Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.4% → 96.5%

Time: 10.4s

Alternatives: 9

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% 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: 96.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -1 \cdot 10^{+147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 3 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -1e+147)
     t_0
     (if (<= M 3e+83)
       (*
        (fma (* M M) -0.5 1.0)
        (exp (- (fabs (- n m)) (+ (pow (fma 0.5 (+ n m) (- M)) 2.0) l))))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -1e+147) {
		tmp = t_0;
	} else if (M <= 3e+83) {
		tmp = fma((M * M), -0.5, 1.0) * exp((fabs((n - m)) - (pow(fma(0.5, (n + m), -M), 2.0) + l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -1e+147)
		tmp = t_0;
	elseif (M <= 3e+83)
		tmp = Float64(fma(Float64(M * M), -0.5, 1.0) * exp(Float64(abs(Float64(n - m)) - Float64((fma(0.5, Float64(n + m), Float64(-M)) ^ 2.0) + l))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -1e+147], t$95$0, If[LessEqual[M, 3e+83], N[(N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[Power[N[(0.5 * N[(n + m), $MachinePrecision] + (-M)), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -9.9999999999999998e146 or 3e83 < M

   1. Initial program 83.3%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 27.1

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

   8. Applied rewrites 27.1%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 27.1%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

          \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       4. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       5. lower-neg.f64 97.3

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

   14. Applied rewrites 97.3%

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

   if -9.9999999999999998e146 < M < 3e83

   1. Initial program 73.3%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(\frac{1}{2}, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {M}^{2}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 92.8%

      \[\leadsto e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+147}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;M \leq 3 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in K around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 94.0%

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

6. Final simplification 94.0%

   \[\leadsto \cos M \cdot e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \]
7. Add Preprocessing

Alternative 3: 89.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -1 \cdot 10^{+147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 3 \cdot 10^{+83}:\\ \;\;\;\;\left(-0.5 \cdot \left(M \cdot M\right)\right) \cdot e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -1e+147)
     t_0
     (if (<= M 3e+83)
       (*
        (* -0.5 (* M M))
        (exp (- (fabs (- n m)) (+ (pow (fma 0.5 (+ n m) (- M)) 2.0) l))))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -1e+147) {
		tmp = t_0;
	} else if (M <= 3e+83) {
		tmp = (-0.5 * (M * M)) * exp((fabs((n - m)) - (pow(fma(0.5, (n + m), -M), 2.0) + l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -1e+147)
		tmp = t_0;
	elseif (M <= 3e+83)
		tmp = Float64(Float64(-0.5 * Float64(M * M)) * exp(Float64(abs(Float64(n - m)) - Float64((fma(0.5, Float64(n + m), Float64(-M)) ^ 2.0) + l))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -1e+147], t$95$0, If[LessEqual[M, 3e+83], N[(N[(-0.5 * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[Power[N[(0.5 * N[(n + m), $MachinePrecision] + (-M)), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -9.9999999999999998e146 or 3e83 < M

   1. Initial program 83.3%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 27.1

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

   8. Applied rewrites 27.1%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 27.1%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

          \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       4. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       5. lower-neg.f64 97.3

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

   14. Applied rewrites 97.3%

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

   if -9.9999999999999998e146 < M < 3e83

   1. Initial program 73.3%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(\frac{1}{2}, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {M}^{2}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 92.8%

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

   9. Taylor expanded in M around inf

      \[\leadsto e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(\frac{1}{2}, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \left(\frac{-1}{2} \cdot {M}^{\color{blue}{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 84.0%

