Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.3% → 96.6%

Time: 10.7s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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(((abs((n - m)) - l) - ((((n + m) / 2.0d0) - m_1) ** 2.0d0))) * cos(m_1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 K around 0

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

   1. cos-neg N/A

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

   2. lower-cos.f64 97.1

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

5. Applied rewrites 97.1%

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

6. Final simplification 97.1%

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

Alternative 2: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n + m, 0.5, -M\right)\\ t_1 := \left|n - m\right|\\ t_2 := e^{-\ell}\\ t_3 := \cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right)\\ t_4 := t\_3 \cdot e^{\left(t\_1 - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}}\\ \mathbf{if}\;t\_4 \leq -0.1:\\ \;\;\;\;\cos \left(\frac{1}{\frac{\frac{\mathsf{fma}\left(-2, \frac{m}{n}, 2\right)}{n}}{K}} - M\right) \cdot t\_2\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\left(\left(-\ell\right) \cdot e^{t\_1 - t\_0 \cdot t\_0}\right) \cdot t\_3\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2 \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot \cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma (+ n m) 0.5 (- M)))
        (t_1 (fabs (- n m)))
        (t_2 (exp (- l)))
        (t_3 (cos (- (/ (* (+ n m) K) 2.0) M)))
        (t_4 (* t_3 (exp (- (- t_1 l) (pow (- (/ (+ n m) 2.0) M) 2.0))))))
   (if (<= t_4 -0.1)
     (* (cos (- (/ 1.0 (/ (/ (fma -2.0 (/ m n) 2.0) n) K)) M)) t_2)
     (if (<= t_4 0.0)
       (* (* (- l) (exp (- t_1 (* t_0 t_0)))) t_3)
       (if (<= t_4 INFINITY)
         (* t_2 (cos M))
         (* (exp (* -0.25 (* m m))) (cos M)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma((n + m), 0.5, -M);
	double t_1 = fabs((n - m));
	double t_2 = exp(-l);
	double t_3 = cos(((((n + m) * K) / 2.0) - M));
	double t_4 = t_3 * exp(((t_1 - l) - pow((((n + m) / 2.0) - M), 2.0)));
	double tmp;
	if (t_4 <= -0.1) {
		tmp = cos(((1.0 / ((fma(-2.0, (m / n), 2.0) / n) / K)) - M)) * t_2;
	} else if (t_4 <= 0.0) {
		tmp = (-l * exp((t_1 - (t_0 * t_0)))) * t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2 * cos(M);
	} else {
		tmp = exp((-0.25 * (m * m))) * cos(M);
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(Float64(n + m), 0.5, Float64(-M))
	t_1 = abs(Float64(n - m))
	t_2 = exp(Float64(-l))
	t_3 = cos(Float64(Float64(Float64(Float64(n + m) * K) / 2.0) - M))
	t_4 = Float64(t_3 * exp(Float64(Float64(t_1 - l) - (Float64(Float64(Float64(n + m) / 2.0) - M) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.1)
		tmp = Float64(cos(Float64(Float64(1.0 / Float64(Float64(fma(-2.0, Float64(m / n), 2.0) / n) / K)) - M)) * t_2);
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(Float64(-l) * exp(Float64(t_1 - Float64(t_0 * t_0)))) * t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(t_2 * cos(M));
	else
		tmp = Float64(exp(Float64(-0.25 * Float64(m * m))) * cos(M));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(n + m), $MachinePrecision] * 0.5 + (-M)), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(N[(N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Exp[N[(N[(t$95$1 - l), $MachinePrecision] - N[Power[N[(N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.1], N[(N[Cos[N[(N[(1.0 / N[(N[(N[(-2.0 * N[(m / n), $MachinePrecision] + 2.0), $MachinePrecision] / n), $MachinePrecision] / K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[((-l) * N[Exp[N[(t$95$1 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$2 * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 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)))))) < -0.10000000000000001

   1. Initial program 34.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 32.1

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 32.1

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in n around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. neg-mul-1 N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-in N/A

