Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.4% → 96.8%

Time: 13.6s

Alternatives: 13

Speedup: 2.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

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

Alternative 2: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ t_1 := t\_0 - \ell\\ t_2 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)\\ t_3 := \left(m + n\right) \cdot \left(m + n\right)\\ \mathbf{if}\;e^{t\_1 - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot t\_2 \leq 0.6:\\ \;\;\;\;t\_2 \cdot e^{t\_1 - M \cdot \mathsf{fma}\left(M, \frac{n + \mathsf{fma}\left(-0.25, \frac{t\_3}{M}, m\right)}{-M}, M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - \mathsf{fma}\left(0.25, t\_3, \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n)))
        (t_1 (- t_0 l))
        (t_2 (cos (- (/ (* K (+ m n)) 2.0) M)))
        (t_3 (* (+ m n) (+ m n))))
   (if (<= (* (exp (- t_1 (pow (- (/ (+ m n) 2.0) M) 2.0))) t_2) 0.6)
     (*
      t_2
      (exp (- t_1 (* M (fma M (/ (+ n (fma -0.25 (/ t_3 M) m)) (- M)) M)))))
     (exp (- t_0 (fma 0.25 t_3 l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double t_1 = t_0 - l;
	double t_2 = cos((((K * (m + n)) / 2.0) - M));
	double t_3 = (m + n) * (m + n);
	double tmp;
	if ((exp((t_1 - pow((((m + n) / 2.0) - M), 2.0))) * t_2) <= 0.6) {
		tmp = t_2 * exp((t_1 - (M * fma(M, ((n + fma(-0.25, (t_3 / M), m)) / -M), M))));
	} else {
		tmp = exp((t_0 - fma(0.25, t_3, l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	t_1 = Float64(t_0 - l)
	t_2 = cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M))
	t_3 = Float64(Float64(m + n) * Float64(m + n))
	tmp = 0.0
	if (Float64(exp(Float64(t_1 - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0))) * t_2) <= 0.6)
		tmp = Float64(t_2 * exp(Float64(t_1 - Float64(M * fma(M, Float64(Float64(n + fma(-0.25, Float64(t_3 / M), m)) / Float64(-M)), M)))));
	else
		tmp = exp(Float64(t_0 - fma(0.25, t_3, l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - l), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Exp[N[(t$95$1 - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], 0.6], N[(t$95$2 * N[Exp[N[(t$95$1 - N[(M * N[(M * N[(N[(n + N[(-0.25 * N[(t$95$3 / M), $MachinePrecision] + m), $MachinePrecision]), $MachinePrecision] / (-M)), $MachinePrecision] + M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$0 - N[(0.25 * t$95$3 + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < 0.599999999999999978

   1. Initial program 97.9%

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

   3. Taylor expanded in M around -inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 97.9%

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

   if 0.599999999999999978 < (*.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 30.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 97.3

