Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.3% → 96.7%

Time: 14.8s

Alternatives: 7

Speedup: 1.0×

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: 75.3% 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: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

2. Taylor expanded in K around 0 98.4%

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

   1. cos-neg 98.4%

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

4. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 74.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{m}\right)}^{\left(m \cdot -0.25\right)}\\ t_1 := \cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -7.4 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M \leq 1.85 \cdot 10^{-196}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 2.4 \cdot 10^{-98}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;M \leq 3 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (pow (exp m) (* m -0.25))) (t_1 (* (cos M) (exp (* M (- M))))))
   (if (<= M -7.4e-32)
     t_1
     (if (<= M 1.85e-196)
       t_0
       (if (<= M 2.4e-98) (exp (- l)) (if (<= M 3e-9) t_0 t_1))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = pow(exp(m), (m * -0.25));
	double t_1 = cos(M) * exp((M * -M));
	double tmp;
	if (M <= -7.4e-32) {
		tmp = t_1;
	} else if (M <= 1.85e-196) {
		tmp = t_0;
	} else if (M <= 2.4e-98) {
		tmp = exp(-l);
	} else if (M <= 3e-9) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = exp(m) ** (m * (-0.25d0))
    t_1 = cos(m_1) * exp((m_1 * -m_1))
    if (m_1 <= (-7.4d-32)) then
        tmp = t_1
    else if (m_1 <= 1.85d-196) then
        tmp = t_0
    else if (m_1 <= 2.4d-98) then
        tmp = exp(-l)
    else if (m_1 <= 3d-9) then
        tmp = t_0
    else
        tmp = t_1
    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.pow(Math.exp(m), (m * -0.25));
	double t_1 = Math.cos(M) * Math.exp((M * -M));
	double tmp;
	if (M <= -7.4e-32) {
		tmp = t_1;
	} else if (M <= 1.85e-196) {
		tmp = t_0;
	} else if (M <= 2.4e-98) {
		tmp = Math.exp(-l);
	} else if (M <= 3e-9) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.pow(math.exp(m), (m * -0.25))
	t_1 = math.cos(M) * math.exp((M * -M))
	tmp = 0
	if M <= -7.4e-32:
		tmp = t_1
	elif M <= 1.85e-196:
		tmp = t_0
	elif M <= 2.4e-98:
		tmp = math.exp(-l)
	elif M <= 3e-9:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(m) ^ Float64(m * -0.25)
	t_1 = Float64(cos(M) * exp(Float64(M * Float64(-M))))
	tmp = 0.0
	if (M <= -7.4e-32)
		tmp = t_1;
	elseif (M <= 1.85e-196)
		tmp = t_0;
	elseif (M <= 2.4e-98)
		tmp = exp(Float64(-l));
	elseif (M <= 3e-9)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(m) ^ (m * -0.25);
	t_1 = cos(M) * exp((M * -M));
	tmp = 0.0;
	if (M <= -7.4e-32)
		tmp = t_1;
	elseif (M <= 1.85e-196)
		tmp = t_0;
	elseif (M <= 2.4e-98)
		tmp = exp(-l);
	elseif (M <= 3e-9)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Power[N[Exp[m], $MachinePrecision], N[(m * -0.25), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -7.4e-32], t$95$1, If[LessEqual[M, 1.85e-196], t$95$0, If[LessEqual[M, 2.4e-98], N[Exp[(-l)], $MachinePrecision], If[LessEqual[M, 3e-9], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -7.4e-32 or 2.99999999999999998e-9 < M

   1. Initial program 76.5%

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

   2. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in M around inf 94.1%

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

      1. mul-1-neg 94.1%

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

      2. unpow2 94.1%

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

      3. distribute-rgt-neg-in 94.1%

         \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   7. Simplified 94.1%

      \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   if -7.4e-32 < M < 1.85000000000000005e-196 or 2.40000000000000005e-98 < M < 2.99999999999999998e-9

   1. Initial program 78.9%

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

   2. Taylor expanded in m around inf 46.3%

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

      1. *-commutative 46.3%

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

      2. unpow2 46.3%

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

   4. Simplified 46.3%

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

   5. Taylor expanded in K around 0 55.6%

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

      1. cos-neg 55.6%

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

      2. *-commutative 55.6%

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

      3. unpow2 55.6%

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

   7. Simplified 55.6%

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

   8. Taylor expanded in M around 0 55.6%

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

      1. *-commutative 55.6%

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

      2. unpow2 55.6%

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

      3. associate-*r* 55.6%

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

      4. exp-prod 55.6%

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

   10. Simplified 55.6%

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

   if 1.85000000000000005e-196 < M < 2.40000000000000005e-98

   1. Initial program 88.2%

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

   2. Taylor expanded in l around inf 48.1%

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

      1. mul-1-neg 48.1%

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

   4. Simplified 48.1%

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

   5. Taylor expanded in K around 0 54.1%

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

      1. cos-neg 54.1%

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

   7. Simplified 54.1%

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

   8. Taylor expanded in M around 0 54.1%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -7.4 \cdot 10^{-32}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq 1.85 \cdot 10^{-196}:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(m \cdot -0.25\right)}\\ \mathbf{elif}\;M \leq 2.4 \cdot 10^{-98}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;M \leq 3 \cdot 10^{-9}:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \end{array} \]