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

      13. neg-mul-1 N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 49.0

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

   7. Applied rewrites 49.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 49.0

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

   9. Applied rewrites 49.0%

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

   10. Taylor expanded in l around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 49.0

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

   12. Applied rewrites 49.0%

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

   if -0.10000000000000001 < (*.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)))))) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 96.7%

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

   if -0.0 < (*.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)))))) < +inf.0

   1. Initial program 78.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

   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. cos-neg N/A

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

      2. lower-cos.f64 79.9

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

   8. Applied rewrites 79.9%

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

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 61.7

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

   8. Applied rewrites 61.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \leq -0.1:\\ \;\;\;\;\cos \left(\frac{1}{\frac{\frac{\mathsf{fma}\left(-2, \frac{m}{n}, 2\right)}{n}}{K}} - M\right) \cdot e^{-\ell}\\ \mathbf{elif}\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \leq 0:\\ \;\;\;\;\left(\left(-\ell\right) \cdot e^{\left|n - m\right| - \mathsf{fma}\left(n + m, 0.5, -M\right) \cdot \mathsf{fma}\left(n + m, 0.5, -M\right)}\right) \cdot \cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right)\\ \mathbf{elif}\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \leq \infty:\\ \;\;\;\;e^{-\ell} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n + m, 0.5, -M\right)\\ t_1 := \left|n - m\right|\\ t_2 := \cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right)\\ t_3 := t\_2 \cdot e^{\left(t\_1 - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\left(e^{t\_1 - t\_0 \cdot t\_0} \cdot \left(1 - \ell\right)\right) \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;e^{-\ell} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot \cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma (+ n m) 0.5 (- M)))
        (t_1 (fabs (- n m)))
        (t_2 (cos (- (/ (* (+ n m) K) 2.0) M)))
        (t_3 (* t_2 (exp (- (- t_1 l) (pow (- (/ (+ n m) 2.0) M) 2.0))))))
   (if (<= t_3 0.0)
     (* (* (exp (- t_1 (* t_0 t_0))) (- 1.0 l)) t_2)
     (if (<= t_3 INFINITY)
       (* (exp (- l)) (cos M))
       (* (exp (* -0.25 (* m m))) (cos M))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma((n + m), 0.5, -M);
	double t_1 = fabs((n - m));
	double t_2 = cos(((((n + m) * K) / 2.0) - M));
	double t_3 = t_2 * exp(((t_1 - l) - pow((((n + m) / 2.0) - M), 2.0)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = (exp((t_1 - (t_0 * t_0))) * (1.0 - l)) * t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = exp(-l) * cos(M);
	} else {
		tmp = exp((-0.25 * (m * m))) * cos(M);
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(Float64(n + m), 0.5, Float64(-M))
	t_1 = abs(Float64(n - m))
	t_2 = cos(Float64(Float64(Float64(Float64(n + m) * K) / 2.0) - M))
	t_3 = Float64(t_2 * exp(Float64(Float64(t_1 - l) - (Float64(Float64(Float64(n + m) / 2.0) - M) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(Float64(exp(Float64(t_1 - Float64(t_0 * t_0))) * Float64(1.0 - l)) * t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(exp(Float64(-l)) * cos(M));
	else
		tmp = Float64(exp(Float64(-0.25 * Float64(m * m))) * cos(M));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(n + m), $MachinePrecision] * 0.5 + (-M)), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Exp[N[(N[(t$95$1 - l), $MachinePrecision] - N[Power[N[(N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[(N[Exp[N[(t$95$1 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Exp[(-l)], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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)))))) < -0.0

   1. Initial program 97.5%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

   5. Applied rewrites 93.4%

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

   if -0.0 < (*.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)))))) < +inf.0

