Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.0% → 96.6%

Time: 23.9s

Alternatives: 9

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: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ t_1 := e^{\left(t_0 - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ t_2 := K \cdot \left(m + n\right)\\ \mathbf{if}\;\cos \left(\frac{t_2}{2} - M\right) \cdot t_1 \leq \infty:\\ \;\;\;\;t_1 \cdot \cos \left(\frac{{\left(\sqrt[3]{t_2}\right)}^{3}}{2} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + {\left(m + n\right)}^{2} \cdot 0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n)))
        (t_1 (exp (- (- t_0 l) (pow (- (/ (+ m n) 2.0) M) 2.0))))
        (t_2 (* K (+ m n))))
   (if (<= (* (cos (- (/ t_2 2.0) M)) t_1) INFINITY)
     (* t_1 (cos (- (/ (pow (cbrt t_2) 3.0) 2.0) M)))
     (exp (- t_0 (+ l (* (pow (+ m n) 2.0) 0.25)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double t_1 = exp(((t_0 - l) - pow((((m + n) / 2.0) - M), 2.0)));
	double t_2 = K * (m + n);
	double tmp;
	if ((cos(((t_2 / 2.0) - M)) * t_1) <= ((double) INFINITY)) {
		tmp = t_1 * cos(((pow(cbrt(t_2), 3.0) / 2.0) - M));
	} else {
		tmp = exp((t_0 - (l + (pow((m + n), 2.0) * 0.25))));
	}
	return tmp;
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((m - n));
	double t_1 = Math.exp(((t_0 - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
	double t_2 = K * (m + n);
	double tmp;
	if ((Math.cos(((t_2 / 2.0) - M)) * t_1) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * Math.cos(((Math.pow(Math.cbrt(t_2), 3.0) / 2.0) - M));
	} else {
		tmp = Math.exp((t_0 - (l + (Math.pow((m + n), 2.0) * 0.25))));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	t_1 = exp(Float64(Float64(t_0 - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	t_2 = Float64(K * Float64(m + n))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(t_2 / 2.0) - M)) * t_1) <= Inf)
		tmp = Float64(t_1 * cos(Float64(Float64((cbrt(t_2) ^ 3.0) / 2.0) - M)));
	else
		tmp = exp(Float64(t_0 - Float64(l + Float64((Float64(m + n) ^ 2.0) * 0.25))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(t$95$0 - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(t$95$2 / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], Infinity], N[(t$95$1 * N[Cos[N[(N[(N[Power[N[Power[t$95$2, 1/3], $MachinePrecision], 3.0], $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$0 - N[(l + N[(N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 93.3%

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

      1. add-cube-cbrt 94.7%

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

      2. pow3 94.8%

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

   3. Applied egg-rr 94.8%

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

   if +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 0.0%

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

   2. Taylor expanded in M around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

   4. Simplified 0.0%

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

   5. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{1} \cdot e^{\left|m - n\right| - \left(\ell + {\left(m + n\right)}^{2} \cdot 0.25\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \leq \infty:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\frac{{\left(\sqrt[3]{K \cdot \left(m + n\right)}\right)}^{3}}{2} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + {\left(m + n\right)}^{2} \cdot 0.25\right)}\\ \end{array} \]

