Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.6% → 96.7%

Time: 15.3s

Alternatives: 8

Speedup: 2.6×

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: 75.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((Math.abs((m - n)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((Math.cos((((K * (m + n)) / 2.0) - M)) * t_0) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * Math.cos(((K / (2.0 / (m + n))) - M));
	} else {
		tmp = t_0 * Math.cos((K * 0.5));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(((math.fabs((m - n)) - l) - math.pow((((m + n) / 2.0) - M), 2.0)))
	tmp = 0
	if (math.cos((((K * (m + n)) / 2.0) - M)) * t_0) <= math.inf:
		tmp = t_0 * math.cos(((K / (2.0 / (m + n))) - M))
	else:
		tmp = t_0 * math.cos((K * 0.5))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((abs((m - n)) - l) - ((((m + n) / 2.0) - M) ^ 2.0)));
	tmp = 0.0;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_0) <= Inf)
		tmp = t_0 * cos(((K / (2.0 / (m + n))) - M));
	else
		tmp = t_0 * cos((K * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], Infinity], N[(t$95$0 * N[Cos[N[(N[(K / N[(2.0 / N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $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 99.0%

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

      1. lift-+.f64 N/A

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

      2. associate-/l* N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   if +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #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 m around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 46.0

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

   5. Simplified 46.0%

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

   6. Taylor expanded in K around inf

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

      1. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

4. Final simplification 99.3%

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

Alternative 2: 96.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(m + n), $MachinePrecision] * 0.5 + (-M)), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(t$95$1 - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], Infinity], N[(N[Exp[(-N[(t$95$0 * t$95$0 + N[(l - t$95$1), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Cos[N[(n * N[(K * 0.5), $MachinePrecision] + (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[Cos[N[(K * 0.5), $MachinePrecision]], $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 99.0%

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

   3. Taylor expanded in m around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 98.5

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

   5. Simplified 98.5%

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

   7. Applied egg-rr 98.5%

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

   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 m around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 46.0

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

   5. Simplified 46.0%

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

   6. Taylor expanded in K around inf

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

      1. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

4. Final simplification 98.8%

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

Alternative 3: 96.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(m + n\right) \cdot 0.5 - M\\ e^{\left|m - n\right| - \mathsf{fma}\left(t\_0, t\_0, \ell\right)} \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* (+ m n) 0.5) M))) (exp (- (fabs (- m n)) (fma t_0 t_0 l)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = ((m + n) * 0.5) - M;
	return exp((fabs((m - n)) - fma(t_0, t_0, l)));
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(Float64(m + n) * 0.5) - M)
	return exp(Float64(abs(Float64(m - n)) - fma(t_0, t_0, l)))
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision]}, N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(t$95$0 * t$95$0 + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 79.7%