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

   5. Simplified 97.3%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 94.3

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

   8. Simplified 94.3%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \leq 0.6:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot \mathsf{fma}\left(M, \frac{n + \mathsf{fma}\left(-0.25, \frac{\left(m + n\right) \cdot \left(m + n\right)}{M}, m\right)}{-M}, M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)\\ t_1 := \left|m - n\right|\\ \mathbf{if}\;e^{\left(t\_1 - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot t\_0 \leq 0.6:\\ \;\;\;\;t\_0 \cdot e^{M \cdot \mathsf{fma}\left(M, \frac{\left(m + n\right) + \frac{t\_1 - \mathsf{fma}\left(m + n, \left(m + n\right) \cdot 0.25, \ell\right)}{M}}{M}, -M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_1 - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (cos (- (/ (* K (+ m n)) 2.0) M))) (t_1 (fabs (- m n))))
   (if (<= (* (exp (- (- t_1 l) (pow (- (/ (+ m n) 2.0) M) 2.0))) t_0) 0.6)
     (*
      t_0
      (exp
       (*
        M
        (fma
         M
         (/ (+ (+ m n) (/ (- t_1 (fma (+ m n) (* (+ m n) 0.25) l)) M)) M)
         (- M)))))
     (exp (- t_1 (fma 0.25 (* (+ m n) (+ m n)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos((((K * (m + n)) / 2.0) - M));
	double t_1 = fabs((m - n));
	double tmp;
	if ((exp(((t_1 - l) - pow((((m + n) / 2.0) - M), 2.0))) * t_0) <= 0.6) {
		tmp = t_0 * exp((M * fma(M, (((m + n) + ((t_1 - fma((m + n), ((m + n) * 0.25), l)) / M)) / M), -M)));
	} else {
		tmp = exp((t_1 - fma(0.25, ((m + n) * (m + n)), l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M))
	t_1 = abs(Float64(m - n))
	tmp = 0.0
	if (Float64(exp(Float64(Float64(t_1 - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0))) * t_0) <= 0.6)
		tmp = Float64(t_0 * exp(Float64(M * fma(M, Float64(Float64(Float64(m + n) + Float64(Float64(t_1 - fma(Float64(m + n), Float64(Float64(m + n) * 0.25), l)) / M)) / M), Float64(-M)))));
	else
		tmp = exp(Float64(t_1 - fma(0.25, Float64(Float64(m + n) * Float64(m + n)), l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Exp[N[(N[(t$95$1 - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], 0.6], N[(t$95$0 * N[Exp[N[(M * N[(M * N[(N[(N[(m + n), $MachinePrecision] + N[(N[(t$95$1 - N[(N[(m + n), $MachinePrecision] * N[(N[(m + n), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision] / M), $MachinePrecision] + (-M)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$1 - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < 0.599999999999999978

   1. Initial program 97.9%

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

   3. Taylor expanded in M around -inf

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

   5. Simplified 97.9%

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

   if 0.599999999999999978 < (*.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 30.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 97.3

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

   5. Simplified 97.3%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 94.3

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

   8. Simplified 94.3%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \leq 0.6:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{M \cdot \mathsf{fma}\left(M, \frac{\left(m + n\right) + \frac{\left|m - n\right| - \mathsf{fma}\left(m + n, \left(m + n\right) \cdot 0.25, \ell\right)}{M}}{M}, -M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -5.15 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (* M (- M))))))
   (if (<= M -5.15e+64)
     t_0
     (if (<= M 27.0)
       (exp (- (fabs (- m n)) (fma 0.25 (* (+ m n) (+ m n)) l)))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp((M * -M));
	double tmp;
	if (M <= -5.15e+64) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = exp((fabs((m - n)) - fma(0.25, ((m + n) * (m + n)), l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(M * Float64(-M))))
	tmp = 0.0
	if (M <= -5.15e+64)
		tmp = t_0;
	elseif (M <= 27.0)
		tmp = exp(Float64(abs(Float64(m - n)) - fma(0.25, Float64(Float64(m + n) * Float64(m + n)), l)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -5.15e+64], t$95$0, If[LessEqual[M, 27.0], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -5.15000000000000009e64 or 27 < M

   1. Initial program 79.7%

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

   3. Taylor expanded in 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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 98.5

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

   8. Simplified 98.5%

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

   if -5.15000000000000009e64 < M < 27

   1. Initial program 81.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 93.9

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

   5. Simplified 93.9%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 91.5

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

   8. Simplified 91.5%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -5.15 \cdot 10^{+64}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -6.6 \cdot 10^{-189}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|m - n\right|\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -53.0)
   (exp (* -0.25 (* m m)))
   (if (<= m -6.6e-189) (exp (- l)) (exp (fma -0.25 (* n n) (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = exp((-0.25 * (m * m)));
	} else if (m <= -6.6e-189) {
		tmp = exp(-l);
	} else {
		tmp = exp(fma(-0.25, (n * n), fabs((m - n))));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -53.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (m <= -6.6e-189)
		tmp = exp(Float64(-l));
	else
		tmp = exp(fma(-0.25, Float64(n * n), abs(Float64(m - n))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -53.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -6.6e-189], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision] + N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -53