   1. Initial program 78.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

   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. cos-neg N/A

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

      2. lower-cos.f64 79.9

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

   8. Applied rewrites 79.9%

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

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 61.7

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

   8. Applied rewrites 61.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \leq 0:\\ \;\;\;\;\left(e^{\left|n - m\right| - \mathsf{fma}\left(n + m, 0.5, -M\right) \cdot \mathsf{fma}\left(n + m, 0.5, -M\right)} \cdot \left(1 - \ell\right)\right) \cdot \cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right)\\ \mathbf{elif}\;\cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \leq \infty:\\ \;\;\;\;e^{-\ell} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6600:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.25 \cdot 10^{-226}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos \left(\frac{K}{\frac{2}{m}} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -6600.0)
   (* 1.0 (exp (* -0.25 (* m m))))
   (if (<= m -1.25e-226)
     (* (exp (* (- M) M)) (cos (- (/ K (/ 2.0 m)) M)))
     (* (exp (* (* n n) -0.25)) (cos M)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -6600.0) {
		tmp = 1.0 * exp((-0.25 * (m * m)));
	} else if (m <= -1.25e-226) {
		tmp = exp((-M * M)) * cos(((K / (2.0 / m)) - M));
	} else {
		tmp = exp(((n * n) * -0.25)) * cos(M);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-6600.0d0)) then
        tmp = 1.0d0 * exp(((-0.25d0) * (m * m)))
    else if (m <= (-1.25d-226)) then
        tmp = exp((-m_1 * m_1)) * cos(((k / (2.0d0 / m)) - m_1))
    else
        tmp = exp(((n * n) * (-0.25d0))) * cos(m_1)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -6600.0) {
		tmp = 1.0 * Math.exp((-0.25 * (m * m)));
	} else if (m <= -1.25e-226) {
		tmp = Math.exp((-M * M)) * Math.cos(((K / (2.0 / m)) - M));
	} else {
		tmp = Math.exp(((n * n) * -0.25)) * Math.cos(M);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -6600.0:
		tmp = 1.0 * math.exp((-0.25 * (m * m)))
	elif m <= -1.25e-226:
		tmp = math.exp((-M * M)) * math.cos(((K / (2.0 / m)) - M))
	else:
		tmp = math.exp(((n * n) * -0.25)) * math.cos(M)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -6600.0)
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(m * m))));
	elseif (m <= -1.25e-226)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * cos(Float64(Float64(K / Float64(2.0 / m)) - M)));
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * cos(M));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -6600.0)
		tmp = 1.0 * exp((-0.25 * (m * m)));
	elseif (m <= -1.25e-226)
		tmp = exp((-M * M)) * cos(((K / (2.0 / m)) - M));
	else
		tmp = exp(((n * n) * -0.25)) * cos(M);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -6600.0], N[(1.0 * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -1.25e-226], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(K / N[(2.0 / m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6600

   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 m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 98.6%

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

   if -6600 < m < -1.2499999999999999e-226

   1. Initial program 80.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 80.4

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in M around inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 57.1

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

   7. Applied rewrites 57.1%

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

   8. Taylor expanded in n around 0

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

      1. lower-/.f64 64.3

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

   10. Applied rewrites 64.3%

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

   if -1.2499999999999999e-226 < m

   1. Initial program 78.6%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 51.1

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

   8. Applied rewrites 51.1%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6600:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.25 \cdot 10^{-226}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos \left(\frac{K}{\frac{2}{m}} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6600:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.55 \cdot 10^{-223}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -6600.0)
   (* 1.0 (exp (* -0.25 (* m m))))
   (if (<= m -1.55e-223)
     (* (exp (* (- M) M)) 1.0)
     (* (exp (* (* n n) -0.25)) (cos M)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -6600.0) {
		tmp = 1.0 * exp((-0.25 * (m * m)));
	} else if (m <= -1.55e-223) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = exp(((n * n) * -0.25)) * cos(M);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-6600.0d0)) then
        tmp = 1.0d0 * exp(((-0.25d0) * (m * m)))
    else if (m <= (-1.55d-223)) then
        tmp = exp((-m_1 * m_1)) * 1.0d0
    else
        tmp = exp(((n * n) * (-0.25d0))) * cos(m_1)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -6600.0) {
		tmp = 1.0 * Math.exp((-0.25 * (m * m)));
	} else if (m <= -1.55e-223) {
		tmp = Math.exp((-M * M)) * 1.0;
	} else {
		tmp = Math.exp(((n * n) * -0.25)) * Math.cos(M);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -6600.0:
		tmp = 1.0 * math.exp((-0.25 * (m * m)))
	elif m <= -1.55e-223:
		tmp = math.exp((-M * M)) * 1.0
	else:
		tmp = math.exp(((n * n) * -0.25)) * math.cos(M)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -6600.0)
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(m * m))));
	elseif (m <= -1.55e-223)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * cos(M));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -6600.0)
		tmp = 1.0 * exp((-0.25 * (m * m)));
	elseif (m <= -1.55e-223)
		tmp = exp((-M * M)) * 1.0;
	else
		tmp = exp(((n * n) * -0.25)) * cos(M);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -6600.0], N[(1.0 * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -1.55e-223], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6600

