Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.4% → 96.6%

Time: 9.9s

Alternatives: 11

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.4% 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: 96.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.2%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 96.2%

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

6. Final simplification 96.2%

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

Alternative 2: 37.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)}\\ \mathbf{if}\;t\_1 \leq 0.4 \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot -0.5\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, M \cdot M, -0.5\right), M \cdot M, 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l)))
        (t_1
         (*
          (cos (- (/ (* K (+ m n)) 2.0) M))
          (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- n m))))))))
   (if (or (<= t_1 0.4) (not (<= t_1 INFINITY)))
     (* (* (* M M) -0.5) t_0)
     (* (fma (fma 0.041666666666666664 (* M M) -0.5) (* M M) 1.0) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double t_1 = cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((n - m)))));
	double tmp;
	if ((t_1 <= 0.4) || !(t_1 <= ((double) INFINITY))) {
		tmp = ((M * M) * -0.5) * t_0;
	} else {
		tmp = fma(fma(0.041666666666666664, (M * M), -0.5), (M * M), 1.0) * t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	t_1 = 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(n - m))))))
	tmp = 0.0
	if ((t_1 <= 0.4) || !(t_1 <= Inf))
		tmp = Float64(Float64(Float64(M * M) * -0.5) * t_0);
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(M * M), -0.5), Float64(M * M), 1.0) * t_0);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = 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[(n - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.4], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(N[(N[(M * M), $MachinePrecision] * -0.5), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(M * M), $MachinePrecision] + -0.5), $MachinePrecision] * N[(M * M), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < 0.40000000000000002 or +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 75.0%

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

   3. Taylor expanded in 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 27.2

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

   5. Applied rewrites 27.2%

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

   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 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 25.1%

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

   12. Taylor expanded in M around inf

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

   14. Applied rewrites 36.4%

       \[\leadsto \left(\left(M \cdot M\right) \cdot -0.5\right) \cdot e^{-\ell} \]

   if 0.40000000000000002 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 92.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 90.3

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

   5. Applied rewrites 90.3%

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

   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 89.8

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in M around 0

      \[\leadsto \left(1 + \color{blue}{{M}^{2} \cdot \left(\frac{1}{24} \cdot {M}^{2} - \frac{1}{2}\right)}\right) \cdot e^{-\ell} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, M \cdot M, -0.5\right), \color{blue}{M \cdot M}, 1\right) \cdot e^{-\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \leq 0.4 \lor \neg \left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \leq \infty\right):\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot -0.5\right) \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, M \cdot M, -0.5\right), M \cdot M, 1\right) \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)}\\ \mathbf{if}\;t\_1 \leq 0.4 \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot -0.5\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l)))
        (t_1
         (*
          (cos (- (/ (* K (+ m n)) 2.0) M))
          (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- n m))))))))
   (if (or (<= t_1 0.4) (not (<= t_1 INFINITY)))
     (* (* (* M M) -0.5) t_0)
     (* (fma (* M M) -0.5 1.0) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double t_1 = cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((n - m)))));
	double tmp;
	if ((t_1 <= 0.4) || !(t_1 <= ((double) INFINITY))) {
		tmp = ((M * M) * -0.5) * t_0;
	} else {
		tmp = fma((M * M), -0.5, 1.0) * t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	t_1 = 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(n - m))))))
	tmp = 0.0
	if ((t_1 <= 0.4) || !(t_1 <= Inf))
		tmp = Float64(Float64(Float64(M * M) * -0.5) * t_0);
	else
		tmp = Float64(fma(Float64(M * M), -0.5, 1.0) * t_0);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = 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[(n - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.4], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(N[(N[(M * M), $MachinePrecision] * -0.5), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < 0.40000000000000002 or +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 75.0%

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

   3. Taylor expanded in 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 27.2

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

   5. Applied rewrites 27.2%

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

   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 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 25.1%

