Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 99.9%

Time: 11.8s

Alternatives: 22

Speedup: 1.5×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Initial Program: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \cos \left(-0.5 \cdot K\right), U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma (* 2.0 (* J (sinh l))) (cos (* -0.5 K)) U))

C

double code(double J, double l, double K, double U) {
	return fma((2.0 * (J * sinh(l))), cos((-0.5 * K)), U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(2.0 * Float64(J * sinh(l))), cos(Float64(-0.5 * K)), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 91.1

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \cos \left(-0.5 \cdot K\right), U\right) \]
6. Add Preprocessing

Alternative 2: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := 2 \cdot \left(J \cdot \sinh \ell\right)\\ \mathbf{if}\;t\_0 \leq -0.17:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\right) \cdot t\_0 + U\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (* 2.0 (* J (sinh l)))))
   (if (<= t_0 -0.17)
     (+ (* (* (* (fma (* l l) 0.3333333333333333 2.0) l) J) t_0) U)
     (if (<= t_0 -0.01)
       (fma t_1 (fma (* K K) -0.125 1.0) U)
       (fma t_1 1.0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = 2.0 * (J * sinh(l));
	double tmp;
	if (t_0 <= -0.17) {
		tmp = (((fma((l * l), 0.3333333333333333, 2.0) * l) * J) * t_0) + U;
	} else if (t_0 <= -0.01) {
		tmp = fma(t_1, fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(t_1, 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(2.0 * Float64(J * sinh(l)))
	tmp = 0.0
	if (t_0 <= -0.17)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J) * t_0) + U);
	elseif (t_0 <= -0.01)
		tmp = fma(t_1, fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(t_1, 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.17], N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(t$95$1 * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(t$95$1 * 1.0 + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.170000000000000012

   1. Initial program 88.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3}, 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. lower-*.f64 88.5

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied rewrites 88.5%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -0.170000000000000012 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 91.3

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.17:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := 2 \cdot \left(J \cdot \sinh \ell\right)\\ \mathbf{if}\;t\_0 \leq -0.17:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (* 2.0 (* J (sinh l)))))
   (if (<= t_0 -0.17)
     (fma (* (* (fma (* l l) 0.3333333333333333 2.0) J) (cos (* 0.5 K))) l U)
     (if (<= t_0 -0.01)
       (fma t_1 (fma (* K K) -0.125 1.0) U)
       (fma t_1 1.0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = 2.0 * (J * sinh(l));
	double tmp;
	if (t_0 <= -0.17) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * J) * cos((0.5 * K))), l, U);
	} else if (t_0 <= -0.01) {
		tmp = fma(t_1, fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(t_1, 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(2.0 * Float64(J * sinh(l)))
	tmp = 0.0
	if (t_0 <= -0.17)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J) * cos(Float64(0.5 * K))), l, U);
	elseif (t_0 <= -0.01)
		tmp = fma(t_1, fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(t_1, 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.17], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(t$95$1 * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(t$95$1 * 1.0 + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.170000000000000012

   1. Initial program 88.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      5. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \ell, U\right)} \]

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   if -0.170000000000000012 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 91.3

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.17:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := 2 \cdot \left(J \cdot \sinh \ell\right)\\ \mathbf{if}\;t\_0 \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (* 2.0 (* J (sinh l)))))
   (if (<= t_0 -0.42)
     (fma (* (* 2.0 J) (cos (* 0.5 K))) l U)
     (if (<= t_0 -0.01)
       (fma t_1 (fma (* K K) -0.125 1.0) U)
       (fma t_1 1.0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = 2.0 * (J * sinh(l));
	double tmp;
	if (t_0 <= -0.42) {
		tmp = fma(((2.0 * J) * cos((0.5 * K))), l, U);
	} else if (t_0 <= -0.01) {
		tmp = fma(t_1, fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(t_1, 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(2.0 * Float64(J * sinh(l)))
	tmp = 0.0
	if (t_0 <= -0.42)
		tmp = fma(Float64(Float64(2.0 * J) * cos(Float64(0.5 * K))), l, U);
	elseif (t_0 <= -0.01)
		tmp = fma(t_1, fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(t_1, 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.42], N[(N[(N[(2.0 * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(t$95$1 * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(t$95$1 * 1.0 + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.419999999999999984