Alternative 2: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

2. Taylor expanded in K around 0 94.0%

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

4. Simplified 94.0%

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

5. Final simplification 94.0%

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

Alternative 3: 94.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -2 \cdot 10^{+103} \lor \neg \left(M \leq 6.5 \cdot 10^{+31}\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + {\left(m + n\right)}^{2} \cdot 0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -2e+103) (not (<= M 6.5e+31)))
   (* (cos M) (exp (- (* M (- M)) l)))
   (exp (- (fabs (- m n)) (+ l (* (pow (+ m n) 2.0) 0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2e+103) || !(M <= 6.5e+31)) {
		tmp = cos(M) * exp(((M * -M) - l));
	} else {
		tmp = exp((fabs((m - n)) - (l + (pow((m + n), 2.0) * 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-2d+103)) .or. (.not. (m_1 <= 6.5d+31))) then
        tmp = cos(m_1) * exp(((m_1 * -m_1) - l))
    else
        tmp = exp((abs((m - n)) - (l + (((m + n) ** 2.0d0) * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2e+103) || !(M <= 6.5e+31)) {
		tmp = Math.cos(M) * Math.exp(((M * -M) - l));
	} else {
		tmp = Math.exp((Math.abs((m - n)) - (l + (Math.pow((m + n), 2.0) * 0.25))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -2e+103) or not (M <= 6.5e+31):
		tmp = math.cos(M) * math.exp(((M * -M) - l))
	else:
		tmp = math.exp((math.fabs((m - n)) - (l + (math.pow((m + n), 2.0) * 0.25))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -2e+103) || !(M <= 6.5e+31))
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(-M)) - l)));
	else
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(l + Float64((Float64(m + n) ^ 2.0) * 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -2e+103) || ~((M <= 6.5e+31)))
		tmp = cos(M) * exp(((M * -M) - l));
	else
		tmp = exp((abs((m - n)) - (l + (((m + n) ^ 2.0) * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -2e+103], N[Not[LessEqual[M, 6.5e+31]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * (-M)), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -2e103 or 6.5000000000000004e31 < M

   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 99.0%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in M around inf 99.0%

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

      1. neg-mul-1 99.0%

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

      2. unpow2 99.0%

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

      3. distribute-rgt-neg-in 99.0%

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

   7. Simplified 99.0%

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

   if -2e103 < M < 6.5000000000000004e31

   1. Initial program 72.5%

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

   2. Taylor expanded in M around 0 71.9%

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

      1. associate-*r* 71.9%

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

      2. *-commutative 71.9%

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

      3. *-commutative 71.9%

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

      4. *-commutative 71.9%

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

   4. Simplified 71.9%

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

   5. Taylor expanded in K around 0 90.6%

      \[\leadsto \color{blue}{1} \cdot e^{\left|m - n\right| - \left(\ell + {\left(m + n\right)}^{2} \cdot 0.25\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2 \cdot 10^{+103} \lor \neg \left(M \leq 6.5 \cdot 10^{+31}\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + {\left(m + n\right)}^{2} \cdot 0.25\right)}\\ \end{array} \]

Alternative 4: 77.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-153}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25 - \ell}\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{-10}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right) - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -8e-153)
   (* (cos M) (exp (- (* (* m m) -0.25) l)))
   (if (<= n 6.8e-10)
     (* (cos M) (exp (- (* M (- M)) l)))
     (* (cos M) (exp (- (* -0.25 (* n n)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -8e-153) {
		tmp = cos(M) * exp((((m * m) * -0.25) - l));
	} else if (n <= 6.8e-10) {
		tmp = cos(M) * exp(((M * -M) - l));
	} else {
		tmp = cos(M) * exp(((-0.25 * (n * 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 (n <= (-8d-153)) then
        tmp = cos(m_1) * exp((((m * m) * (-0.25d0)) - l))
    else if (n <= 6.8d-10) then
        tmp = cos(m_1) * exp(((m_1 * -m_1) - l))
    else
        tmp = cos(m_1) * exp((((-0.25d0) * (n * 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 (n <= -8e-153) {
		tmp = Math.cos(M) * Math.exp((((m * m) * -0.25) - l));
	} else if (n <= 6.8e-10) {
		tmp = Math.cos(M) * Math.exp(((M * -M) - l));
	} else {
		tmp = Math.cos(M) * Math.exp(((-0.25 * (n * n)) - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -8e-153:
		tmp = math.cos(M) * math.exp((((m * m) * -0.25) - l))
	elif n <= 6.8e-10:
		tmp = math.cos(M) * math.exp(((M * -M) - l))
	else:
		tmp = math.cos(M) * math.exp(((-0.25 * (n * n)) - l))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -8e-153)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(m * m) * -0.25) - l)));
	elseif (n <= 6.8e-10)
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(-M)) - l)));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(-0.25 * Float64(n * n)) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -8e-153)
		tmp = cos(M) * exp((((m * m) * -0.25) - l));
	elseif (n <= 6.8e-10)
		tmp = cos(M) * exp(((M * -M) - l));
	else
		tmp = cos(M) * exp(((-0.25 * (n * n)) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -8e-153], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6.8e-10], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * (-M)), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.00000000000000031e-153