   1. Initial program 76.6%

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in m around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 100.0

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

   11. Simplified 100.0%

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

   if -53 < m < -6.6000000000000002e-189

   1. Initial program 88.9%

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

   3. Taylor expanded in 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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 83.8

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

   8. Simplified 83.8%

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

   9. Taylor expanded in l around inf

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

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 51.4

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

   11. Simplified 51.4%

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

   if -6.6000000000000002e-189 < m

   1. Initial program 80.3%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 95.2

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

   5. Simplified 95.2%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 82.0

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

   8. Simplified 82.0%

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

   9. Taylor expanded in n around inf

      \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{{n}^{2}}, \ell\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{n \cdot n}, \ell\right)} \]

       2. lower-*.f64 63.7

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \color{blue}{n \cdot n}, \ell\right)} \]

   11. Simplified 63.7%

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \color{blue}{n \cdot n}, \ell\right)} \]

   12. Taylor expanded in l around 0

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

       1. lower-exp.f64 N/A

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

       2. cancel-sign-sub-inv N/A

          \[\leadsto e^{\color{blue}{\left|n - m\right| + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot {n}^{2}}} \]

       3. metadata-eval N/A

          \[\leadsto e^{\left|n - m\right| + \color{blue}{\frac{-1}{4}} \cdot {n}^{2}} \]

       4. +-commutative N/A

          \[\leadsto e^{\color{blue}{\frac{-1}{4} \cdot {n}^{2} + \left|n - m\right|}} \]

       5. lower-fma.f64 N/A

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {n}^{2}, \left|n - m\right|\right)}} \]

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-fabs.f64 N/A

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

       9. lower--.f64 44.9

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

   14. Simplified 44.9%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -6.6 \cdot 10^{-189}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|m - n\right|\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -56:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, n \cdot n, \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -56.0)
   (exp (* -0.25 (* m m)))
   (exp (- (fabs (- m n)) (fma 0.25 (* n n) l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -56.0) {
		tmp = exp((-0.25 * (m * m)));
	} else {
		tmp = exp((fabs((m - n)) - fma(0.25, (n * n), l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -56.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	else
		tmp = exp(Float64(abs(Float64(m - n)) - fma(0.25, Float64(n * n), l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -56.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(n * n), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -56

   1. Initial program 76.6%

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in m around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 100.0

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

   11. Simplified 100.0%

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

   if -56 < m

   1. Initial program 81.9%

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

   3. Taylor expanded in 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 96.1

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

   5. Simplified 96.1%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 82.3

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

   8. Simplified 82.3%

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

   9. Taylor expanded in n around inf

      \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{{n}^{2}}, \ell\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{n \cdot n}, \ell\right)} \]

       2. lower-*.f64 67.5

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(0.25, \color{blue}{n \cdot n}, \ell\right)} \]

   11. Simplified 67.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -56:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, n \cdot n, \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- (fabs (- m n)) (fma 0.25 (* (+ m n) (+ m n)) l))))

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((m - n)) - fma(0.25, ((m + n) * (m + n)), l)));
}

Julia

function code(K, m, n, M, l)
	return exp(Float64(abs(Float64(m - n)) - fma(0.25, Float64(Float64(m + n) * Float64(m + n)), l)))
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 80.6%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 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. Simplified 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. Taylor expanded in M around 0

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. fabs-sub N/A

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

   4. sub-neg N/A

      \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

   5. mul-1-neg N/A

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

   6. lower-fabs.f64 N/A

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

   7. mul-1-neg N/A

      \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

   8. sub-neg N/A

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

   9. lower--.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

       \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

   12. unpow2 N/A

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

   14. +-commutative N/A

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

   15. lower-+.f64 N/A

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

   16. +-commutative N/A

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

   17. lower-+.f64 86.8

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

8. Simplified 86.8%

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

9. Final simplification 86.8%

   \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)} \]
10. Add Preprocessing