   1. Initial program 88.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      5. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \ell, U\right)} \]

   5. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\cos \left(K \cdot \frac{1}{2}\right) \cdot \left(J \cdot 2\right), \ell, U\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 77.6%

      \[\leadsto \mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right), \ell, U\right) \]

   if -0.419999999999999984 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 93.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 93.7

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.5

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 82.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 91.3

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.0001984126984126984\right) \cdot \ell, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.42)
     (fma (* (* 2.0 J) (cos (* 0.5 K))) l U)
     (if (<= t_0 -0.01)
       (fma
        (*
         (*
          (fma (* (* (* (* (* l l) l) J) 0.0001984126984126984) l) (* l l) J)
          l)
         2.0)
        (fma (* K K) -0.125 1.0)
        U)
       (fma (* 2.0 (* J (sinh l))) 1.0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.42) {
		tmp = fma(((2.0 * J) * cos((0.5 * K))), l, U);
	} else if (t_0 <= -0.01) {
		tmp = fma(((fma((((((l * l) * l) * J) * 0.0001984126984126984) * l), (l * l), J) * l) * 2.0), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((2.0 * (J * sinh(l))), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.42)
		tmp = fma(Float64(Float64(2.0 * J) * cos(Float64(0.5 * K))), l, U);
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(l * l) * l) * J) * 0.0001984126984126984) * l), Float64(l * l), J) * l) * 2.0), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.42], N[(N[(N[(2.0 * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * l), $MachinePrecision] * N[(l * l), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.419999999999999984

   1. Initial program 88.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      5. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \ell, U\right)} \]

   5. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\cos \left(K \cdot \frac{1}{2}\right) \cdot \left(J \cdot 2\right), \ell, U\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 77.6%

      \[\leadsto \mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right), \ell, U\right) \]

   if -0.419999999999999984 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 93.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 93.7

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.5

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 82.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right)\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

   10. Applied rewrites 73.9%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \left(\ell \cdot J\right) \cdot \ell, 0.008333333333333333 \cdot J\right) \cdot \ell, \ell, 0.16666666666666666 \cdot J\right), \ell \cdot \ell, J\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{4}\right), \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 73.9%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.0001984126984126984\right) \cdot \ell, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 91.3

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.0001984126984126984\right) \cdot \ell, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right), J, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.0001984126984126984\right) \cdot \ell, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.42)
     (fma (* (* 2.0 l) (cos (* 0.5 K))) J U)
     (if (<= t_0 -0.01)
       (fma
        (*
         (*
          (fma (* (* (* (* (* l l) l) J) 0.0001984126984126984) l) (* l l) J)
          l)
         2.0)
        (fma (* K K) -0.125 1.0)
        U)
       (fma (* 2.0 (* J (sinh l))) 1.0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.42) {
		tmp = fma(((2.0 * l) * cos((0.5 * K))), J, U);
	} else if (t_0 <= -0.01) {
		tmp = fma(((fma((((((l * l) * l) * J) * 0.0001984126984126984) * l), (l * l), J) * l) * 2.0), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((2.0 * (J * sinh(l))), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.42)
		tmp = fma(Float64(Float64(2.0 * l) * cos(Float64(0.5 * K))), J, U);
	elseif (t_0 <= -0.01)
		tmp = fma(Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(l * l) * l) * J) * 0.0001984126984126984) * l), Float64(l * l), J) * l) * 2.0), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.42], N[(N[(N[(2.0 * l), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * l), $MachinePrecision] * N[(l * l), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.419999999999999984

   1. Initial program 88.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(2 \cdot \ell\right)}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot J, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

      12. lower-*.f64 77.6

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot 0.5\right)}, U\right) \]

   5. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(K \cdot 0.5\right), U\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell\right), J, U\right)} \]

   if -0.419999999999999984 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 93.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 93.7

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.5

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 82.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right)\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

   10. Applied rewrites 73.9%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \left(\ell \cdot J\right) \cdot \ell, 0.008333333333333333 \cdot J\right) \cdot \ell, \ell, 0.16666666666666666 \cdot J\right), \ell \cdot \ell, J\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{4}\right), \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 73.9%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.0001984126984126984\right) \cdot \ell, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 91.3