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

   12. Taylor expanded in M around inf

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

   14. Applied rewrites 36.4%

       \[\leadsto \left(\left(M \cdot M\right) \cdot -0.5\right) \cdot e^{-\ell} \]

   if 0.40000000000000002 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 92.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 90.3

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

   5. Applied rewrites 90.3%

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

   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 89.8

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

   8. Applied rewrites 89.8%

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

   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 80.4%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \leq 0.4 \lor \neg \left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \leq \infty\right):\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot -0.5\right) \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right) - M\\ t_1 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)\\ t_2 := \left|n - m\right|\\ \mathbf{if}\;t\_1 \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - t\_2\right)} \leq \infty:\\ \;\;\;\;t\_1 \cdot e^{\mathsf{fma}\left(t\_0, -t\_0, \left(-\ell\right) + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot \cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (+ n m)) M))
        (t_1 (cos (- (/ (* K (+ m n)) 2.0) M)))
        (t_2 (fabs (- n m))))
   (if (<=
        (* t_1 (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l t_2))))
        INFINITY)
     (* t_1 (exp (fma t_0 (- t_0) (+ (- l) t_2))))
     (* (exp (* (* m m) -0.25)) (cos M)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (n + m)) - M;
	double t_1 = cos((((K * (m + n)) / 2.0) - M));
	double t_2 = fabs((n - m));
	double tmp;
	if ((t_1 * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - t_2)))) <= ((double) INFINITY)) {
		tmp = t_1 * exp(fma(t_0, -t_0, (-l + t_2)));
	} else {
		tmp = exp(((m * m) * -0.25)) * cos(M);
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * Float64(n + m)) - M)
	t_1 = cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M))
	t_2 = abs(Float64(n - m))
	tmp = 0.0
	if (Float64(t_1 * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - t_2)))) <= Inf)
		tmp = Float64(t_1 * exp(fma(t_0, Float64(-t_0), Float64(Float64(-l) + t_2))));
	else
		tmp = Float64(exp(Float64(Float64(m * m) * -0.25)) * cos(M));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[Exp[N[(t$95$0 * (-t$95$0) + N[((-l) + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 94.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 94.5%

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

   if +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.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. 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 100.0%

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

   6. Taylor expanded in m around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {m}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 65.0%

      \[\leadsto e^{\left(m \cdot m\right) \cdot -0.25} \cdot \cos M \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)} \leq \infty:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\mathsf{fma}\left(0.5 \cdot \left(n + m\right) - M, -\left(0.5 \cdot \left(n + m\right) - M\right), \left(-\ell\right) + \left|n - m\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -4.5 \cdot 10^{+23} \lor \neg \left(M \leq 1.7 \cdot 10^{-5}\right):\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -4.5e+23) (not (<= M 1.7e-5)))
   (* (exp (* (- M) M)) (cos M))
   (*
    (exp (- (fabs (- n m)) (+ (pow (fma 0.5 (+ n m) (- M)) 2.0) l)))
    (fma (* (* M (+ n m)) K) 0.5 1.0))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -4.5e+23) || !(M <= 1.7e-5)) {
		tmp = exp((-M * M)) * cos(M);
	} else {
		tmp = exp((fabs((n - m)) - (pow(fma(0.5, (n + m), -M), 2.0) + l))) * fma(((M * (n + m)) * K), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -4.5e+23) || !(M <= 1.7e-5))
		tmp = Float64(exp(Float64(Float64(-M) * M)) * cos(M));
	else
		tmp = Float64(exp(Float64(abs(Float64(n - m)) - Float64((fma(0.5, Float64(n + m), Float64(-M)) ^ 2.0) + l))) * fma(Float64(Float64(M * Float64(n + m)) * K), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -4.5e+23], N[Not[LessEqual[M, 1.7e-5]], $MachinePrecision]], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(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] * N[(N[(N[(M * N[(n + m), $MachinePrecision]), $MachinePrecision] * K), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -4.49999999999999979e23 or 1.7e-5 < M

   1. Initial program 81.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 100.0%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 98.3%