Alternative 8: 63.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -6.6 \cdot 10^{-189}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -53.0)
   (exp (* -0.25 (* m m)))
   (if (<= m -6.6e-189) (exp (- l)) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = exp((-0.25 * (m * m)));
	} else if (m <= -6.6e-189) {
		tmp = exp(-l);
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	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 <= (-53.0d0)) then
        tmp = exp(((-0.25d0) * (m * m)))
    else if (m <= (-6.6d-189)) then
        tmp = exp(-l)
    else
        tmp = exp(((-0.25d0) * (n * n)))
    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 <= -53.0) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else if (m <= -6.6e-189) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -53.0:
		tmp = math.exp((-0.25 * (m * m)))
	elif m <= -6.6e-189:
		tmp = math.exp(-l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -53.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (m <= -6.6e-189)
		tmp = exp(Float64(-l));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -53.0)
		tmp = exp((-0.25 * (m * m)));
	elseif (m <= -6.6e-189)
		tmp = exp(-l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -53.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -6.6e-189], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -53

   1. Initial program 76.6%

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in m around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 100.0

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

   11. Simplified 100.0%

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

   if -53 < m < -6.6000000000000002e-189

   1. Initial program 88.9%

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

   3. Taylor expanded in 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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 83.8

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

   8. Simplified 83.8%

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

   9. Taylor expanded in l around inf

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

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 51.4

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

   11. Simplified 51.4%

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

   if -6.6000000000000002e-189 < m

   1. Initial program 80.3%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 95.2

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

   5. Simplified 95.2%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 82.0

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

   8. Simplified 82.0%

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

   9. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 49.0

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

   11. Simplified 49.0%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -6.6 \cdot 10^{-189}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{if}\;m \leq -53:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 54:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (* m m)))))
   (if (<= m -53.0) t_0 (if (<= m 54.0) (exp (- l)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * (m * m)));
	double tmp;
	if (m <= -53.0) {
		tmp = t_0;
	} else if (m <= 54.0) {
		tmp = 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(((-0.25d0) * (m * m)))
    if (m <= (-53.0d0)) then
        tmp = t_0
    else if (m <= 54.0d0) then
        tmp = 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((-0.25 * (m * m)));
	double tmp;
	if (m <= -53.0) {
		tmp = t_0;
	} else if (m <= 54.0) {
		tmp = Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * (m * m)))
	tmp = 0
	if m <= -53.0:
		tmp = t_0
	elif m <= 54.0:
		tmp = math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * Float64(m * m)))
	tmp = 0.0
	if (m <= -53.0)
		tmp = t_0;
	elseif (m <= 54.0)
		tmp = exp(Float64(-l));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-0.25 * (m * m)));
	tmp = 0.0;
	if (m <= -53.0)
		tmp = t_0;
	elseif (m <= 54.0)
		tmp = exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -53.0], t$95$0, If[LessEqual[m, 54.0], N[Exp[(-l)], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -53 or 54 < m

   1. Initial program 75.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 97.7

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

   8. Simplified 97.7%

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

   9. Taylor expanded in m around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 98.4

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

   11. Simplified 98.4%

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

   if -53 < m < 54

   1. Initial program 85.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 94.9

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

   5. Simplified 94.9%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 76.0

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

   8. Simplified 76.0%

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

   9. Taylor expanded in l around inf

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

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 43.7

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

   11. Simplified 43.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 54:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 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. Simplified 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. Taylor expanded in M around 0

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. fabs-sub N/A

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

   4. sub-neg N/A

      \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

   5. mul-1-neg N/A

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

   6. lower-fabs.f64 N/A

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

   7. mul-1-neg N/A

      \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

   8. sub-neg N/A

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

   9. lower--.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

       \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

   12. unpow2 N/A

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

   14. +-commutative N/A

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

   15. lower-+.f64 N/A

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

   16. +-commutative N/A

       \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

   17. lower-+.f64 86.8

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

8. Simplified 86.8%

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

9. Taylor expanded in l around inf

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

    1. neg-mul-1 N/A

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

    2. lower-neg.f64 36.2

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

11. Simplified 36.2%

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

Alternative 11: 9.0% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(K \cdot K, \left(m \cdot m\right) \cdot -0.125, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right), -1\right), 1\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (fma (* K K) (* (* m m) -0.125) 1.0)
  (fma l (fma l (fma l -0.16666666666666666 0.5) -1.0) 1.0)))