       \[\leadsto e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \left(\left(M \cdot M\right) \cdot -0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+147}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;M \leq 3 \cdot 10^{+83}:\\ \;\;\;\;\left(-0.5 \cdot \left(M \cdot M\right)\right) \cdot e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -85000:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot 1\\ \mathbf{elif}\;m \leq -8.5 \cdot 10^{-73}:\\ \;\;\;\;e^{-\ell} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;m \leq -1.66 \cdot 10^{-183}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -85000.0)
   (* (exp (* -0.25 (* m m))) 1.0)
   (if (<= m -8.5e-73)
     (* (exp (- l)) (fma (* M M) -0.5 1.0))
     (if (<= m -1.66e-183)
       (* (exp (* (- M) M)) 1.0)
       (* (exp (* (* n n) -0.25)) (cos M))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -85000.0) {
		tmp = exp((-0.25 * (m * m))) * 1.0;
	} else if (m <= -8.5e-73) {
		tmp = exp(-l) * fma((M * M), -0.5, 1.0);
	} else if (m <= -1.66e-183) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = exp(((n * n) * -0.25)) * cos(M);
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -85000.0)
		tmp = Float64(exp(Float64(-0.25 * Float64(m * m))) * 1.0);
	elseif (m <= -8.5e-73)
		tmp = Float64(exp(Float64(-l)) * fma(Float64(M * M), -0.5, 1.0));
	elseif (m <= -1.66e-183)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * cos(M));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -85000.0], N[(N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[m, -8.5e-73], N[(N[Exp[(-l)], $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -1.66e-183], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -85000

   1. Initial program 68.6%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 17.1

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

   5. Applied rewrites 17.1%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 23.6

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

   8. Applied rewrites 23.6%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 23.6%

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

   12. Taylor expanded in m around inf

       \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot {m}^{2}}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 94.2

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

   14. Applied rewrites 94.2%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]

   if -85000 < m < -8.4999999999999996e-73

   1. Initial program 74.7%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 33.8

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

   5. Applied rewrites 33.8%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 33.9

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

   8. Applied rewrites 33.9%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 38.9%

       \[\leadsto \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \cdot e^{-\ell} \]

   if -8.4999999999999996e-73 < m < -1.66e-183

   1. Initial program 82.4%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.9

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

   5. Applied rewrites 30.9%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 32.0

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

   8. Applied rewrites 32.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 32.0%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

          \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       4. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       5. lower-neg.f64 72.2

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

   14. Applied rewrites 72.2%

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

   if -1.66e-183 < m

   1. Initial program 78.0%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{{n}^{2} \cdot \frac{-1}{4}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{{n}^{2} \cdot \frac{-1}{4}}} \]

      3. unpow2 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot \frac{-1}{4}} \]

      4. lower-*.f64 44.1

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(n \cdot n\right) \cdot \frac{-1}{4}} \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(n \cdot n\right) \cdot \frac{-1}{4}} \]