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right), J, U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.0001984126984126984\right) \cdot \ell, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.98)
   (fma
    (cos (* 0.5 K))
    (*
     (*
      (fma (fma 0.016666666666666666 (* l l) 0.3333333333333333) (* l l) 2.0)
      J)
     l)
    U)
   (fma (* 2.0 (* J (sinh l))) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.98) {
		tmp = fma(cos((0.5 * K)), ((fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * J) * l), U);
	} else {
		tmp = fma((2.0 * (J * sinh(l))), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.98)
		tmp = fma(cos(Float64(0.5 * K)), Float64(Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * J) * l), U);
	else
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.98], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.97999999999999998

   1. Initial program 90.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{60} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{3} \cdot J\right)\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{60} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{3} \cdot J\right)\right) \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{60} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{3} \cdot J\right)\right) \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{60}, \ell \cdot \ell, \frac{1}{3}\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{60}, \ell \cdot \ell, \frac{1}{3}\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{60}, \ell \cdot \ell, \frac{1}{3}\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell\right)} + U \]

      4. lower-fma.f64 89.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{K}{2}\right), \mathsf{fma}\left(J \cdot \mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2 \cdot J\right) \cdot \ell, U\right)} \]

   7. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), \left(J \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right)\right) \cdot \ell, U\right)} \]

   if 0.97999999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 91.5

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 99.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right), \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot J, \ell, U\right)\\ \mathbf{elif}\;t\_0 \leq 0.055:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.42)
     (fma (* 2.0 J) l U)
     (if (<= t_0 0.055)
       (* (* (* (* K K) l) J) -0.25)
       (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.42) {
		tmp = fma((2.0 * J), l, U);
	} else if (t_0 <= 0.055) {
		tmp = (((K * K) * l) * J) * -0.25;
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.42)
		tmp = fma(Float64(2.0 * J), l, U);
	elseif (t_0 <= 0.055)
		tmp = Float64(Float64(Float64(Float64(K * K) * l) * J) * -0.25);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.42], N[(N[(2.0 * J), $MachinePrecision] * l + U), $MachinePrecision], If[LessEqual[t$95$0, 0.055], N[(N[(N[(N[(K * K), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * -0.25), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.419999999999999984

   1. Initial program 88.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(2 \cdot \ell\right)}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot J, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

      12. lower-*.f64 77.6

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot 0.5\right)}, U\right) \]

   5. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(K \cdot 0.5\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.7%

      \[\leadsto \mathsf{fma}\left(J \cdot 2, \color{blue}{\ell}, U\right) \]

   if -0.419999999999999984 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0550000000000000003

   1. Initial program 93.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(2 \cdot \ell\right)}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot J, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

      12. lower-*.f64 41.5

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot 0.5\right)}, U\right) \]

   5. Applied rewrites 41.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(K \cdot 0.5\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.6%

      \[\leadsto \mathsf{fma}\left(J \cdot 2, \color{blue}{\ell}, U\right) \]

   9. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 54.0%

       \[\leadsto \mathsf{fma}\left(\ell \cdot 2, \color{blue}{J}, \mathsf{fma}\left(-0.25 \cdot \left(\ell \cdot J\right), K \cdot K, U\right)\right) \]

   12. Taylor expanded in K around inf

       \[\leadsto \frac{-1}{4} \cdot \left(J \cdot \color{blue}{\left({K}^{2} \cdot \ell\right)}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 49.7%

       \[\leadsto \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.25 \]

   if 0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      5. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \ell, U\right)} \]

   5. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, \color{blue}{J}, U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot J, \ell, U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.055:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.0001984126984126984\right) \cdot \ell, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma
    (*
     (* (fma (* (* (* (* (* l l) l) J) 0.0001984126984126984) l) (* l l) J) l)
     2.0)
    (fma (* K K) -0.125 1.0)
    U)
   (fma (* 2.0 (* J (sinh l))) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(((fma((((((l * l) * l) * J) * 0.0001984126984126984) * l), (l * l), J) * l) * 2.0), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((2.0 * (J * sinh(l))), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(l * l) * l) * J) * 0.0001984126984126984) * l), Float64(l * l), J) * l) * 2.0), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * l), $MachinePrecision] * N[(l * l), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 90.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 90.4