   1. Initial program 65.3%

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

   2. Taylor expanded in K around 0 91.1%

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

   4. Simplified 91.1%

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

   5. Taylor expanded in m around inf 70.0%

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

      1. *-commutative 70.0%

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

      2. unpow2 70.0%

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

   7. Simplified 70.0%

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

   if -8.00000000000000031e-153 < n < 6.8000000000000003e-10

   1. Initial program 84.3%

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

   2. Taylor expanded in K around 0 93.4%

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

   4. Simplified 93.4%

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

   5. Taylor expanded in M around inf 82.5%

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

      1. neg-mul-1 82.5%

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

      2. unpow2 82.5%

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

      3. distribute-rgt-neg-in 82.5%

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

   7. Simplified 82.5%

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

   if 6.8000000000000003e-10 < n

   1. Initial program 66.7%

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

   2. Taylor expanded in K around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in n around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. unpow2 89.7%

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

   7. Simplified 89.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-153}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25 - \ell}\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{-10}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right) - \ell}\\ \end{array} \]

Alternative 5: 75.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.5 \cdot 10^{-153} \lor \neg \left(n \leq 32000\right):\\ \;\;\;\;e^{n \cdot \left(m \cdot -0.5 + n \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right) - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -8.5e-153) (not (<= n 32000.0)))
   (exp (* n (+ (* m -0.5) (* n -0.25))))
   (* (cos M) (exp (- (* M (- M)) l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -8.5e-153) || !(n <= 32000.0)) {
		tmp = exp((n * ((m * -0.5) + (n * -0.25))));
	} else {
		tmp = cos(M) * exp(((M * -M) - l));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((n <= (-8.5d-153)) .or. (.not. (n <= 32000.0d0))) then
        tmp = exp((n * ((m * (-0.5d0)) + (n * (-0.25d0)))))
    else
        tmp = cos(m_1) * exp(((m_1 * -m_1) - l))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -8.5e-153) || !(n <= 32000.0)) {
		tmp = Math.exp((n * ((m * -0.5) + (n * -0.25))));
	} else {
		tmp = Math.cos(M) * Math.exp(((M * -M) - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -8.5e-153) or not (n <= 32000.0):
		tmp = math.exp((n * ((m * -0.5) + (n * -0.25))))
	else:
		tmp = math.cos(M) * math.exp(((M * -M) - l))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -8.5e-153) || !(n <= 32000.0))
		tmp = exp(Float64(n * Float64(Float64(m * -0.5) + Float64(n * -0.25))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(-M)) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -8.5e-153) || ~((n <= 32000.0)))
		tmp = exp((n * ((m * -0.5) + (n * -0.25))));
	else
		tmp = cos(M) * exp(((M * -M) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -8.5e-153], N[Not[LessEqual[n, 32000.0]], $MachinePrecision]], N[Exp[N[(n * N[(N[(m * -0.5), $MachinePrecision] + N[(n * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * (-M)), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -8.4999999999999996e-153 or 32000 < n

   1. Initial program 65.5%

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

   2. Taylor expanded in M around 0 61.7%

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

      1. associate-*r* 61.7%

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

      2. *-commutative 61.7%

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

      3. *-commutative 61.7%

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

      4. *-commutative 61.7%

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

   4. Simplified 61.7%

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

   5. Taylor expanded in n around inf 42.1%

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

      1. +-commutative 42.1%

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

      2. *-commutative 42.1%

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

      3. fma-def 42.1%

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

      4. unpow2 42.1%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(\color{blue}{n \cdot n}, -0.25, -0.5 \cdot \left(m \cdot n\right)\right)} \]