C

double code(double K, double m, double n, double M, double l) {
	return fma((K * K), ((m * m) * -0.125), 1.0) * fma(l, fma(l, fma(l, -0.16666666666666666, 0.5), -1.0), 1.0);
}

Julia

function code(K, m, n, M, l)
	return Float64(fma(Float64(K * K), Float64(Float64(m * m) * -0.125), 1.0) * fma(l, fma(l, fma(l, -0.16666666666666666, 0.5), -1.0), 1.0))
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[(N[(K * K), $MachinePrecision] * N[(N[(m * m), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(l * N[(l * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.6%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 34.2

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

5. Simplified 34.2%

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

6. Taylor expanded in m around inf

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 36.3

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

8. Simplified 36.3%

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

9. Taylor expanded in l around 0

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

    1. +-commutative N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(m \cdot K\right)\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1\right) + 1\right)} \]

    2. lower-fma.f64 N/A

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. lower-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f64 10.1

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

11. Simplified 10.1%

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

12. Taylor expanded in m around 0

    \[\leadsto \color{blue}{\left(1 + \frac{-1}{8} \cdot \left({K}^{2} \cdot {m}^{2}\right)\right)} \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]
13. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \left(1 + \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot {m}^{2}}\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    2. +-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot {m}^{2} + 1\right)} \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    3. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left({K}^{2} \cdot \frac{-1}{8}\right)} \cdot {m}^{2} + 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    4. associate-*r* N/A

       \[\leadsto \left(\color{blue}{{K}^{2} \cdot \left(\frac{-1}{8} \cdot {m}^{2}\right)} + 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{8} \cdot {m}^{2}, 1\right)} \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{8} \cdot {m}^{2}, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    7. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{8} \cdot {m}^{2}, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    8. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(K \cdot K, \color{blue}{\frac{-1}{8} \cdot {m}^{2}}, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(K \cdot K, \frac{-1}{8} \cdot \color{blue}{\left(m \cdot m\right)}, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right) \]

    10. lower-*.f64 8.9

        \[\leadsto \mathsf{fma}\left(K \cdot K, -0.125 \cdot \color{blue}{\left(m \cdot m\right)}, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right), -1\right), 1\right) \]

14. Simplified 8.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125 \cdot \left(m \cdot m\right), 1\right)} \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right), -1\right), 1\right) \]

15. Final simplification 8.9%

    \[\leadsto \mathsf{fma}\left(K \cdot K, \left(m \cdot m\right) \cdot -0.125, 1\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, -0.16666666666666666, 0.5\right), -1\right), 1\right) \]
16. Add Preprocessing

Alternative 12: 10.0% accurate, 18.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.6%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 34.2

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

5. Simplified 34.2%

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

6. Taylor expanded in m around inf

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 36.3

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

8. Simplified 36.3%

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

9. Taylor expanded in l around 0

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

    1. +-commutative N/A

       \[\leadsto \cos \left(\frac{1}{2} \cdot \left(m \cdot K\right)\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \ell\right) - 1\right) + 1\right)} \]

    2. lower-fma.f64 N/A

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. lower-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f64 10.1

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

11. Simplified 10.1%

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

12. Taylor expanded in m around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. lower-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f64 9.3

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

14. Simplified 9.3%

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

Alternative 13: 7.2% accurate, 359.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

3. Taylor expanded in n around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 42.3

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

5. Simplified 42.3%

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

6. Taylor expanded in n around 0

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

   1. lower-cos.f64 N/A

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

   2. sub-neg N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-neg.f64 5.8

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

8. Simplified 5.8%

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

9. Taylor expanded in K around 0

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

    1. cos-neg N/A

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

    2. lower-cos.f64 5.8

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

11. Simplified 5.8%

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

12. Taylor expanded in M around 0

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

14. Simplified 5.8%

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

Reproduce

herbie shell --seed 2024210 
(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)))))))