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.5

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 70.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right)\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

   10. Applied rewrites 63.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \left(\ell \cdot J\right) \cdot \ell, 0.008333333333333333 \cdot J\right) \cdot \ell, \ell, 0.16666666666666666 \cdot J\right), \ell \cdot \ell, J\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{4}\right), \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 63.6%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.0001984126984126984\right) \cdot \ell, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 91.3

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{1}, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.0001984126984126984\right) \cdot \ell, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(-0.5 \cdot K\right), J, U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma (* (* 2.0 (sinh l)) (cos (* -0.5 K))) J U))

C

double code(double J, double l, double K, double U) {
	return fma(((2.0 * sinh(l)) * cos((-0.5 * K))), J, U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(Float64(2.0 * sinh(l)) * cos(Float64(-0.5 * K))), J, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot -0.5\right) \cdot \left(\sinh \ell \cdot 2\right), J, U\right)} \]

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(-0.5 \cdot K\right), J, U\right) \]
6. Add Preprocessing

Alternative 11: 83.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.0001984126984126984\right) \cdot \ell, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma
    (*
     (* (fma (* (* (* (* (* l l) l) J) 0.0001984126984126984) l) (* l l) J) l)
     2.0)
    (fma (* K K) -0.125 1.0)
    U)
   (fma
    (*
     1.0
     (*
      (fma
       (fma
        (fma (* l l) 0.0001984126984126984 0.008333333333333333)
        (* l l)
        0.16666666666666666)
       (* l l)
       1.0)
      l))
    (* 2.0 J)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(((fma((((((l * l) * l) * J) * 0.0001984126984126984) * l), (l * l), J) * l) * 2.0), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((1.0 * (fma(fma(fma((l * l), 0.0001984126984126984, 0.008333333333333333), (l * l), 0.16666666666666666), (l * l), 1.0) * l)), (2.0 * J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(l * l) * l) * J) * 0.0001984126984126984) * l), Float64(l * l), J) * l) * 2.0), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(1.0 * Float64(fma(fma(fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l)), Float64(2.0 * J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * l), $MachinePrecision] * N[(l * l), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(1.0 * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 90.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 90.4

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.5

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 70.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right)\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

   10. Applied rewrites 63.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \left(\ell \cdot J\right) \cdot \ell, 0.008333333333333333 \cdot J\right) \cdot \ell, \ell, 0.16666666666666666 \cdot J\right), \ell \cdot \ell, J\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{4}\right), \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 63.6%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.0001984126984126984\right) \cdot \ell, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 91.3

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right)} \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\left(\color{blue}{\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}} + 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{2}, 1\right)} \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{{\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right) + \frac{1}{6}}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right) \cdot {\ell}^{2}} + \frac{1}{6}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}, {\ell}^{2}, \frac{1}{6}\right)}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {\ell}^{2} + \frac{1}{120}}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5040}, {\ell}^{2}, \frac{1}{120}\right)}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{\ell \cdot \ell}, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{\ell \cdot \ell}, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      16. lower-*.f64 95.8

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right) \]

   7. Applied rewrites 95.8%

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2\right) \cdot \cos \left(K \cdot \frac{-1}{2}\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \color{blue}{\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2\right)} + U \]

      4. lift-*.f64 N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right)} \cdot 2\right) + U \]

      5. associate-*l* N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot \left(J \cdot 2\right)\right)} + U \]

      6. lift-*.f64 N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot \color{blue}{\left(J \cdot 2\right)}\right) + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right)\right) \cdot \left(J \cdot 2\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right), J \cdot 2, U\right)} \]

   9. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right)} \]