      2. lower-cos.f64 55.4

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

   8. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(n \cdot n\right) \cdot -0.25} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -85000:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot 1\\ \mathbf{elif}\;m \leq -8.5 \cdot 10^{-73}:\\ \;\;\;\;e^{-\ell} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;m \leq -1.66 \cdot 10^{-183}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ t_1 := e^{-0.25 \cdot \left(m \cdot m\right)} \cdot 1\\ \mathbf{if}\;m \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;m \leq -7.1 \cdot 10^{-233}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 10^{-282}:\\ \;\;\;\;e^{-\ell} \cdot 1\\ \mathbf{elif}\;m \leq 4.8 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)) (t_1 (* (exp (* -0.25 (* m m))) 1.0)))
   (if (<= m -3.6e+14)
     t_1
     (if (<= m -7.1e-233)
       t_0
       (if (<= m 1e-282) (* (exp (- l)) 1.0) (if (<= m 4.8e-11) t_0 t_1))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double t_1 = exp((-0.25 * (m * m))) * 1.0;
	double tmp;
	if (m <= -3.6e+14) {
		tmp = t_1;
	} else if (m <= -7.1e-233) {
		tmp = t_0;
	} else if (m <= 1e-282) {
		tmp = exp(-l) * 1.0;
	} else if (m <= 4.8e-11) {
		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_1 * m_1)) * 1.0d0
    t_1 = exp(((-0.25d0) * (m * m))) * 1.0d0
    if (m <= (-3.6d+14)) then
        tmp = t_1
    else if (m <= (-7.1d-233)) then
        tmp = t_0
    else if (m <= 1d-282) then
        tmp = exp(-l) * 1.0d0
    else if (m <= 4.8d-11) 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.exp((-M * M)) * 1.0;
	double t_1 = Math.exp((-0.25 * (m * m))) * 1.0;
	double tmp;
	if (m <= -3.6e+14) {
		tmp = t_1;
	} else if (m <= -7.1e-233) {
		tmp = t_0;
	} else if (m <= 1e-282) {
		tmp = Math.exp(-l) * 1.0;
	} else if (m <= 4.8e-11) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-M * M)) * 1.0
	t_1 = math.exp((-0.25 * (m * m))) * 1.0
	tmp = 0
	if m <= -3.6e+14:
		tmp = t_1
	elif m <= -7.1e-233:
		tmp = t_0
	elif m <= 1e-282:
		tmp = math.exp(-l) * 1.0
	elif m <= 4.8e-11:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	t_1 = Float64(exp(Float64(-0.25 * Float64(m * m))) * 1.0)
	tmp = 0.0
	if (m <= -3.6e+14)
		tmp = t_1;
	elseif (m <= -7.1e-233)
		tmp = t_0;
	elseif (m <= 1e-282)
		tmp = Float64(exp(Float64(-l)) * 1.0);
	elseif (m <= 4.8e-11)
		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)) * 1.0;
	t_1 = exp((-0.25 * (m * m))) * 1.0;
	tmp = 0.0;
	if (m <= -3.6e+14)
		tmp = t_1;
	elseif (m <= -7.1e-233)
		tmp = t_0;
	elseif (m <= 1e-282)
		tmp = exp(-l) * 1.0;
	elseif (m <= 4.8e-11)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[m, -3.6e+14], t$95$1, If[LessEqual[m, -7.1e-233], t$95$0, If[LessEqual[m, 1e-282], N[(N[Exp[(-l)], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[m, 4.8e-11], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.6e14 or 4.8000000000000002e-11 < m

   1. Initial program 71.9%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 22.9

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

   5. Applied rewrites 22.9%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 32.7

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

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 32.0%

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

   12. Taylor expanded in m around inf

       \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot {m}^{2}}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 93.6

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

   14. Applied rewrites 93.6%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]

   if -3.6e14 < m < -7.1e-233 or 1e-282 < m < 4.8000000000000002e-11

   1. Initial program 76.8%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 35.4

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

   5. Applied rewrites 35.4%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 37.7

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

   8. Applied rewrites 37.7%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 37.7%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

          \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       4. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       5. lower-neg.f64 50.3

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

   14. Applied rewrites 50.3%

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

   if -7.1e-233 < m < 1e-282

   1. Initial program 93.8%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 48.4

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

   8. Applied rewrites 48.4%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 48.4%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot 1\\ \mathbf{elif}\;m \leq -7.1 \cdot 10^{-233}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;m \leq 10^{-282}:\\ \;\;\;\;e^{-\ell} \cdot 1\\ \mathbf{elif}\;m \leq 4.8 \cdot 10^{-11}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -85000:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot 1\\ \mathbf{elif}\;m \leq -8.5 \cdot 10^{-73}:\\ \;\;\;\;e^{-\ell} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;m \leq -1 \cdot 10^{-185}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -85000.0)
   (* (exp (* -0.25 (* m m))) 1.0)
   (if (<= m -8.5e-73)
     (* (exp (- l)) (fma (* M M) -0.5 1.0))
     (if (<= m -1e-185)
       (* (exp (* (- M) M)) 1.0)
       (* (exp (* (* n n) -0.25)) 1.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -85000.0) {
		tmp = exp((-0.25 * (m * m))) * 1.0;
	} else if (m <= -8.5e-73) {
		tmp = exp(-l) * fma((M * M), -0.5, 1.0);
	} else if (m <= -1e-185) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = exp(((n * n) * -0.25)) * 1.0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -85000.0)
		tmp = Float64(exp(Float64(-0.25 * Float64(m * m))) * 1.0);
	elseif (m <= -8.5e-73)
		tmp = Float64(exp(Float64(-l)) * fma(Float64(M * M), -0.5, 1.0));
	elseif (m <= -1e-185)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * 1.0);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -85000.0], N[(N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[m, -8.5e-73], N[(N[Exp[(-l)], $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -1e-185], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -85000