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

   if -4.49999999999999979e23 < M < 1.7e-5

   1. Initial program 73.8%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 85.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -4.5 \cdot 10^{+23} \lor \neg \left(M \leq 1.7 \cdot 10^{-5}\right):\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \mathsf{fma}\left(\left(M \cdot \left(n + m\right)\right) \cdot K, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(m \cdot m\right) \cdot -0.25} \cdot \cos M\\ t_1 := e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{if}\;M \leq -0.0075:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M \leq -9.6 \cdot 10^{-137}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 8.2 \cdot 10^{-202}:\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot -0.5\right) \cdot e^{-\ell}\\ \mathbf{elif}\;M \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (* m m) -0.25)) (cos M)))
        (t_1 (* (exp (* (- M) M)) (cos M))))
   (if (<= M -0.0075)
     t_1
     (if (<= M -9.6e-137)
       t_0
       (if (<= M 8.2e-202)
         (* (* (* M M) -0.5) (exp (- l)))
         (if (<= M 5.6e-8) t_0 t_1))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((m * m) * -0.25)) * cos(M);
	double t_1 = exp((-M * M)) * cos(M);
	double tmp;
	if (M <= -0.0075) {
		tmp = t_1;
	} else if (M <= -9.6e-137) {
		tmp = t_0;
	} else if (M <= 8.2e-202) {
		tmp = ((M * M) * -0.5) * exp(-l);
	} else if (M <= 5.6e-8) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((m * m) * (-0.25d0))) * cos(m_1)
    t_1 = exp((-m_1 * m_1)) * cos(m_1)
    if (m_1 <= (-0.0075d0)) then
        tmp = t_1
    else if (m_1 <= (-9.6d-137)) then
        tmp = t_0
    else if (m_1 <= 8.2d-202) then
        tmp = ((m_1 * m_1) * (-0.5d0)) * exp(-l)
    else if (m_1 <= 5.6d-8) 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) * -0.25)) * Math.cos(M);
	double t_1 = Math.exp((-M * M)) * Math.cos(M);
	double tmp;
	if (M <= -0.0075) {
		tmp = t_1;
	} else if (M <= -9.6e-137) {
		tmp = t_0;
	} else if (M <= 8.2e-202) {
		tmp = ((M * M) * -0.5) * Math.exp(-l);
	} else if (M <= 5.6e-8) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(((m * m) * -0.25)) * math.cos(M)
	t_1 = math.exp((-M * M)) * math.cos(M)
	tmp = 0
	if M <= -0.0075:
		tmp = t_1
	elif M <= -9.6e-137:
		tmp = t_0
	elif M <= 8.2e-202:
		tmp = ((M * M) * -0.5) * math.exp(-l)
	elif M <= 5.6e-8:
		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) * -0.25)) * cos(M))
	t_1 = Float64(exp(Float64(Float64(-M) * M)) * cos(M))
	tmp = 0.0
	if (M <= -0.0075)
		tmp = t_1;
	elseif (M <= -9.6e-137)
		tmp = t_0;
	elseif (M <= 8.2e-202)
		tmp = Float64(Float64(Float64(M * M) * -0.5) * exp(Float64(-l)));
	elseif (M <= 5.6e-8)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((m * m) * -0.25)) * cos(M);
	t_1 = exp((-M * M)) * cos(M);
	tmp = 0.0;
	if (M <= -0.0075)
		tmp = t_1;
	elseif (M <= -9.6e-137)
		tmp = t_0;
	elseif (M <= 8.2e-202)
		tmp = ((M * M) * -0.5) * exp(-l);
	elseif (M <= 5.6e-8)
		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[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -0.0075], t$95$1, If[LessEqual[M, -9.6e-137], t$95$0, If[LessEqual[M, 8.2e-202], N[(N[(N[(M * M), $MachinePrecision] * -0.5), $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 5.6e-8], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -0.0074999999999999997 or 5.5999999999999999e-8 < M

   1. Initial program 81.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 100.0%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 96.1%