   10. Taylor expanded in K around 0

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 93.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 82.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 20000:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(-0.5 \cdot K\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 20000.0)
   (fma (* 2.0 (* J (sinh l))) (fma (* K K) -0.125 1.0) U)
   (fma
    (*
     (*
      (*
       (fma
        (fma
         (fma 0.0001984126984126984 (* l l) 0.008333333333333333)
         (* l l)
         0.16666666666666666)
        (* l l)
        1.0)
       l)
      J)
     2.0)
    (cos (* -0.5 K))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 20000.0) {
		tmp = fma((2.0 * (J * sinh(l))), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((((fma(fma(fma(0.0001984126984126984, (l * l), 0.008333333333333333), (l * l), 0.16666666666666666), (l * l), 1.0) * l) * J) * 2.0), cos((-0.5 * K)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 20000.0)
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(l * l), 0.008333333333333333), Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l) * J) * 2.0), cos(Float64(-0.5 * K)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 20000.0], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(l * l), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 2e4

   1. Initial program 90.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 90.7

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 86.4

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 86.4%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   if 2e4 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 92.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 92.3

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right)} \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\left(\color{blue}{\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}} + 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{2}, 1\right)} \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{{\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right) + \frac{1}{6}}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right) \cdot {\ell}^{2}} + \frac{1}{6}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}, {\ell}^{2}, \frac{1}{6}\right)}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {\ell}^{2} + \frac{1}{120}}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5040}, {\ell}^{2}, \frac{1}{120}\right)}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{\ell \cdot \ell}, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{\ell \cdot \ell}, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      16. lower-*.f64 96.0

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right) \]

   7. Applied rewrites 96.0%

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 20000:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(-0.5 \cdot K\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 20000:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot \cos \left(-0.5 \cdot K\right), 2 \cdot J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 20000.0)
   (fma (* 2.0 (* J (sinh l))) (fma (* K K) -0.125 1.0) U)
   (fma
    (*
     (*
      (fma
       (fma
        (fma (* l l) 0.0001984126984126984 0.008333333333333333)
        (* l l)
        0.16666666666666666)
       (* l l)
       1.0)
      l)
     (cos (* -0.5 K)))
    (* 2.0 J)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 20000.0) {
		tmp = fma((2.0 * (J * sinh(l))), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(((fma(fma(fma((l * l), 0.0001984126984126984, 0.008333333333333333), (l * l), 0.16666666666666666), (l * l), 1.0) * l) * cos((-0.5 * K))), (2.0 * J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 20000.0)
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(Float64(fma(fma(fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l) * cos(Float64(-0.5 * K))), Float64(2.0 * J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 20000.0], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 2e4

   1. Initial program 90.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 90.7

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 86.4

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 86.4%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   if 2e4 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 92.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 92.3

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right)} \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\left(\color{blue}{\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}} + 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{2}, 1\right)} \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{{\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right) + \frac{1}{6}}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right) \cdot {\ell}^{2}} + \frac{1}{6}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}, {\ell}^{2}, \frac{1}{6}\right)}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {\ell}^{2} + \frac{1}{120}}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5040}, {\ell}^{2}, \frac{1}{120}\right)}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{\ell \cdot \ell}, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{\ell \cdot \ell}, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      16. lower-*.f64 96.0

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right) \]

   7. Applied rewrites 96.0%

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2\right) \cdot \cos \left(K \cdot \frac{-1}{2}\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \color{blue}{\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2\right)} + U \]

      4. lift-*.f64 N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right)} \cdot 2\right) + U \]

      5. associate-*l* N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot \left(J \cdot 2\right)\right)} + U \]

      6. lift-*.f64 N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot \color{blue}{\left(J \cdot 2\right)}\right) + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right)\right) \cdot \left(J \cdot 2\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right), J \cdot 2, U\right)} \]

   9. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 20000:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot \cos \left(-0.5 \cdot K\right), 2 \cdot J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 83.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma
    (*
     (*
      (*
       (fma (fma 0.008333333333333333 (* l l) 0.16666666666666666) (* l l) 1.0)
       l)
      J)
     2.0)
    (fma (* K K) -0.125 1.0)
    U)
   (fma
    (*
     1.0
     (*
      (fma
       (fma
        (fma (* l l) 0.0001984126984126984 0.008333333333333333)
        (* l l)
        0.16666666666666666)
       (* l l)
       1.0)
      l))
    (* 2.0 J)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((((fma(fma(0.008333333333333333, (l * l), 0.16666666666666666), (l * l), 1.0) * l) * J) * 2.0), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((1.0 * (fma(fma(fma((l * l), 0.0001984126984126984, 0.008333333333333333), (l * l), 0.16666666666666666), (l * l), 1.0) * l)), (2.0 * J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l) * J) * 2.0), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(1.0 * Float64(fma(fma(fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l)), Float64(2.0 * J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(N[(N[(N[(0.008333333333333333 * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(1.0 * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 90.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 90.4