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

3. Taylor expanded in m around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 88.3

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

5. Simplified 88.3%

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

6. Taylor expanded in K around inf

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

   1. lower-*.f64 96.8

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

8. Simplified 96.8%

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

9. Taylor expanded in K around 0

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

11. Simplified 97.2%

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

12. Taylor expanded in m around inf

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

    1. lower-exp.f64 N/A

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

    2. lower--.f64 N/A

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

    3. sub-neg N/A

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

    4. mul-1-neg N/A

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

    5. lower-fabs.f64 N/A

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

    6. mul-1-neg N/A

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

    7. sub-neg N/A

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

    8. lower--.f64 N/A

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

    9. +-commutative N/A

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

    10. unpow2 N/A

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

    11. lower-fma.f64 N/A

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

    12. lower--.f64 N/A

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

    13. lower-*.f64 N/A

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

    14. +-commutative N/A

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

    15. lower-+.f64 N/A

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

    16. lower--.f64 N/A

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

    17. lower-*.f64 N/A

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

    18. +-commutative N/A

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

    19. lower-+.f64 97.2

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

14. Simplified 97.2%

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

15. Final simplification 97.2%

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

Alternative 4: 42.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ \mathbf{if}\;M \leq -5.5 \cdot 10^{-47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 4.1 \cdot 10^{-101}:\\ \;\;\;\;t\_0 \cdot \left(-0.5 \cdot \left(M \cdot \left(M \cdot M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l))))
   (if (<= M -5.5e-47)
     t_0
     (if (<= M 4.1e-101) (* t_0 (* -0.5 (* M (* M M)))) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double tmp;
	if (M <= -5.5e-47) {
		tmp = t_0;
	} else if (M <= 4.1e-101) {
		tmp = t_0 * (-0.5 * (M * (M * M)));
	} 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(-l)
    if (m_1 <= (-5.5d-47)) then
        tmp = t_0
    else if (m_1 <= 4.1d-101) then
        tmp = t_0 * ((-0.5d0) * (m_1 * (m_1 * m_1)))
    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(-l);
	double tmp;
	if (M <= -5.5e-47) {
		tmp = t_0;
	} else if (M <= 4.1e-101) {
		tmp = t_0 * (-0.5 * (M * (M * M)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-l)
	tmp = 0
	if M <= -5.5e-47:
		tmp = t_0
	elif M <= 4.1e-101:
		tmp = t_0 * (-0.5 * (M * (M * M)))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	tmp = 0.0
	if (M <= -5.5e-47)
		tmp = t_0;
	elseif (M <= 4.1e-101)
		tmp = Float64(t_0 * Float64(-0.5 * Float64(M * Float64(M * M))));
	else
		tmp = 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 <= -5.5e-47)
		tmp = t_0;
	elseif (M <= 4.1e-101)
		tmp = t_0 * (-0.5 * (M * (M * M)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[M, -5.5e-47], t$95$0, If[LessEqual[M, 4.1e-101], N[(t$95$0 * N[(-0.5 * N[(M * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -5.5000000000000002e-47 or 4.10000000000000026e-101 < M

   1. Initial program 81.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 31.7

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

   5. Simplified 31.7%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lower-exp.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 34.7

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

   8. Simplified 34.7%

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

   9. Taylor expanded in M around 0

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

       1. neg-mul-1 N/A

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

       2. lower-exp.f64 N/A

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

       3. neg-mul-1 N/A

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

       4. lower-neg.f64 34.0

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

   11. Simplified 34.0%

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

   if -5.5000000000000002e-47 < M < 4.10000000000000026e-101

   1. Initial program 77.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 33.4

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

   5. Simplified 33.4%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lower-exp.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 37.3

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

   8. Simplified 37.3%

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

   9. Taylor expanded in M around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 37.3

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

   11. Simplified 37.3%

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

   12. Taylor expanded in M around inf

       \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {M}^{3}\right)} \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {M}^{3}\right)} \cdot e^{\mathsf{neg}\left(\ell\right)} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 57.0

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

   14. Simplified 57.0%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -5.5 \cdot 10^{-47}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;M \leq 4.1 \cdot 10^{-101}:\\ \;\;\;\;e^{-\ell} \cdot \left(-0.5 \cdot \left(M \cdot \left(M \cdot M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 36.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

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

   1. lower-*.f64 N/A

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

   2. cos-neg N/A

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

   3. lower-cos.f64 N/A

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

   4. neg-mul-1 N/A

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

   5. lower-exp.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. lower-neg.f64 35.7

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

8. Simplified 35.7%

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

9. Taylor expanded in M around 0

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

    1. neg-mul-1 N/A

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

    2. lower-exp.f64 N/A

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

    3. neg-mul-1 N/A

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

    4. lower-neg.f64 35.3

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

11. Simplified 35.3%

    \[\leadsto \color{blue}{e^{-\ell}} \]
12. Add Preprocessing

Alternative 6: 9.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, M \cdot M, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(-0.16666666666666666, \ell, 0.5\right), -1\right), 1\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (fma -0.5 (* M M) 1.0)
  (fma l (fma l (fma -0.16666666666666666 l 0.5) -1.0) 1.0)))

C

double code(double K, double m, double n, double M, double l) {
	return fma(-0.5, (M * M), 1.0) * fma(l, fma(l, fma(-0.16666666666666666, l, 0.5), -1.0), 1.0);
}

Julia

function code(K, m, n, M, l)
	return Float64(fma(-0.5, Float64(M * M), 1.0) * fma(l, fma(l, fma(-0.16666666666666666, l, 0.5), -1.0), 1.0))
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[(-0.5 * N[(M * M), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(l * N[(-0.16666666666666666 * l + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.7%

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

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

   1. lower-*.f64 N/A

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

   2. cos-neg N/A

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

   3. lower-cos.f64 N/A

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

   4. neg-mul-1 N/A

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

   5. lower-exp.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. lower-neg.f64 35.7

