Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 77.4% → 95.1%

Time: 13.8s

Alternatives: 9

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: 77.4% 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: 95.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -4 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 2.55 \cdot 10^{+76}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (* M (- M))))))
   (if (<= M -4e+22)
     t_0
     (if (<= M 2.55e+76)
       (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 <= -4e+22) {
		tmp = t_0;
	} else if (M <= 2.55e+76) {
		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 <= -4e+22)
		tmp = t_0;
	elseif (M <= 2.55e+76)
		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, -4e+22], t$95$0, If[LessEqual[M, 2.55e+76], 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 < -4e22 or 2.5500000000000001e76 < M

   1. Initial program 80.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. cos-lowering-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. *-lowering-*.f64 N/A

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

      5. neg-lowering-neg.f64 100.0

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

   8. Simplified 100.0%

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

   if -4e22 < M < 2.5500000000000001e76

   1. Initial program 71.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. cos-lowering-cos.f64 90.6

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

   5. Simplified 90.6%

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

   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. exp-lowering-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. --lowering--.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-lowering-fabs.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 90.6

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

   8. Simplified 90.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq 2.55 \cdot 10^{+76}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 0.5× speedup

Math

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

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) 0.0)
     (*
      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) <= 0.0) {
		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) <= 0.0)
		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], 0.0], 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)))))) < 0.0

   1. Initial program 95.0%

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

   3. Taylor expanded in M around -inf

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\mathsf{neg}\left(\color{blue}{{M}^{2} \cdot \left(1 + -1 \cdot \frac{m + \left(n + \frac{-1}{4} \cdot \frac{{\left(m + n\right)}^{2}}{M}\right)}{M}\right)}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. *-lowering-*.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. accelerator-lowering-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 94.2%

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

   if 0.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 27.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. cos-lowering-cos.f64 91.6

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

   5. Simplified 91.6%

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

   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. exp-lowering-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. --lowering--.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-lowering-fabs.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 92.9

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

   8. Simplified 92.9%

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

4. Final simplification 93.8%

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

Alternative 3: 97.1% 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 75.3%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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. cos-lowering-cos.f64 94.2

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

5. Simplified 94.2%

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

6. Final simplification 94.2%

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

Alternative 4: 88.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;M \leq -1 \cdot 10^{+55}:\\ \;\;\;\;e^{t\_0 - \mathsf{fma}\left(0.25, n \cdot \left(n \cdot \mathsf{fma}\left(m, \frac{m}{n \cdot n} + \frac{2}{n}, 1\right)\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (<= M -1e+55)
     (exp
      (- t_0 (fma 0.25 (* n (* n (fma m (+ (/ m (* n n)) (/ 2.0 n)) 1.0))) l)))
     (exp (- t_0 (fma 0.25 (* (+ m n) (+ m n)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double tmp;
	if (M <= -1e+55) {
		tmp = exp((t_0 - fma(0.25, (n * (n * fma(m, ((m / (n * n)) + (2.0 / n)), 1.0))), l)));
	} else {
		tmp = exp((t_0 - fma(0.25, ((m + n) * (m + n)), l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if (M <= -1e+55)
		tmp = exp(Float64(t_0 - fma(0.25, Float64(n * Float64(n * fma(m, Float64(Float64(m / Float64(n * n)) + Float64(2.0 / n)), 1.0))), l)));
	else
		tmp = exp(Float64(t_0 - 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[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -1e+55], N[Exp[N[(t$95$0 - N[(0.25 * N[(n * N[(n * N[(m * N[(N[(m / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(2.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 - 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 M < -1.00000000000000001e55

   1. Initial program 83.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. cos-lowering-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. exp-lowering-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. --lowering--.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-lowering-fabs.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 74.1