      5. associate-*r* 42.1%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, \color{blue}{\left(-0.5 \cdot m\right) \cdot n}\right)} \]

      6. metadata-eval 42.1%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, \left(\color{blue}{\left(-0.5\right)} \cdot m\right) \cdot n\right)} \]

      7. *-commutative 42.1%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, \color{blue}{n \cdot \left(\left(-0.5\right) \cdot m\right)}\right)} \]

      8. metadata-eval 42.1%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(\color{blue}{-0.5} \cdot m\right)\right)} \]

      9. *-commutative 42.1%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \color{blue}{\left(m \cdot -0.5\right)}\right)} \]

   7. Simplified 42.1%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\color{blue}{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(m \cdot -0.5\right)\right)}} \]

   8. Taylor expanded in K around 0 58.4%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(m \cdot n\right) + -0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r* 58.4%

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

      2. *-commutative 58.4%

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

      3. *-commutative 58.4%

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

      4. unpow2 58.4%

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

      5. associate-*l* 58.4%

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

      6. distribute-lft-out 63.1%

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

   10. Simplified 63.1%

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

   if -8.4999999999999996e-153 < n < 32000

   1. Initial program 83.9%

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

   2. Taylor expanded in K around 0 93.6%

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

   4. Simplified 93.6%

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

   5. Taylor expanded in M around inf 82.3%

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

      1. neg-mul-1 82.3%

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

      2. unpow2 82.3%

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

      3. distribute-rgt-neg-in 82.3%

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

   7. Simplified 82.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.5 \cdot 10^{-153} \lor \neg \left(n \leq 32000\right):\\ \;\;\;\;e^{n \cdot \left(m \cdot -0.5 + n \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right) - \ell}\\ \end{array} \]

Alternative 6: 72.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -86000000000000:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25 - \ell}\\ \mathbf{elif}\;m \leq 4.8 \cdot 10^{-180}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(m \cdot -0.5 + n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -86000000000000.0)
   (* (cos M) (exp (- (* (* m m) -0.25) l)))
   (if (<= m 4.8e-180)
     (* (cos M) (exp (- (* M (- M)) l)))
     (exp (* n (+ (* m -0.5) (* n -0.25)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -86000000000000.0) {
		tmp = cos(M) * exp((((m * m) * -0.25) - l));
	} else if (m <= 4.8e-180) {
		tmp = cos(M) * exp(((M * -M) - l));
	} else {
		tmp = exp((n * ((m * -0.5) + (n * -0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-86000000000000.0d0)) then
        tmp = cos(m_1) * exp((((m * m) * (-0.25d0)) - l))
    else if (m <= 4.8d-180) then
        tmp = cos(m_1) * exp(((m_1 * -m_1) - l))
    else
        tmp = exp((n * ((m * (-0.5d0)) + (n * (-0.25d0)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -86000000000000.0) {
		tmp = Math.cos(M) * Math.exp((((m * m) * -0.25) - l));
	} else if (m <= 4.8e-180) {
		tmp = Math.cos(M) * Math.exp(((M * -M) - l));
	} else {
		tmp = Math.exp((n * ((m * -0.5) + (n * -0.25))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -86000000000000.0:
		tmp = math.cos(M) * math.exp((((m * m) * -0.25) - l))
	elif m <= 4.8e-180:
		tmp = math.cos(M) * math.exp(((M * -M) - l))
	else:
		tmp = math.exp((n * ((m * -0.5) + (n * -0.25))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -86000000000000.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(m * m) * -0.25) - l)));
	elseif (m <= 4.8e-180)
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(-M)) - l)));
	else
		tmp = exp(Float64(n * Float64(Float64(m * -0.5) + Float64(n * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -86000000000000.0)
		tmp = cos(M) * exp((((m * m) * -0.25) - l));
	elseif (m <= 4.8e-180)
		tmp = cos(M) * exp(((M * -M) - l));
	else
		tmp = exp((n * ((m * -0.5) + (n * -0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -86000000000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.8e-180], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * (-M)), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(n * N[(N[(m * -0.5), $MachinePrecision] + N[(n * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.6e13