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.5

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 70.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right)\right)} \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) + 1\right)} \cdot \ell\right) \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\left(\color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right) \cdot {\ell}^{2}} + 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}, {\ell}^{2}, 1\right)} \cdot \ell\right) \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {\ell}^{2} + \frac{1}{6}}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {\ell}^{2}, \frac{1}{6}\right)}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \ell \cdot \ell, \frac{1}{6}\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      11. lower-*.f64 62.2

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   10. Applied rewrites 62.2%

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 91.3

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right)} \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\left(\color{blue}{\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}} + 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{2}, 1\right)} \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{{\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right) + \frac{1}{6}}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right) \cdot {\ell}^{2}} + \frac{1}{6}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}, {\ell}^{2}, \frac{1}{6}\right)}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {\ell}^{2} + \frac{1}{120}}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5040}, {\ell}^{2}, \frac{1}{120}\right)}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{\ell \cdot \ell}, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{\ell \cdot \ell}, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      16. lower-*.f64 95.8

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right) \]

   7. Applied rewrites 95.8%

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2\right) \cdot \cos \left(K \cdot \frac{-1}{2}\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \color{blue}{\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2\right)} + U \]

      4. lift-*.f64 N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right)} \cdot 2\right) + U \]

      5. associate-*l* N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot \left(J \cdot 2\right)\right)} + U \]

      6. lift-*.f64 N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot \color{blue}{\left(J \cdot 2\right)}\right) + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right)\right) \cdot \left(J \cdot 2\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right), J \cdot 2, U\right)} \]

   9. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right)} \]

   10. Taylor expanded in K around 0

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 93.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 82.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666 \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma
    (* (* (fma (* 0.16666666666666666 J) (* l l) J) l) 2.0)
    (fma (* K K) -0.125 1.0)
    U)
   (fma
    (*
     1.0
     (*
      (fma
       (fma
        (fma (* l l) 0.0001984126984126984 0.008333333333333333)
        (* l l)
        0.16666666666666666)
       (* l l)
       1.0)
      l))
    (* 2.0 J)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(((fma((0.16666666666666666 * J), (l * l), J) * l) * 2.0), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((1.0 * (fma(fma(fma((l * l), 0.0001984126984126984, 0.008333333333333333), (l * l), 0.16666666666666666), (l * l), 1.0) * l)), (2.0 * J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(Float64(fma(Float64(0.16666666666666666 * J), Float64(l * l), J) * l) * 2.0), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(1.0 * Float64(fma(fma(fma(Float64(l * l), 0.0001984126984126984, 0.008333333333333333), Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l)), Float64(2.0 * J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(N[(N[(0.16666666666666666 * J), $MachinePrecision] * N[(l * l), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(1.0 * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 90.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 90.4

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.5

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 70.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right)\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

   10. Applied rewrites 63.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \left(\ell \cdot J\right) \cdot \ell, 0.008333333333333333 \cdot J\right) \cdot \ell, \ell, 0.16666666666666666 \cdot J\right), \ell \cdot \ell, J\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   11. Taylor expanded in l around 0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{6} \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 60.7%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666 \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 91.3

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) + 1\right)} \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\left(\color{blue}{\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}} + 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right), {\ell}^{2}, 1\right)} \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{{\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right) + \frac{1}{6}}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right) \cdot {\ell}^{2}} + \frac{1}{6}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}, {\ell}^{2}, \frac{1}{6}\right)}, {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {\ell}^{2} + \frac{1}{120}}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5040}, {\ell}^{2}, \frac{1}{120}\right)}, {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{\ell \cdot \ell}, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{\ell \cdot \ell}, \frac{1}{120}\right), {\ell}^{2}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      16. lower-*.f64 95.8

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \color{blue}{\ell \cdot \ell}, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right) \]

   7. Applied rewrites 95.8%

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right)} \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2\right) \cdot \cos \left(K \cdot \frac{-1}{2}\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \color{blue}{\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right) \cdot 2\right)} + U \]