   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 m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 98.6%

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

   if -6600 < m < -1.55000000000000009e-223

   1. Initial program 82.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 35.8

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

   5. Applied rewrites 35.8%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 43.7

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

   8. Applied rewrites 43.7%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 43.7%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 66.4

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

   14. Applied rewrites 66.4%

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

   if -1.55000000000000009e-223 < m

   1. Initial program 77.6%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 51.2

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

   8. Applied rewrites 51.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6600:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.55 \cdot 10^{-223}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -0.08:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 3.7 \cdot 10^{-268}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -0.08)
     t_0
     (if (<= M 3.7e-268)
       (* 1.0 (exp (* -0.25 (* m m))))
       (if (<= M 27.0) (* 1.0 (exp (- l))) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -0.08) {
		tmp = t_0;
	} else if (M <= 3.7e-268) {
		tmp = 1.0 * exp((-0.25 * (m * m)));
	} else if (M <= 27.0) {
		tmp = 1.0 * exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((-m_1 * m_1)) * 1.0d0
    if (m_1 <= (-0.08d0)) then
        tmp = t_0
    else if (m_1 <= 3.7d-268) then
        tmp = 1.0d0 * exp(((-0.25d0) * (m * m)))
    else if (m_1 <= 27.0d0) then
        tmp = 1.0d0 * exp(-l)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -0.08) {
		tmp = t_0;
	} else if (M <= 3.7e-268) {
		tmp = 1.0 * Math.exp((-0.25 * (m * m)));
	} else if (M <= 27.0) {
		tmp = 1.0 * Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-M * M)) * 1.0
	tmp = 0
	if M <= -0.08:
		tmp = t_0
	elif M <= 3.7e-268:
		tmp = 1.0 * math.exp((-0.25 * (m * m)))
	elif M <= 27.0:
		tmp = 1.0 * math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -0.08)
		tmp = t_0;
	elseif (M <= 3.7e-268)
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(m * m))));
	elseif (M <= 27.0)
		tmp = Float64(1.0 * exp(Float64(-l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-M * M)) * 1.0;
	tmp = 0.0;
	if (M <= -0.08)
		tmp = t_0;
	elseif (M <= 3.7e-268)
		tmp = 1.0 * exp((-0.25 * (m * m)));
	elseif (M <= 27.0)
		tmp = 1.0 * exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -0.08], t$95$0, If[LessEqual[M, 3.7e-268], N[(1.0 * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 27.0], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if M < -0.0800000000000000017 or 27 < M

   1. Initial program 80.4%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 21.4

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

   5. Applied rewrites 21.4%

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

   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. cos-neg N/A

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

      2. lower-cos.f64 25.2

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

   8. Applied rewrites 25.2%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 24.5%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 96.8

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

   14. Applied rewrites 96.8%

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

   if -0.0800000000000000017 < M < 3.70000000000000018e-268

   1. Initial program 70.9%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 68.9

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

   8. Applied rewrites 68.9%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 68.9%

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

   if 3.70000000000000018e-268 < M < 27

   1. Initial program 77.5%

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

   3. Taylor expanded in 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 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 56.5