   1. Initial program 70.7%

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

   2. Taylor expanded in K around 0 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in m around inf 96.6%

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

      1. *-commutative 96.6%

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

      2. unpow2 96.6%

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

   7. Simplified 96.6%

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

   if -8.6e13 < m < 4.79999999999999959e-180

   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 87.7%

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

   4. Simplified 87.7%

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

   5. Taylor expanded in M around inf 75.9%

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

      1. neg-mul-1 75.9%

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

      2. unpow2 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

   7. Simplified 75.9%

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

   if 4.79999999999999959e-180 < m

   1. Initial program 70.2%

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

   2. Taylor expanded in M around 0 62.2%

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

      1. associate-*r* 62.2%

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

      2. *-commutative 62.2%

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

      3. *-commutative 62.2%

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

      4. *-commutative 62.2%

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

   4. Simplified 62.2%

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

   5. Taylor expanded in n around inf 26.3%

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

      1. +-commutative 26.3%

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

      2. *-commutative 26.3%

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

      3. fma-def 26.3%

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

      4. unpow2 26.3%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(\color{blue}{n \cdot n}, -0.25, -0.5 \cdot \left(m \cdot n\right)\right)} \]

      5. associate-*r* 26.3%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, \color{blue}{\left(-0.5 \cdot m\right) \cdot n}\right)} \]

      6. metadata-eval 26.3%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, \left(\color{blue}{\left(-0.5\right)} \cdot m\right) \cdot n\right)} \]

      7. *-commutative 26.3%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, \color{blue}{n \cdot \left(\left(-0.5\right) \cdot m\right)}\right)} \]

      8. metadata-eval 26.3%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(\color{blue}{-0.5} \cdot m\right)\right)} \]

      9. *-commutative 26.3%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \color{blue}{\left(m \cdot -0.5\right)}\right)} \]

   7. Simplified 26.3%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\color{blue}{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(m \cdot -0.5\right)\right)}} \]

   8. Taylor expanded in K around 0 42.9%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(m \cdot n\right) + -0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r* 42.9%

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

      2. *-commutative 42.9%

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

      3. *-commutative 42.9%

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

      4. unpow2 42.9%

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

      5. associate-*l* 42.9%

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

      6. distribute-lft-out 45.7%

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

   10. Simplified 45.7%

       \[\leadsto \color{blue}{e^{n \cdot \left(-0.5 \cdot m + n \cdot -0.25\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -86000000000000:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot m\right) \cdot -0.25 - \ell}\\ \mathbf{elif}\;m \leq 4.8 \cdot 10^{-180}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(m \cdot -0.5 + n \cdot -0.25\right)}\\ \end{array} \]

Alternative 7: 62.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.25 \cdot 10^{-167} \lor \neg \left(n \leq 54\right):\\ \;\;\;\;e^{n \cdot \left(m \cdot -0.5 + n \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -3.25e-167) (not (<= n 54.0)))
   (exp (* n (+ (* m -0.5) (* n -0.25))))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -3.25e-167) || !(n <= 54.0)) {
		tmp = exp((n * ((m * -0.5) + (n * -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 ((n <= (-3.25d-167)) .or. (.not. (n <= 54.0d0))) then
        tmp = exp((n * ((m * (-0.5d0)) + (n * (-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 ((n <= -3.25e-167) || !(n <= 54.0)) {
		tmp = Math.exp((n * ((m * -0.5) + (n * -0.25))));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -3.25e-167) or not (n <= 54.0):
		tmp = math.exp((n * ((m * -0.5) + (n * -0.25))))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -3.25e-167) || !(n <= 54.0))
		tmp = exp(Float64(n * Float64(Float64(m * -0.5) + Float64(n * -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 ((n <= -3.25e-167) || ~((n <= 54.0)))
		tmp = exp((n * ((m * -0.5) + (n * -0.25))));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -3.25e-167], N[Not[LessEqual[n, 54.0]], $MachinePrecision]], N[Exp[N[(n * N[(N[(m * -0.5), $MachinePrecision] + N[(n * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.24999999999999987e-167 or 54 < n