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

   if -0.0074999999999999997 < M < -9.6000000000000002e-137 or 8.2000000000000008e-202 < M < 5.5999999999999999e-8

   1. Initial program 68.7%

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

   3. Taylor expanded in 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 92.5%

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

   6. Taylor expanded in m around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {m}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 57.8%

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

   if -9.6000000000000002e-137 < M < 8.2000000000000008e-202

   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 39.9

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

   5. Applied rewrites 39.9%

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

   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 42.8

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

   8. Applied rewrites 42.8%

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

   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 42.8%

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

   12. Taylor expanded in M around inf

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

   14. Applied rewrites 65.2%

       \[\leadsto \left(\left(M \cdot M\right) \cdot -0.5\right) \cdot e^{-\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 72.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{if}\;M \leq -1.05 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M \leq 1.35 \cdot 10^{-200}:\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot -0.5\right) \cdot t\_0\\ \mathbf{elif}\;M \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(M \cdot M, -0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (* (exp (* (- M) M)) (cos M))))
   (if (<= M -1.05e-20)
     t_1
     (if (<= M 1.35e-200)
       (* (* (* M M) -0.5) t_0)
       (if (<= M 1.7e-5) (* (fma (* M M) -0.5 1.0) t_0) t_1)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double t_1 = exp((-M * M)) * cos(M);
	double tmp;
	if (M <= -1.05e-20) {
		tmp = t_1;
	} else if (M <= 1.35e-200) {
		tmp = ((M * M) * -0.5) * t_0;
	} else if (M <= 1.7e-5) {
		tmp = fma((M * M), -0.5, 1.0) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	t_1 = Float64(exp(Float64(Float64(-M) * M)) * cos(M))
	tmp = 0.0
	if (M <= -1.05e-20)
		tmp = t_1;
	elseif (M <= 1.35e-200)
		tmp = Float64(Float64(Float64(M * M) * -0.5) * t_0);
	elseif (M <= 1.7e-5)
		tmp = Float64(fma(Float64(M * M), -0.5, 1.0) * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -1.05e-20], t$95$1, If[LessEqual[M, 1.35e-200], N[(N[(N[(M * M), $MachinePrecision] * -0.5), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[M, 1.7e-5], N[(N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -1.0499999999999999e-20 or 1.7e-5 < M

   1. Initial program 80.8%

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

   3. Taylor expanded in 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 100.0%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 96.1%

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

   if -1.0499999999999999e-20 < M < 1.3500000000000001e-200

   1. Initial program 73.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 37.9

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

   5. Applied rewrites 37.9%

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

   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 41.8

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

   8. Applied rewrites 41.8%

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

   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 41.8%

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

   12. Taylor expanded in M around inf

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

   14. Applied rewrites 55.6%

       \[\leadsto \left(\left(M \cdot M\right) \cdot -0.5\right) \cdot e^{-\ell} \]