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

   8. Simplified 74.1%

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

   9. Taylor expanded in n around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. associate-+r+ N/A

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

       6. +-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-/l* N/A

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

       9. accelerator-lowering-fma.f64 N/A

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

       10. /-lowering-/.f64 N/A

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. +-commutative N/A

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

       14. associate-*r/ N/A

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

       15. *-commutative N/A

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

       16. associate-/l* N/A

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

       17. accelerator-lowering-fma.f64 N/A

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

       18. /-lowering-/.f64 82.1

           \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(0.25, n \cdot \left(n \cdot \mathsf{fma}\left(m, \frac{m}{n \cdot n}, \mathsf{fma}\left(m, \color{blue}{\frac{2}{n}}, 1\right)\right)\right), \ell\right)} \]

   11. Simplified 82.1%

       \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \color{blue}{n \cdot \left(n \cdot \mathsf{fma}\left(m, \frac{m}{n \cdot n}, \mathsf{fma}\left(m, \frac{2}{n}, 1\right)\right)\right)}, \ell\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. associate-+r+ N/A

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

       5. distribute-lft-out N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. +-lowering-+.f64 N/A

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

       8. /-lowering-/.f64 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. /-lowering-/.f64 84.2

           \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(n \cdot \mathsf{fma}\left(m, \frac{m}{n \cdot n} + \color{blue}{\frac{2}{n}}, 1\right)\right) \cdot n, \ell\right)} \]

   13. Applied egg-rr 84.2%

       \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \color{blue}{\left(n \cdot \mathsf{fma}\left(m, \frac{m}{n \cdot n} + \frac{2}{n}, 1\right)\right) \cdot n}, \ell\right)} \]

   if -1.00000000000000001e55 < M

   1. Initial program 73.3%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. cos-lowering-cos.f64 92.9

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

   5. Simplified 92.9%

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

   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. exp-lowering-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. --lowering--.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-lowering-fabs.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 88.7

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

   8. Simplified 88.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+55}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, n \cdot \left(n \cdot \mathsf{fma}\left(m, \frac{m}{n \cdot n} + \frac{2}{n}, 1\right)\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% 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 75.3%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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. cos-lowering-cos.f64 94.2

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

5. Simplified 94.2%

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

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. exp-lowering-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. --lowering--.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-lowering-fabs.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-commutative N/A

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

   10. +-lowering-+.f64 N/A

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

   11. +-commutative N/A

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

   12. +-lowering-+.f64 85.9

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

8. Simplified 85.9%

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

9. Final simplification 85.9%

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

Alternative 6: 63.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.6 \cdot 10^{-43}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.6e-43)
   (exp (* m (* m -0.25)))
   (if (<= n 54.0) (exp (- l)) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.6e-43) {
		tmp = exp((m * (m * -0.25)));
	} else if (n <= 54.0) {
		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 (n <= 1.6d-43) then
        tmp = exp((m * (m * (-0.25d0))))
    else if (n <= 54.0d0) 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 (n <= 1.6e-43) {
		tmp = Math.exp((m * (m * -0.25)));
	} else if (n <= 54.0) {
		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 n <= 1.6e-43:
		tmp = math.exp((m * (m * -0.25)))
	elif n <= 54.0:
		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 (n <= 1.6e-43)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	elseif (n <= 54.0)
		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 (n <= 1.6e-43)
		tmp = exp((m * (m * -0.25)));
	elseif (n <= 54.0)
		tmp = exp(-l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.6e-43], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 54.0], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.59999999999999992e-43

   1. Initial program 76.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. cos-lowering-cos.f64 93.7

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

   5. Simplified 93.7%

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

   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. exp-lowering-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. --lowering--.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-lowering-fabs.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 83.4

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

   8. Simplified 83.4%

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

   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. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 56.7

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

   11. Simplified 56.7%

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

   if 1.59999999999999992e-43 < n < 54

   1. Initial program 89.0%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. cos-lowering-cos.f64 80.3

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

   5. Simplified 80.3%

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

   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. exp-lowering-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. --lowering--.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-lowering-fabs.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 66.6