      4. lift-*.f64 N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot J\right)} \cdot 2\right) + U \]

      5. associate-*l* N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot \left(J \cdot 2\right)\right)} + U \]

      6. lift-*.f64 N/A

         \[\leadsto \cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right) \cdot \color{blue}{\left(J \cdot 2\right)}\right) + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right)\right) \cdot \left(J \cdot 2\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot \frac{-1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \ell \cdot \ell, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right), J \cdot 2, U\right)} \]

   9. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right)} \]

   10. Taylor expanded in K around 0

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{5040}, \frac{1}{120}\right), \ell \cdot \ell, \frac{1}{6}\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 93.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0001984126984126984, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right), 2 \cdot J, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 77.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666 \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma
    (* (* (fma (* 0.16666666666666666 J) (* l l) J) l) 2.0)
    (fma (* K K) -0.125 1.0)
    U)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(((fma((0.16666666666666666 * J), (l * l), J) * l) * 2.0), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(Float64(fma(Float64(0.16666666666666666 * J), Float64(l * l), J) * l) * 2.0), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(N[(N[(0.16666666666666666 * J), $MachinePrecision] * N[(l * l), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 90.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 90.4

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.5

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 70.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right)\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{6} \cdot J + {\ell}^{2} \cdot \left(\frac{1}{5040} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{120} \cdot J\right)\right)\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]

   10. Applied rewrites 63.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \left(\ell \cdot J\right) \cdot \ell, 0.008333333333333333 \cdot J\right) \cdot \ell, \ell, 0.16666666666666666 \cdot J\right), \ell \cdot \ell, J\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   11. Taylor expanded in l around 0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{6} \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, \frac{-1}{8}, 1\right), U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 60.7%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666 \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      5. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \ell, U\right)} \]

   5. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, \color{blue}{J}, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 76.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.055:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666, \left(J \cdot \ell\right) \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.055)
   (fma
    (* (* (fma 0.16666666666666666 (* (* J l) l) J) l) 2.0)
    (fma (* K K) -0.125 1.0)
    U)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.055) {
		tmp = fma(((fma(0.16666666666666666, ((J * l) * l), J) * l) * 2.0), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.055)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(Float64(J * l) * l), J) * l) * 2.0), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.055], N[(N[(N[(N[(0.16666666666666666 * N[(N[(J * l), $MachinePrecision] * l), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0550000000000000003

   1. Initial program 90.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 90.5

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right)} \]

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.5

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right), U\right) \]

   7. Applied rewrites 69.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 58.6

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{\left(\ell \cdot J\right)} \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   10. Applied rewrites 58.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, \left(\ell \cdot J\right) \cdot \ell, J\right) \cdot \ell\right)} \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right) \]

   if 0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      5. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \ell, U\right)} \]

   5. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, \color{blue}{J}, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.055:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666, \left(J \cdot \ell\right) \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 76.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(t\_0 \cdot J\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (* l l) 0.3333333333333333 2.0)))
   (if (<= (cos (/ K 2.0)) -0.01)
     (fma (* (fma (* K K) -0.125 1.0) (* t_0 J)) l U)
     (fma (* t_0 l) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((l * l), 0.3333333333333333, 2.0);
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (t_0 * J)), l, U);
	} else {
		tmp = fma((t_0 * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(l * l), 0.3333333333333333, 2.0)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(t_0 * J)), l, U);
	else
		tmp = fma(Float64(t_0 * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(t$95$0 * l), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 90.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      5. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \ell, U\right)} \]

   5. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right)\right), \ell, U\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 59.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right) \]

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      5. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \ell, U\right)} \]

   5. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, \color{blue}{J}, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 74.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.055:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.055)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.055) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.055)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.055], N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0550000000000000003

   1. Initial program 90.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(2 \cdot \ell\right)}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot J, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

      12. lower-*.f64 65.7

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot 0.5\right)}, U\right) \]

   5. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(K \cdot 0.5\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.7%

      \[\leadsto \mathsf{fma}\left(\ell \cdot J, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.25, 2\right)}, U\right) \]

   if 0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      5. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \ell, U\right)} \]