   if 1.3500000000000001e-200 < M < 1.7e-5

   1. Initial program 73.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 44.5

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

   5. Applied rewrites 44.5%

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

   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 49.0

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

   8. Applied rewrites 49.0%

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

   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 49.0%

       \[\leadsto \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \cdot e^{-\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 63.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 8 \cdot 10^{-226}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot \cos M\\ \mathbf{elif}\;n \leq 3.55:\\ \;\;\;\;e^{\left|n - m\right| - M \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 8e-226)
   (* (exp (* (* m m) -0.25)) (cos M))
   (if (<= n 3.55)
     (* (exp (- (fabs (- n m)) (* M M))) (cos M))
     (* (cos M) (exp (* (* n n) -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 8e-226) {
		tmp = exp(((m * m) * -0.25)) * cos(M);
	} else if (n <= 3.55) {
		tmp = exp((fabs((n - m)) - (M * M))) * cos(M);
	} else {
		tmp = cos(M) * exp(((n * 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 (n <= 8d-226) then
        tmp = exp(((m * m) * (-0.25d0))) * cos(m_1)
    else if (n <= 3.55d0) then
        tmp = exp((abs((n - m)) - (m_1 * m_1))) * cos(m_1)
    else
        tmp = cos(m_1) * exp(((n * 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 (n <= 8e-226) {
		tmp = Math.exp(((m * m) * -0.25)) * Math.cos(M);
	} else if (n <= 3.55) {
		tmp = Math.exp((Math.abs((n - m)) - (M * M))) * Math.cos(M);
	} else {
		tmp = Math.cos(M) * Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 8e-226:
		tmp = math.exp(((m * m) * -0.25)) * math.cos(M)
	elif n <= 3.55:
		tmp = math.exp((math.fabs((n - m)) - (M * M))) * math.cos(M)
	else:
		tmp = math.cos(M) * math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 8e-226)
		tmp = Float64(exp(Float64(Float64(m * m) * -0.25)) * cos(M));
	elseif (n <= 3.55)
		tmp = Float64(exp(Float64(abs(Float64(n - m)) - Float64(M * M))) * cos(M));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(n * n) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 8e-226)
		tmp = exp(((m * m) * -0.25)) * cos(M);
	elseif (n <= 3.55)
		tmp = exp((abs((n - m)) - (M * M))) * cos(M);
	else
		tmp = cos(M) * exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 8e-226], N[(N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.55], N[(N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 7.99999999999999937e-226

   1. Initial program 79.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 96.0%

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

   6. Taylor expanded in m around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {m}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 55.5%

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

   if 7.99999999999999937e-226 < n < 3.5499999999999998

   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. 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 93.5%

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

   6. Taylor expanded in M around inf

      \[\leadsto e^{\left|m - n\right| - {M}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 58.5%

      \[\leadsto e^{\left|m - n\right| - M \cdot M} \cdot \cos M \]

   if 3.5499999999999998 < n

   1. Initial program 72.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

   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 98.5

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

   8. Applied rewrites 98.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 8 \cdot 10^{-226}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot \cos M\\ \mathbf{elif}\;n \leq 3.55:\\ \;\;\;\;e^{\left|n - m\right| - M \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 5.6 \cdot 10^{-226}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25} \cdot \cos M\\ \mathbf{elif}\;n \leq 3.55:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 5.6e-226)
   (* (exp (* (* m m) -0.25)) (cos M))
   (if (<= n 3.55)
     (* (exp (* (- M) M)) (cos M))
     (* (cos M) (exp (* (* n n) -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 5.6e-226) {
		tmp = exp(((m * m) * -0.25)) * cos(M);
	} else if (n <= 3.55) {
		tmp = exp((-M * M)) * cos(M);
	} else {
		tmp = cos(M) * exp(((n * 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 (n <= 5.6d-226) then
        tmp = exp(((m * m) * (-0.25d0))) * cos(m_1)
    else if (n <= 3.55d0) then
        tmp = exp((-m_1 * m_1)) * cos(m_1)
    else
        tmp = cos(m_1) * exp(((n * 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 (n <= 5.6e-226) {
		tmp = Math.exp(((m * m) * -0.25)) * Math.cos(M);
	} else if (n <= 3.55) {
		tmp = Math.exp((-M * M)) * Math.cos(M);
	} else {
		tmp = Math.cos(M) * Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 5.6e-226:
		tmp = math.exp(((m * m) * -0.25)) * math.cos(M)
	elif n <= 3.55:
		tmp = math.exp((-M * M)) * math.cos(M)
	else:
		tmp = math.cos(M) * math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 5.6e-226)
		tmp = Float64(exp(Float64(Float64(m * m) * -0.25)) * cos(M));
	elseif (n <= 3.55)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * cos(M));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(n * n) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 5.6e-226)
		tmp = exp(((m * m) * -0.25)) * cos(M);
	elseif (n <= 3.55)
		tmp = exp((-M * M)) * cos(M);
	else
		tmp = cos(M) * exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 5.6e-226], N[(N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.55], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 5.60000000000000016e-226