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

   8. Applied rewrites 56.5%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 56.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -0.08:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;M \leq 3.7 \cdot 10^{-268}:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6600:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.5 \cdot 10^{-223}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -6600.0)
   (* 1.0 (exp (* -0.25 (* m m))))
   (if (<= m -1.5e-223)
     (* (exp (* (- M) M)) 1.0)
     (* (exp (* (* n n) -0.25)) 1.0))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -6600.0) {
		tmp = 1.0 * exp((-0.25 * (m * m)));
	} else if (m <= -1.5e-223) {
		tmp = exp((-M * M)) * 1.0;
	} else {
		tmp = exp(((n * n) * -0.25)) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-6600.0d0)) then
        tmp = 1.0d0 * exp(((-0.25d0) * (m * m)))
    else if (m <= (-1.5d-223)) then
        tmp = exp((-m_1 * m_1)) * 1.0d0
    else
        tmp = exp(((n * n) * (-0.25d0))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -6600.0) {
		tmp = 1.0 * Math.exp((-0.25 * (m * m)));
	} else if (m <= -1.5e-223) {
		tmp = Math.exp((-M * M)) * 1.0;
	} else {
		tmp = Math.exp(((n * n) * -0.25)) * 1.0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -6600.0:
		tmp = 1.0 * math.exp((-0.25 * (m * m)))
	elif m <= -1.5e-223:
		tmp = math.exp((-M * M)) * 1.0
	else:
		tmp = math.exp(((n * n) * -0.25)) * 1.0
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -6600.0)
		tmp = Float64(1.0 * exp(Float64(-0.25 * Float64(m * m))));
	elseif (m <= -1.5e-223)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -6600.0)
		tmp = 1.0 * exp((-0.25 * (m * m)));
	elseif (m <= -1.5e-223)
		tmp = exp((-M * M)) * 1.0;
	else
		tmp = exp(((n * n) * -0.25)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -6600.0], N[(1.0 * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -1.5e-223], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6600

   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 m around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 98.6%

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

   if -6600 < m < -1.49999999999999996e-223

   1. Initial program 82.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 35.8

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

   5. Applied rewrites 35.8%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 43.7

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

   8. Applied rewrites 43.7%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 43.7%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 66.4

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

   14. Applied rewrites 66.4%

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

   if -1.49999999999999996e-223 < m

   1. Initial program 77.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 34.6

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

   5. Applied rewrites 34.6%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 37.7

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

   8. Applied rewrites 37.7%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 37.0%

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

   12. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 51.2

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

   14. Applied rewrites 51.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6600:\\ \;\;\;\;1 \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.5 \cdot 10^{-223}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{if}\;M \leq -2.9 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) 1.0)))
   (if (<= M -2.9e-67) t_0 (if (<= M 27.0) (* 1.0 (exp (- l))) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -2.9e-67) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = 1.0 * exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((-m_1 * m_1)) * 1.0d0
    if (m_1 <= (-2.9d-67)) then
        tmp = t_0
    else if (m_1 <= 27.0d0) then
        tmp = 1.0d0 * exp(-l)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((-M * M)) * 1.0;
	double tmp;
	if (M <= -2.9e-67) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = 1.0 * Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-M * M)) * 1.0
	tmp = 0
	if M <= -2.9e-67:
		tmp = t_0
	elif M <= 27.0:
		tmp = 1.0 * math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * 1.0)
	tmp = 0.0
	if (M <= -2.9e-67)
		tmp = t_0;
	elseif (M <= 27.0)
		tmp = Float64(1.0 * exp(Float64(-l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-M * M)) * 1.0;
	tmp = 0.0;
	if (M <= -2.9e-67)
		tmp = t_0;
	elseif (M <= 27.0)
		tmp = 1.0 * exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[M, -2.9e-67], t$95$0, If[LessEqual[M, 27.0], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -2.90000000000000005e-67 or 27 < M

   1. Initial program 79.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 20.4

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

   5. Applied rewrites 20.4%

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

   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. cos-neg N/A

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

      2. lower-cos.f64 24.0

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

   8. Applied rewrites 24.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 23.4%

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

   12. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 90.9

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

   14. Applied rewrites 90.9%

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

   if -2.90000000000000005e-67 < M < 27

   1. Initial program 75.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 46.9

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 53.6

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 53.6%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.9 \cdot 10^{-67}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 31.0

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

5. Applied rewrites 31.0%

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

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. cos-neg N/A

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

   2. lower-cos.f64 35.9

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

8. Applied rewrites 35.9%

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

9. Taylor expanded in M around 0

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

11. Applied rewrites 35.6%

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

Reproduce

herbie shell --seed 2024235 
(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)))))))