   1. Initial program 65.5%

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

   2. Taylor expanded in M around 0 61.7%

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

      1. associate-*r* 61.7%

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

      2. *-commutative 61.7%

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

      3. *-commutative 61.7%

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

      4. *-commutative 61.7%

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

   4. Simplified 61.7%

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

   5. Taylor expanded in n around inf 42.1%

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

      1. +-commutative 42.1%

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

      2. *-commutative 42.1%

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

      3. fma-def 42.1%

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

      4. unpow2 42.1%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(\color{blue}{n \cdot n}, -0.25, -0.5 \cdot \left(m \cdot n\right)\right)} \]

      5. associate-*r* 42.1%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, \color{blue}{\left(-0.5 \cdot m\right) \cdot n}\right)} \]

      6. metadata-eval 42.1%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, \left(\color{blue}{\left(-0.5\right)} \cdot m\right) \cdot n\right)} \]

      7. *-commutative 42.1%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, \color{blue}{n \cdot \left(\left(-0.5\right) \cdot m\right)}\right)} \]

      8. metadata-eval 42.1%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(\color{blue}{-0.5} \cdot m\right)\right)} \]

      9. *-commutative 42.1%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \color{blue}{\left(m \cdot -0.5\right)}\right)} \]

   7. Simplified 42.1%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\color{blue}{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(m \cdot -0.5\right)\right)}} \]

   8. Taylor expanded in K around 0 58.4%

      \[\leadsto \color{blue}{e^{-0.5 \cdot \left(m \cdot n\right) + -0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r* 58.4%

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

      2. *-commutative 58.4%

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

      3. *-commutative 58.4%

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

      4. unpow2 58.4%

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

      5. associate-*l* 58.4%

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

      6. distribute-lft-out 63.1%

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

   10. Simplified 63.1%

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

   if -3.24999999999999987e-167 < n < 54

   1. Initial program 83.9%

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

   2. Taylor expanded in M around 0 75.4%

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

      1. associate-*r* 75.4%

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

      2. *-commutative 75.4%

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

      3. *-commutative 75.4%

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

      4. *-commutative 75.4%

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

   4. Simplified 75.4%

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

   5. Taylor expanded in K around 0 86.1%

      \[\leadsto \color{blue}{1} \cdot e^{\left|m - n\right| - \left(\ell + {\left(m + n\right)}^{2} \cdot 0.25\right)} \]

   6. Taylor expanded in l around inf 45.4%

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

      1. neg-mul-1 45.4%

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

   8. Simplified 45.4%

      \[\leadsto 1 \cdot e^{\color{blue}{-\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.25 \cdot 10^{-167} \lor \neg \left(n \leq 54\right):\\ \;\;\;\;e^{n \cdot \left(m \cdot -0.5 + n \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 8: 35.8% 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 72.9%

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

2. Taylor expanded in M around 0 67.2%

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

   1. associate-*r* 67.2%

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

   2. *-commutative 67.2%

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

   3. *-commutative 67.2%

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

   4. *-commutative 67.2%

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

4. Simplified 67.2%

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

5. Taylor expanded in K around 0 88.2%

   \[\leadsto \color{blue}{1} \cdot e^{\left|m - n\right| - \left(\ell + {\left(m + n\right)}^{2} \cdot 0.25\right)} \]

6. Taylor expanded in l around inf 37.8%

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

   1. neg-mul-1 37.8%

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

8. Simplified 37.8%

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

9. Final simplification 37.8%

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

Alternative 9: 6.4% accurate, 38.6× speedup

Math

\[\begin{array}{l} \\ 1 + -0.125 \cdot \left(\left(m \cdot m\right) \cdot \left(K \cdot K\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (+ 1.0 (* -0.125 (* (* m m) (* K K)))))

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 + (-0.125 * ((m * m) * (K * K)));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0 + ((-0.125d0) * ((m * m) * (k * k)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 + (-0.125 * ((m * m) * (K * K)));
}