   1. Initial program 79.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 96.0%

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

   6. Taylor expanded in m around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {m}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 55.5%

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

   if 5.60000000000000016e-226 < n < 3.5499999999999998

   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. 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 93.5%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 58.1%

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

   if 3.5499999999999998 < n

   1. Initial program 72.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

   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 98.5

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

   8. Applied rewrites 98.5%

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

Alternative 10: 39.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ \mathbf{if}\;M \leq -7.5 \cdot 10^{-165} \lor \neg \left(M \leq 1.35 \cdot 10^{-200}\right):\\ \;\;\;\;\cos M \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot -0.5\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l))))
   (if (or (<= M -7.5e-165) (not (<= M 1.35e-200)))
     (* (cos M) t_0)
     (* (* (* M M) -0.5) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double tmp;
	if ((M <= -7.5e-165) || !(M <= 1.35e-200)) {
		tmp = cos(M) * t_0;
	} else {
		tmp = ((M * M) * -0.5) * 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(-l)
    if ((m_1 <= (-7.5d-165)) .or. (.not. (m_1 <= 1.35d-200))) then
        tmp = cos(m_1) * t_0
    else
        tmp = ((m_1 * m_1) * (-0.5d0)) * 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(-l);
	double tmp;
	if ((M <= -7.5e-165) || !(M <= 1.35e-200)) {
		tmp = Math.cos(M) * t_0;
	} else {
		tmp = ((M * M) * -0.5) * t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-l)
	tmp = 0
	if (M <= -7.5e-165) or not (M <= 1.35e-200):
		tmp = math.cos(M) * t_0
	else:
		tmp = ((M * M) * -0.5) * t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	tmp = 0.0
	if ((M <= -7.5e-165) || !(M <= 1.35e-200))
		tmp = Float64(cos(M) * t_0);
	else
		tmp = Float64(Float64(Float64(M * M) * -0.5) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-l);
	tmp = 0.0;
	if ((M <= -7.5e-165) || ~((M <= 1.35e-200)))
		tmp = cos(M) * t_0;
	else
		tmp = ((M * M) * -0.5) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, If[Or[LessEqual[M, -7.5e-165], N[Not[LessEqual[M, 1.35e-200]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(M * M), $MachinePrecision] * -0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -7.5000000000000002e-165 or 1.3500000000000001e-200 < M

   1. Initial program 77.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 35.2

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

   5. Applied rewrites 35.2%

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

   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 41.5

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

   8. Applied rewrites 41.5%

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

   if -7.5000000000000002e-165 < M < 1.3500000000000001e-200

   1. Initial program 75.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 34.8

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

   5. Applied rewrites 34.8%

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

   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 38.3

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

   8. Applied rewrites 38.3%

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

   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 38.3%

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

   12. Taylor expanded in M around inf

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

   14. Applied rewrites 66.5%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -7.5 \cdot 10^{-165} \lor \neg \left(M \leq 1.35 \cdot 10^{-200}\right):\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot -0.5\right) \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.0% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (K m n M l) :precision binary64 (* (* (* M M) -0.5) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	return ((M * M) * -0.5) * 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 = ((m_1 * m_1) * (-0.5d0)) * exp(-l)
end function

Java

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

Python

def code(K, m, n, M, l):
	return ((M * M) * -0.5) * math.exp(-l)

Julia

function code(K, m, n, M, l)
	return Float64(Float64(Float64(M * M) * -0.5) * exp(Float64(-l)))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = ((M * M) * -0.5) * exp(-l);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[(N[(M * M), $MachinePrecision] * -0.5), $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 35.1

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

5. Applied rewrites 35.1%

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

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 40.9

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

8. Applied rewrites 40.9%

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

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 32.0%

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

12. Taylor expanded in M around inf

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

14. Applied rewrites 32.0%

    \[\leadsto \left(\left(M \cdot M\right) \cdot -0.5\right) \cdot e^{-\ell} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(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)))))))