Alternative 3: 66.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -195:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq -7 \cdot 10^{-241}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -195.0)
   (pow (exp m) (* m -0.25))
   (if (<= m -7e-241)
     (* (cos M) (exp (* M (- M))))
     (* (cos M) (exp (* -0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -195.0) {
		tmp = pow(exp(m), (m * -0.25));
	} else if (m <= -7e-241) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-195.0d0)) then
        tmp = exp(m) ** (m * (-0.25d0))
    else if (m <= (-7d-241)) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -195.0) {
		tmp = Math.pow(Math.exp(m), (m * -0.25));
	} else if (m <= -7e-241) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -195.0:
		tmp = math.pow(math.exp(m), (m * -0.25))
	elif m <= -7e-241:
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -195.0)
		tmp = exp(m) ^ Float64(m * -0.25);
	elseif (m <= -7e-241)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -195.0)
		tmp = exp(m) ^ (m * -0.25);
	elseif (m <= -7e-241)
		tmp = cos(M) * exp((M * -M));
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -195.0], N[Power[N[Exp[m], $MachinePrecision], N[(m * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -7e-241], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -195

   1. Initial program 73.8%

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

   2. Taylor expanded in m around inf 73.8%

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

      1. *-commutative 73.8%

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

      2. unpow2 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

      2. *-commutative 96.8%

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

      3. unpow2 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in M around 0 96.8%

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

      1. *-commutative 96.8%

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

      2. unpow2 96.8%

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

      3. associate-*r* 96.8%

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

      4. exp-prod 96.8%

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

   10. Simplified 96.8%

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

   if -195 < m < -6.9999999999999998e-241

   1. Initial program 85.0%

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

   2. Taylor expanded in K around 0 98.1%

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

      1. cos-neg 98.1%

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

   4. Simplified 98.1%

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

   5. Taylor expanded in M around inf 53.4%

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

      1. mul-1-neg 53.4%

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

      2. unpow2 53.4%

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

      3. distribute-rgt-neg-in 53.4%

         \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   7. Simplified 53.4%

      \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   if -6.9999999999999998e-241 < m

   1. Initial program 77.8%

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

   2. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

   4. Simplified 98.6%

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

   5. Taylor expanded in n around inf 51.9%

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

      1. *-commutative 51.9%

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

      2. unpow2 51.9%

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

   7. Simplified 51.9%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -195:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq -7 \cdot 10^{-241}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 4: 73.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.000155:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 720:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -0.000155)
   (* (cos M) (exp l))
   (if (<= l 720.0) (pow (exp m) (* m -0.25)) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -0.000155) {
		tmp = cos(M) * exp(l);
	} else if (l <= 720.0) {
		tmp = pow(exp(m), (m * -0.25));
	} else {
		tmp = 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 <= (-0.000155d0)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 720.0d0) then
        tmp = exp(m) ** (m * (-0.25d0))
    else
        tmp = 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 <= -0.000155) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 720.0) {
		tmp = Math.pow(Math.exp(m), (m * -0.25));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -0.000155:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 720.0:
		tmp = math.pow(math.exp(m), (m * -0.25))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -0.000155)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 720.0)
		tmp = exp(m) ^ Float64(m * -0.25);
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -0.000155)
		tmp = cos(M) * exp(l);
	elseif (l <= 720.0)
		tmp = exp(m) ^ (m * -0.25);
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -0.000155], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 720.0], N[Power[N[Exp[m], $MachinePrecision], N[(m * -0.25), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.55e-4

   1. Initial program 82.1%

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

   2. Taylor expanded in l around inf 21.9%

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

      1. mul-1-neg 21.9%

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

   4. Simplified 21.9%

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

   5. Taylor expanded in K around 0 22.1%

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

      1. cos-neg 22.1%

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

   7. Simplified 22.1%

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

      1. expm1-log1p-u 17.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)\right)} \]

      2. expm1-udef 17.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)} - 1} \]

      3. add-sqr-sqrt 17.3%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

      4. sqrt-unprod 17.3%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

      5. sqr-neg 17.3%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

      6. sqrt-unprod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

      7. add-sqr-sqrt 73.6%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   9. Applied egg-rr 73.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 73.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)\right)} \]