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

   8. Simplified 66.6%

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

   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. neg-lowering-neg.f64 52.8

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

   11. Simplified 52.8%

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

   if 54 < n

   1. Initial program 67.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. cos-lowering-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. exp-lowering-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. --lowering--.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-lowering-fabs.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 100.0

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

   8. Simplified 100.0%

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

   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. *-lowering-*.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. *-lowering-*.f64 96.3

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

   11. Simplified 96.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.6 \cdot 10^{-43}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{if}\;m \leq -4.6 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 7.6 \cdot 10^{-34}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* m (* m -0.25)))))
   (if (<= m -4.6e-27) t_0 (if (<= m 7.6e-34) (exp (- l)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((m * (m * -0.25)));
	double tmp;
	if (m <= -4.6e-27) {
		tmp = t_0;
	} else if (m <= 7.6e-34) {
		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((m * (m * (-0.25d0))))
    if (m <= (-4.6d-27)) then
        tmp = t_0
    else if (m <= 7.6d-34) 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((m * (m * -0.25)));
	double tmp;
	if (m <= -4.6e-27) {
		tmp = t_0;
	} else if (m <= 7.6e-34) {
		tmp = Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((m * (m * -0.25)))
	tmp = 0
	if m <= -4.6e-27:
		tmp = t_0
	elif m <= 7.6e-34:
		tmp = math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(m * Float64(m * -0.25)))
	tmp = 0.0
	if (m <= -4.6e-27)
		tmp = t_0;
	elseif (m <= 7.6e-34)
		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((m * (m * -0.25)));
	tmp = 0.0;
	if (m <= -4.6e-27)
		tmp = t_0;
	elseif (m <= 7.6e-34)
		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[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -4.6e-27], t$95$0, If[LessEqual[m, 7.6e-34], N[Exp[(-l)], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -4.5999999999999999e-27 or 7.6000000000000002e-34 < m

   1. Initial program 73.1%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. cos-lowering-cos.f64 95.5

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

   5. Simplified 95.5%

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

   6. Taylor expanded in 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. exp-lowering-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. --lowering--.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-lowering-fabs.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 90.9

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

   8. Simplified 90.9%

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

   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. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 87.0

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

   11. Simplified 87.0%

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

   if -4.5999999999999999e-27 < m < 7.6000000000000002e-34

   1. Initial program 78.2%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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. cos-lowering-cos.f64 92.5

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

   5. Simplified 92.5%

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

   6. Taylor expanded in 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. exp-lowering-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. --lowering--.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-lowering-fabs.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 79.2

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

   8. Simplified 79.2%

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

   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. neg-lowering-neg.f64 47.0

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

   11. Simplified 47.0%

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

Alternative 8: 35.2% 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 75.3%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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. cos-lowering-cos.f64 94.2

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

5. Simplified 94.2%

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

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. exp-lowering-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. --lowering--.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-lowering-fabs.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-commutative N/A

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

   10. +-lowering-+.f64 N/A

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

   11. +-commutative N/A

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

   12. +-lowering-+.f64 85.9

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

8. Simplified 85.9%

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

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. neg-lowering-neg.f64 31.5

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

11. Simplified 31.5%

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

Alternative 9: 6.7% 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 75.3%

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

3. Taylor expanded in l around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 27.0

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

5. Simplified 27.0%

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

6. Taylor expanded in l around 0

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

   1. cos-lowering-cos.f64 N/A

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

   2. sub-neg N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 N/A

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

   6. +-commutative N/A

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

   7. +-lowering-+.f64 N/A

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

   8. neg-lowering-neg.f64 7.1

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

8. Simplified 7.1%

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

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. cos-lowering-cos.f64 7.1

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

11. Simplified 7.1%

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

12. Taylor expanded in M around 0

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

14. Simplified 7.1%

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

Reproduce

herbie shell --seed 2024204 
(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)))))))