 87.2%

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

10. Final simplification 87.2%

    \[\leadsto e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)} \]

Alternative 5: 44.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-31}:\\ \;\;\;\;e^{0.25 \cdot {m}^{2} - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 7.2e-31)
   (exp (- (* 0.25 (pow m 2.0)) (+ n l)))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 7.2e-31) {
		tmp = exp(((0.25 * pow(m, 2.0)) - (n + l)));
	} 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 (l <= 7.2d-31) then
        tmp = exp(((0.25d0 * (m ** 2.0d0)) - (n + l)))
    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 (l <= 7.2e-31) {
		tmp = Math.exp(((0.25 * Math.pow(m, 2.0)) - (n + l)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 7.2e-31:
		tmp = math.exp(((0.25 * math.pow(m, 2.0)) - (n + l)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 7.2e-31)
		tmp = exp(Float64(Float64(0.25 * (m ^ 2.0)) - Float64(n + l)));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 7.2e-31)
		tmp = exp(((0.25 * (m ^ 2.0)) - (n + l)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 7.2e-31], N[Exp[N[(N[(0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision] - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.20000000000000007e-31

   1. Initial program 72.7%

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

   3. Applied egg-rr 10.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(K \cdot \left(m + n\right), 0.5, -M\right)\right) \cdot e^{\left(m - n\right) + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right)}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 10.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(K \cdot \left(m + n\right), 0.5, -M\right)\right) \cdot e^{\left(m - n\right) + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right)}\right)\right)} \]

      2. expm1-log1p 12.1%

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

      3. fma-neg 12.1%

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

      4. associate-*r* 12.1%

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

      5. +-commutative 12.1%

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

      6. associate-+l- 12.1%

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

      7. associate--r- 12.1%

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

      8. +-commutative 12.1%

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

   5. Simplified 12.1%

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

   6. Taylor expanded in K around 0 12.1%

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

      1. cos-neg 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in M around 0 11.1%

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

   10. Taylor expanded in m around inf 24.6%

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

   if 7.20000000000000007e-31 < l

   1. Initial program 83.0%

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

   3. Applied egg-rr 30.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(K \cdot \left(m + n\right), 0.5, -M\right)\right) \cdot e^{\left(m - n\right) + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right)}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 30.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(K \cdot \left(m + n\right), 0.5, -M\right)\right) \cdot e^{\left(m - n\right) + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right)}\right)\right)} \]

      2. expm1-log1p 30.8%

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

      3. fma-neg 30.8%

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

      4. associate-*r* 30.8%

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

      5. +-commutative 30.8%

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

      6. associate-+l- 30.8%

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

      7. associate--r- 30.8%

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

      8. +-commutative 30.8%

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

   5. Simplified 30.8%

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

   6. Taylor expanded in K around 0 30.7%

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

      1. cos-neg 30.7%

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

   8. Simplified 30.7%

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

   9. Taylor expanded in m around 0 40.5%

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

       1. *-commutative 40.5%

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

       2. +-commutative 40.5%

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

   11. Simplified 40.5%

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

   12. Taylor expanded in l around inf 91.6%

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

       1. mul-1-neg 91.6%

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

   14. Simplified 91.6%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-31}:\\ \;\;\;\;e^{0.25 \cdot {m}^{2} - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 6: 35.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Applied egg-rr 16.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(K \cdot \left(m + n\right), 0.5, -M\right)\right) \cdot e^{\left(m - n\right) + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right)}\right)} - 1} \]
4. Step-by-step derivation

   1. expm1-def 16.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(K \cdot \left(m + n\right), 0.5, -M\right)\right) \cdot e^{\left(m - n\right) + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right)}\right)\right)} \]

   2. expm1-log1p 18.0%

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

   3. fma-neg 18.0%

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

   4. associate-*r* 18.0%

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

   5. +-commutative 18.0%

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

   6. associate-+l- 18.0%

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

   7. associate--r- 18.0%

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

   8. +-commutative 18.0%

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

5. Simplified 18.0%

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

6. Taylor expanded in K around 0 18.0%

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

   1. cos-neg 18.0%

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

8. Simplified 18.0%

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

9. Taylor expanded in m around 0 21.2%

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

    1. *-commutative 21.2%

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

    2. +-commutative 21.2%

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

11. Simplified 21.2%

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

12. Taylor expanded in l around inf 38.0%

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

    1. mul-1-neg 38.0%

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

14. Simplified 38.0%

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

15. Final simplification 38.0%

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

Alternative 7: 20.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ e^{\left(m + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right) - \left(n + \ell\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- (+ m (* 0.25 (* (+ m n) (+ m n)))) (+ n l))))

C

double code(double K, double m, double n, double M, double l) {
	return exp(((m + (0.25 * ((m + n) * (m + n)))) - (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 = exp(((m + (0.25d0 * ((m + n) * (m + n)))) - (n + l)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp(((m + (0.25 * ((m + n) * (m + n)))) - (n + l)));
}

Python

def code(K, m, n, M, l):
	return math.exp(((m + (0.25 * ((m + n) * (m + n)))) - (n + l)))

Julia

function code(K, m, n, M, l)
	return exp(Float64(Float64(m + Float64(0.25 * Float64(Float64(m + n) * Float64(m + n)))) - Float64(n + l)))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(((m + (0.25 * ((m + n) * (m + n)))) - (n + l)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(N[(m + N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(n + l), $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. Applied egg-rr 16.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(K \cdot \left(m + n\right), 0.5, -M\right)\right) \cdot e^{\left(m - n\right) + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right)}\right)} - 1} \]
4. Step-by-step derivation

   1. expm1-def 16.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(K \cdot \left(m + n\right), 0.5, -M\right)\right) \cdot e^{\left(m - n\right) + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right)}\right)\right)} \]

   2. expm1-log1p 18.0%

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

   3. fma-neg 18.0%

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

   4. associate-*r* 18.0%

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

   5. +-commutative 18.0%

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

   6. associate-+l- 18.0%

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

   7. associate--r- 18.0%

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

   8. +-commutative 18.0%

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

5. Simplified 18.0%

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

6. Taylor expanded in K around 0 18.0%

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

   1. cos-neg 18.0%

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

8. Simplified 18.0%

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

9. Taylor expanded in M around 0 22.7%

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

    1. pow2 22.7%

       \[\leadsto e^{\left(m + 0.25 \cdot \color{blue}{\left(\left(m + n\right) \cdot \left(m + n\right)\right)}\right) - \left(\ell + n\right)} \]

11. Applied egg-rr 22.7%

    \[\leadsto e^{\left(m + 0.25 \cdot \color{blue}{\left(\left(m + n\right) \cdot \left(m + n\right)\right)}\right) - \left(\ell + n\right)} \]

12. Final simplification 22.7%

    \[\leadsto e^{\left(m + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right) - \left(n + \ell\right)} \]

Reproduce

herbie shell --seed 2023338 
(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)))))))