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

8. Simplified 35.7%

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

9. Taylor expanded in M around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 27.9

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

11. Simplified 27.9%

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

12. Taylor expanded in l around 0

    \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, M \cdot M, 1\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1\right)\right)} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, M \cdot M, 1\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1\right) + 1\right)} \]

    2. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, M \cdot M, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1, 1\right)} \]

    3. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, M \cdot M, 1\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, M \cdot M, 1\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) + \color{blue}{-1}, 1\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, M \cdot M, 1\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\mathsf{fma}\left(\ell, \frac{1}{2} + \frac{-1}{6} \cdot \ell, -1\right)}, 1\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, M \cdot M, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \color{blue}{\frac{-1}{6} \cdot \ell + \frac{1}{2}}, -1\right), 1\right) \]

    7. lower-fma.f64 10.4

       \[\leadsto \mathsf{fma}\left(-0.5, M \cdot M, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \ell, 0.5\right)}, -1\right), 1\right) \]

14. Simplified 10.4%

    \[\leadsto \mathsf{fma}\left(-0.5, M \cdot M, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(-0.16666666666666666, \ell, 0.5\right), -1\right), 1\right)} \]
15. Add Preprocessing

Alternative 7: 8.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, M \cdot M, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(0.5, \ell, -1\right), 1\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (fma -0.5 (* M M) 1.0) (fma l (fma 0.5 l -1.0) 1.0)))

C

double code(double K, double m, double n, double M, double l) {
	return fma(-0.5, (M * M), 1.0) * fma(l, fma(0.5, l, -1.0), 1.0);
}

Julia

function code(K, m, n, M, l)
	return Float64(fma(-0.5, Float64(M * M), 1.0) * fma(l, fma(0.5, l, -1.0), 1.0))
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[(-0.5 * N[(M * M), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(0.5 * l + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.7%

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

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

   1. lower-*.f64 N/A

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

   2. cos-neg N/A

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

   3. lower-cos.f64 N/A

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

   4. neg-mul-1 N/A

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

   5. lower-exp.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. lower-neg.f64 35.7

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

8. Simplified 35.7%

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

9. Taylor expanded in M around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 27.9

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

11. Simplified 27.9%

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

12. Taylor expanded in l around 0

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

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, M \cdot M, 1\right) \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{2} \cdot \ell - 1\right) + 1\right)} \]

    2. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, M \cdot M, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \frac{1}{2} \cdot \ell - 1, 1\right)} \]

    3. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, M \cdot M, 1\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\frac{1}{2} \cdot \ell + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, M \cdot M, 1\right) \cdot \mathsf{fma}\left(\ell, \frac{1}{2} \cdot \ell + \color{blue}{-1}, 1\right) \]

    5. lower-fma.f64 9.7

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

14. Simplified 9.7%

    \[\leadsto \mathsf{fma}\left(-0.5, M \cdot M, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \mathsf{fma}\left(0.5, \ell, -1\right), 1\right)} \]
15. Add Preprocessing

Alternative 8: 6.8% accurate, 359.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return 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 = 1.0d0
end function

Java

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

Python

def code(K, m, n, M, l):
	return 1.0

Julia

function code(K, m, n, M, l)
	return 1.0
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 79.7%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. 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. Simplified 32.4%

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

6. Taylor expanded in l around 0

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

   1. lower-cos.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, K \cdot \left(m + n\right), \mathsf{neg}\left(M\right)\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{K \cdot \left(m + n\right)}, \mathsf{neg}\left(M\right)\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{2}, K \cdot \color{blue}{\left(m + n\right)}, \mathsf{neg}\left(M\right)\right)\right) \]

   6. lower-neg.f64 8.3

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

8. Simplified 8.3%

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

9. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \]
10. Step-by-step derivation

    1. cos-neg N/A

       \[\leadsto \color{blue}{\cos M} \]

    2. lower-cos.f64 8.7

       \[\leadsto \color{blue}{\cos M} \]

11. Simplified 8.7%

    \[\leadsto \color{blue}{\cos M} \]

12. Taylor expanded in M around 0

    \[\leadsto \color{blue}{1} \]
13. Step-by-step derivation

14. Simplified 8.7%

    \[\leadsto \color{blue}{1} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))))))