   1. Initial program 68.6%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 17.1

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

   5. Applied rewrites 17.1%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 23.6

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

   8. Applied rewrites 23.6%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 23.6%

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

   12. Taylor expanded in m around inf

       \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot {m}^{2}}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 94.2

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

   14. Applied rewrites 94.2%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]

   if -85000 < m < -8.4999999999999996e-73

   1. Initial program 74.7%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 33.8

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

   5. Applied rewrites 33.8%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 33.9

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

   8. Applied rewrites 33.9%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 38.9%

       \[\leadsto \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \cdot e^{-\ell} \]

   if -8.4999999999999996e-73 < m < -9.9999999999999999e-186

   1. Initial program 82.4%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.9

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

   5. Applied rewrites 30.9%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 32.0

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

   8. Applied rewrites 32.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 32.0%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

          \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       4. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       5. lower-neg.f64 72.2

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

   14. Applied rewrites 72.2%

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

   if -9.9999999999999999e-186 < m

   1. Initial program 78.0%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{{n}^{2} \cdot \frac{-1}{4}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{{n}^{2} \cdot \frac{-1}{4}}} \]

      3. unpow2 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot \frac{-1}{4}} \]

      4. lower-*.f64 44.1

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(n \cdot n\right) \cdot \frac{-1}{4}} \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(n \cdot n\right) \cdot \frac{-1}{4}} \]

      2. lower-cos.f64 55.4

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

   8. Applied rewrites 55.4%

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

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\left(n \cdot n\right) \cdot \frac{-1}{4}} \]
   10. Step-by-step derivation

   11. Applied rewrites 55.3%

       \[\leadsto 1 \cdot e^{\left(n \cdot n\right) \cdot -0.25} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -85000:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot 1\\ \mathbf{elif}\;m \leq -8.5 \cdot 10^{-73}:\\ \;\;\;\;e^{-\ell} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{elif}\;m \leq -1 \cdot 10^{-185}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot 1\\ \mathbf{elif}\;m \leq -1 \cdot 10^{-185}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -3.6e+14)
   (* (exp (* -0.25 (* m m))) 1.0)
   (if (<= m -1e-185)
     (* (exp (* (- M) M)) 1.0)
     (* (exp (* (* n n) -0.25)) 1.0))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -3.6e+14) {
		tmp = exp((-0.25 * (m * m))) * 1.0;
	} else if (m <= -1e-185) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = exp(((n * n) * -0.25)) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-3.6d+14)) then
        tmp = exp(((-0.25d0) * (m * m))) * 1.0d0
    else if (m <= (-1d-185)) then
        tmp = exp((-m_1 * m_1)) * 1.0d0
    else
        tmp = exp(((n * n) * (-0.25d0))) * 1.0d0
    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 <= -3.6e+14) {
		tmp = Math.exp((-0.25 * (m * m))) * 1.0;
	} else if (m <= -1e-185) {
		tmp = Math.exp((-M * M)) * 1.0;
	} else {
		tmp = Math.exp(((n * n) * -0.25)) * 1.0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -3.6e+14:
		tmp = math.exp((-0.25 * (m * m))) * 1.0
	elif m <= -1e-185:
		tmp = math.exp((-M * M)) * 1.0
	else:
		tmp = math.exp(((n * n) * -0.25)) * 1.0
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -3.6e+14)
		tmp = Float64(exp(Float64(-0.25 * Float64(m * m))) * 1.0);
	elseif (m <= -1e-185)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -3.6e+14)
		tmp = exp((-0.25 * (m * m))) * 1.0;
	elseif (m <= -1e-185)
		tmp = exp((-M * M)) * 1.0;
	else
		tmp = exp(((n * n) * -0.25)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -3.6e+14], N[(N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[m, -1e-185], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.6e14