   5. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, \color{blue}{J}, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.055:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 52.6% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 3.6e+94) (fma (* 2.0 J) l U) (* (* (* (* K K) l) J) -0.25)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 3.6e+94) {
		tmp = fma((2.0 * J), l, U);
	} else {
		tmp = (((K * K) * l) * J) * -0.25;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 3.6e+94)
		tmp = fma(Float64(2.0 * J), l, U);
	else
		tmp = Float64(Float64(Float64(Float64(K * K) * l) * J) * -0.25);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 3.6e+94], N[(N[(2.0 * J), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(N[(K * K), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * -0.25), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.59999999999999992e94

   1. Initial program 89.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(2 \cdot \ell\right)}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot J, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

      12. lower-*.f64 75.4

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot 0.5\right)}, U\right) \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(K \cdot 0.5\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.8%

      \[\leadsto \mathsf{fma}\left(J \cdot 2, \color{blue}{\ell}, U\right) \]

   if 3.59999999999999992e94 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(2 \cdot \ell\right)}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot J, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

      12. lower-*.f64 29.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot 0.5\right)}, U\right) \]

   5. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(K \cdot 0.5\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.4%

      \[\leadsto \mathsf{fma}\left(J \cdot 2, \color{blue}{\ell}, U\right) \]

   9. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 26.1%

       \[\leadsto \mathsf{fma}\left(\ell \cdot 2, \color{blue}{J}, \mathsf{fma}\left(-0.25 \cdot \left(\ell \cdot J\right), K \cdot K, U\right)\right) \]

   12. Taylor expanded in K around inf

       \[\leadsto \frac{-1}{4} \cdot \left(J \cdot \color{blue}{\left({K}^{2} \cdot \ell\right)}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 29.2%

       \[\leadsto \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.25 \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 53.3% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2 \cdot J, \ell, U\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (fma (* 2.0 J) l U))

C

double code(double J, double l, double K, double U) {
	return fma((2.0 * J), l, U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(2.0 * J), l, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(2.0 * J), $MachinePrecision] * l + U), $MachinePrecision]

Derivation

1. Initial program 91.1%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(2 \cdot \ell\right)}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot J, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   10. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

   12. lower-*.f64 67.1

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot 0.5\right)}, U\right) \]

5. Applied rewrites 67.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(K \cdot 0.5\right), U\right)} \]

6. Taylor expanded in K around 0

   \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
7. Step-by-step derivation

8. Applied rewrites 58.4%

   \[\leadsto \mathsf{fma}\left(J \cdot 2, \color{blue}{\ell}, U\right) \]

9. Final simplification 58.4%

   \[\leadsto \mathsf{fma}\left(2 \cdot J, \ell, U\right) \]
10. Add Preprocessing

Alternative 22: 19.2% accurate, 30.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \ell\right) \cdot 2 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (* (* J l) 2.0))

C

double code(double J, double l, double K, double U) {
	return (J * l) * 2.0;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = (j * l) * 2.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return (J * l) * 2.0;
}

Python

def code(J, l, K, U):
	return (J * l) * 2.0

Julia

function code(J, l, K, U)
	return Float64(Float64(J * l) * 2.0)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = (J * l) * 2.0;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(J * l), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 91.1%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(2 \cdot \ell\right)}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right) \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot \ell\right)} \cdot J, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   10. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \color{blue}{\cos \left(\frac{1}{2} \cdot K\right)}, U\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

   12. lower-*.f64 67.1

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \color{blue}{\left(K \cdot 0.5\right)}, U\right) \]

5. Applied rewrites 67.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(K \cdot 0.5\right), U\right)} \]

6. Taylor expanded in K around 0

   \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
7. Step-by-step derivation

8. Applied rewrites 58.4%

   \[\leadsto \mathsf{fma}\left(J \cdot 2, \color{blue}{\ell}, U\right) \]

9. Taylor expanded in J around inf

   \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\ell}\right) \]
10. Step-by-step derivation

11. Applied rewrites 18.3%

    \[\leadsto \left(\ell \cdot J\right) \cdot 2 \]

12. Final simplification 18.3%

    \[\leadsto \left(J \cdot \ell\right) \cdot 2 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))