       2. expm1-log1p 73.6%

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

   11. Simplified 73.6%

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

   if -1.55e-4 < l < 720

   1. Initial program 75.3%

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

   2. Taylor expanded in m around inf 44.7%

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

      1. *-commutative 44.7%

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

      2. unpow2 44.7%

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

   4. Simplified 44.7%

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

   5. Taylor expanded in K around 0 58.0%

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

      1. cos-neg 58.0%

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

      2. *-commutative 58.0%

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

      3. unpow2 58.0%

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

   7. Simplified 58.0%

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

   8. Taylor expanded in M around 0 58.0%

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

      1. *-commutative 58.0%

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

      2. unpow2 58.0%

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

      3. associate-*r* 58.0%

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

      4. exp-prod 58.0%

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

   10. Simplified 58.0%

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

   if 720 < l

   1. Initial program 79.7%

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

   2. Taylor expanded in l around inf 79.7%

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

      1. mul-1-neg 79.7%

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

   4. Simplified 79.7%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0 100.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.000155:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 720:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(m \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 5: 48.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{-12}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 1.55e-12) (* (cos M) (exp l)) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 1.55e-12) {
		tmp = cos(M) * exp(l);
	} else {
		tmp = 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 <= 1.55d-12) then
        tmp = cos(m_1) * exp(l)
    else
        tmp = 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 <= 1.55e-12) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 1.55e-12:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 1.55e-12)
		tmp = Float64(cos(M) * exp(l));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 1.55e-12)
		tmp = cos(M) * exp(l);
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 1.55e-12], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5500000000000001e-12

   1. Initial program 77.6%

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

   2. Taylor expanded in l around inf 14.3%

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

      1. mul-1-neg 14.3%

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

   4. Simplified 14.3%

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

   5. Taylor expanded in K around 0 14.6%

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

      1. cos-neg 14.6%

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

   7. Simplified 14.6%

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

      1. expm1-log1p-u 12.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)\right)} \]

      2. expm1-udef 12.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)} - 1} \]

      3. add-sqr-sqrt 10.4%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

      4. sqrt-unprod 12.9%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

      5. sqr-neg 12.9%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

      6. sqrt-unprod 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

      7. add-sqr-sqrt 33.2%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   9. Applied egg-rr 33.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 33.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)\right)} \]

       2. expm1-log1p 33.2%

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

   11. Simplified 33.2%

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

   if 1.5500000000000001e-12 < l

   1. Initial program 80.0%

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

   2. Taylor expanded in l around inf 78.6%

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

      1. mul-1-neg 78.6%

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

   4. Simplified 78.6%

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

   5. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in M around 0 98.6%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{-12}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 6: 48.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-266}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -5e-266) (* (cos M) (exp l)) (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -5e-266) {
		tmp = cos(M) * exp(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 <= (-5d-266)) then
        tmp = cos(m_1) * exp(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 <= -5e-266) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -5e-266:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -5e-266)
		tmp = Float64(cos(M) * exp(l));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -5e-266)
		tmp = cos(M) * exp(l);
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -5e-266], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.99999999999999992e-266

   1. Initial program 78.3%

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

   2. Taylor expanded in l around inf 17.6%

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

      1. mul-1-neg 17.6%

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

   4. Simplified 17.6%

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

   5. Taylor expanded in K around 0 17.9%

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

      1. cos-neg 17.9%

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

   7. Simplified 17.9%

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

      1. expm1-log1p-u 15.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)\right)} \]

      2. expm1-udef 15.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)} - 1} \]

      3. add-sqr-sqrt 15.2%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

      4. sqrt-unprod 15.2%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

      5. sqr-neg 15.2%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

      6. sqrt-unprod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

      7. add-sqr-sqrt 46.9%

         \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   9. Applied egg-rr 46.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 46.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)\right)} \]

       2. expm1-log1p 46.9%

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

   11. Simplified 46.9%

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

   if -4.99999999999999992e-266 < l

   1. Initial program 78.2%

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

   2. Taylor expanded in l around inf 44.3%

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

      1. mul-1-neg 44.3%

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

   4. Simplified 44.3%

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

   5. Taylor expanded in K around 0 54.7%

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

      1. cos-neg 54.7%

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

   7. Simplified 54.7%

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

   8. Taylor expanded in l around -inf 54.7%

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

      1. neg-mul-1 54.7%

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

      2. exp-neg 54.7%

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

      3. associate-*r/ 54.7%

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

      4. *-rgt-identity 54.7%

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

   10. Simplified 54.7%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-266}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]

Alternative 7: 34.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

2. Taylor expanded in l around inf 31.9%

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

   1. mul-1-neg 31.9%

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

4. Simplified 31.9%

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

5. Taylor expanded in K around 0 37.6%

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

   1. cos-neg 37.6%

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

7. Simplified 37.6%

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

8. Taylor expanded in M around 0 36.4%

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

9. Final simplification 36.4%

   \[\leadsto e^{-\ell} \]

Reproduce

herbie shell --seed 2023291 
(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)))))))