   1. Initial program 68.6%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 17.1

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

   5. Applied rewrites 17.1%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 23.6

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

   8. Applied rewrites 23.6%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 23.6%

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

   12. Taylor expanded in m around inf

       \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot {m}^{2}}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 94.2

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

   14. Applied rewrites 94.2%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]

   if -3.6e14 < m < -9.9999999999999999e-186

   1. Initial program 78.4%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 32.4

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 33.0

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

   8. Applied rewrites 33.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 33.0%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

          \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       4. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       5. lower-neg.f64 49.5

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

   14. Applied rewrites 49.5%

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

   if -9.9999999999999999e-186 < m

   1. Initial program 78.0%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{{n}^{2} \cdot \frac{-1}{4}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{{n}^{2} \cdot \frac{-1}{4}}} \]

      3. unpow2 N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot \frac{-1}{4}} \]

      4. lower-*.f64 44.1

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(n \cdot n\right) \cdot \frac{-1}{4}} \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(n \cdot n\right) \cdot \frac{-1}{4}} \]

      2. lower-cos.f64 55.4

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

   8. Applied rewrites 55.4%

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

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\left(n \cdot n\right) \cdot \frac{-1}{4}} \]
   10. Step-by-step derivation

   11. Applied rewrites 55.3%

       \[\leadsto 1 \cdot e^{\left(n \cdot n\right) \cdot -0.25} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot 1\\ \mathbf{elif}\;m \leq -1 \cdot 10^{-185}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -27:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 2 \cdot 10^{-17}:\\ \;\;\;\;e^{-\ell} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -27.0) t_0 (if (<= M 2e-17) (* (exp (- l)) 1.0) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -27.0) {
		tmp = t_0;
	} else if (M <= 2e-17) {
		tmp = exp(-l) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((-m_1 * m_1)) * 1.0d0
    if (m_1 <= (-27.0d0)) then
        tmp = t_0
    else if (m_1 <= 2d-17) then
        tmp = exp(-l) * 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -27.0) {
		tmp = t_0;
	} else if (M <= 2e-17) {
		tmp = Math.exp(-l) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-M * M)) * 1.0
	tmp = 0
	if M <= -27.0:
		tmp = t_0
	elif M <= 2e-17:
		tmp = math.exp(-l) * 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -27.0)
		tmp = t_0;
	elseif (M <= 2e-17)
		tmp = Float64(exp(Float64(-l)) * 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-M * M)) * 1.0;
	tmp = 0.0;
	if (M <= -27.0)
		tmp = t_0;
	elseif (M <= 2e-17)
		tmp = exp(-l) * 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -27.0], t$95$0, If[LessEqual[M, 2e-17], N[(N[Exp[(-l)], $MachinePrecision] * 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -27 or 2.00000000000000014e-17 < M

   1. Initial program 74.2%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 16.6

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

   5. Applied rewrites 16.6%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 25.3

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

   8. Applied rewrites 25.3%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 24.5%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

          \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       4. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(\mathsf{neg}\left(M\right)\right) \cdot M}} \]

       5. lower-neg.f64 93.5

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

   14. Applied rewrites 93.5%

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

   if -27 < M < 2.00000000000000014e-17

   1. Initial program 77.9%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 42.8

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 45.9

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

   8. Applied rewrites 45.9%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 45.9%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -27:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;M \leq 2 \cdot 10^{-17}:\\ \;\;\;\;e^{-\ell} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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 = exp(-l) * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in l around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 30.5

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

5. Applied rewrites 30.5%

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

6. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. lower-cos.f64 36.2

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

8. Applied rewrites 36.2%

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

9. Taylor expanded in M around 0

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

11. Applied rewrites 35.9%

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

12. Final simplification 35.9%

    \[\leadsto e^{-\ell} \cdot 1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(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)))))))