Python

def code(K, m, n, M, l):
	return 1.0 + (-0.125 * ((m * m) * (K * K)))

Julia

function code(K, m, n, M, l)
	return Float64(1.0 + Float64(-0.125 * Float64(Float64(m * m) * Float64(K * K))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0 + (-0.125 * ((m * m) * (K * K)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 + N[(-0.125 * N[(N[(m * m), $MachinePrecision] * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

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

2. Taylor expanded in M around 0 67.2%

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

   1. associate-*r* 67.2%

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

   2. *-commutative 67.2%

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

   3. *-commutative 67.2%

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

   4. *-commutative 67.2%

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

4. Simplified 67.2%

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

5. Taylor expanded in n around inf 29.9%

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

   1. +-commutative 29.9%

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

   2. *-commutative 29.9%

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

   3. fma-def 29.9%

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

   4. unpow2 29.9%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(\color{blue}{n \cdot n}, -0.25, -0.5 \cdot \left(m \cdot n\right)\right)} \]

   5. associate-*r* 29.9%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, \color{blue}{\left(-0.5 \cdot m\right) \cdot n}\right)} \]

   6. metadata-eval 29.9%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, \left(\color{blue}{\left(-0.5\right)} \cdot m\right) \cdot n\right)} \]

   7. *-commutative 29.9%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, \color{blue}{n \cdot \left(\left(-0.5\right) \cdot m\right)}\right)} \]

   8. metadata-eval 29.9%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(\color{blue}{-0.5} \cdot m\right)\right)} \]

   9. *-commutative 29.9%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \color{blue}{\left(m \cdot -0.5\right)}\right)} \]

7. Simplified 29.9%

   \[\leadsto \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\color{blue}{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(m \cdot -0.5\right)\right)}} \]

8. Taylor expanded in K around 0 13.6%

   \[\leadsto \color{blue}{\left(1 + -0.125 \cdot \left({K}^{2} \cdot {\left(m + n\right)}^{2}\right)\right)} \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(m \cdot -0.5\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 13.6%

      \[\leadsto \left(1 + -0.125 \cdot \color{blue}{\left({\left(m + n\right)}^{2} \cdot {K}^{2}\right)}\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(m \cdot -0.5\right)\right)} \]

   2. +-commutative 13.6%

      \[\leadsto \left(1 + -0.125 \cdot \left({\color{blue}{\left(n + m\right)}}^{2} \cdot {K}^{2}\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(m \cdot -0.5\right)\right)} \]

   3. unpow2 13.6%

      \[\leadsto \left(1 + -0.125 \cdot \left({\left(n + m\right)}^{2} \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(m \cdot -0.5\right)\right)} \]

10. Simplified 13.6%

    \[\leadsto \color{blue}{\left(1 + -0.125 \cdot \left({\left(n + m\right)}^{2} \cdot \left(K \cdot K\right)\right)\right)} \cdot e^{\mathsf{fma}\left(n \cdot n, -0.25, n \cdot \left(m \cdot -0.5\right)\right)} \]

11. Taylor expanded in n around 0 5.8%

    \[\leadsto \color{blue}{1 + -0.125 \cdot \left({K}^{2} \cdot {m}^{2}\right)} \]
12. Step-by-step derivation

    1. unpow2 5.8%

       \[\leadsto 1 + -0.125 \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot {m}^{2}\right) \]

    2. unpow2 5.8%

       \[\leadsto 1 + -0.125 \cdot \left(\left(K \cdot K\right) \cdot \color{blue}{\left(m \cdot m\right)}\right) \]

13. Simplified 5.8%

    \[\leadsto \color{blue}{1 + -0.125 \cdot \left(\left(K \cdot K\right) \cdot \left(m \cdot m\right)\right)} \]

14. Final simplification 5.8%

    \[\leadsto 1 + -0.125 \cdot \left(\left(m \cdot m\right) \cdot \left(K \cdot K\right)\right) \]

Reproduce

herbie shell --seed 2023277 
(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)))))))