Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.9%

Time: 13.2s

Alternatives: 20

Speedup: 1.5×

• 49.4% 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: 86.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

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

Alternative 2: 94.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.995:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.995)
     (+ U (* t_0 (* J (* l (fma l (* l 0.3333333333333333) 2.0)))))
     (fma (* (sinh l) J) 2.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.995) {
		tmp = U + (t_0 * (J * (l * fma(l, (l * 0.3333333333333333), 2.0))));
	} else {
		tmp = fma((sinh(l) * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.995)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)))));
	else
		tmp = fma(Float64(sinh(l) * J), 2.0, 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, 0.995], N[(U + N[(t$95$0 * N[(J * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.994999999999999996

   1. Initial program 80.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if 0.994999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.9%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 89.9

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

   5. Applied rewrites 89.9%

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

   7. Applied rewrites 99.1%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.995:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.995:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.995)
   (fma (* l (* (cos (* K 0.5)) (fma l (* l 0.3333333333333333) 2.0))) J U)
   (fma (* (sinh l) J) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.995) {
		tmp = fma((l * (cos((K * 0.5)) * fma(l, (l * 0.3333333333333333), 2.0))), J, U);
	} else {
		tmp = fma((sinh(l) * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.995)
		tmp = fma(Float64(l * Float64(cos(Float64(K * 0.5)) * fma(l, Float64(l * 0.3333333333333333), 2.0))), J, U);
	else
		tmp = fma(Float64(sinh(l) * J), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.995], N[(N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.994999999999999996

   1. Initial program 80.4%

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

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

   7. Applied rewrites 91.9%

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

   if 0.994999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.9%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 89.9

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

   5. Applied rewrites 89.9%

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

   7. Applied rewrites 99.1%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.995:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.995:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.995)
   (fma l (* (fma l (* l 0.3333333333333333) 2.0) (* (cos (* K 0.5)) J)) U)
   (fma (* (sinh l) J) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.995) {
		tmp = fma(l, (fma(l, (l * 0.3333333333333333), 2.0) * (cos((K * 0.5)) * J)), U);
	} else {
		tmp = fma((sinh(l) * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.995)
		tmp = fma(l, Float64(fma(l, Float64(l * 0.3333333333333333), 2.0) * Float64(cos(Float64(K * 0.5)) * J)), U);
	else
		tmp = fma(Float64(sinh(l) * J), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.995], N[(l * N[(N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.994999999999999996

   1. Initial program 80.4%

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

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 89.3%

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

   if 0.994999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.9%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 89.9

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

   5. Applied rewrites 89.9%

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

   7. Applied rewrites 99.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.995:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(2 \cdot \ell\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot J, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (cos (* K 0.5)) (* J (* 2.0 l)) U)
   (fma (* (sinh l) J) 2.0 U)))

C

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

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(cos(Float64(K * 0.5)), Float64(J * Float64(2.0 * l)), U);
	else
		tmp = fma(Float64(sinh(l) * J), 2.0, 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[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

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

   1. Initial program 88.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 95.2%

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

4. Final simplification 91.2%

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

Alternative 6: 84.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(l * fma(K, Float64(K * Float64(J * -0.125)), J)), 2.0, U);
	else
		tmp = fma(Float64(sinh(l) * J), 2.0, 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[(l * N[(K * N[(K * N[(J * -0.125), $MachinePrecision]), $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 48.7%

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

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

   1. Initial program 88.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 95.2%

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

4. Final simplification 86.3%

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

Alternative 7: 83.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e-115)
   (fma (* (* 2.0 (sinh l)) (fma -0.125 (* K K) 1.0)) J U)
   (+
    U
    (*
     (*
      J
      (*
       l
       (fma
        (* l l)
        (fma
         l
         (* l (fma (* l l) 0.0003968253968253968 0.016666666666666666))
         0.3333333333333333)
        2.0)))
     (cos (/ K 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e-115) {
		tmp = fma(((2.0 * sinh(l)) * fma(-0.125, (K * K), 1.0)), J, U);
	} else {
		tmp = U + ((J * (l * fma((l * l), fma(l, (l * fma((l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0))) * cos((K / 2.0)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e-115)
		tmp = fma(Float64(Float64(2.0 * sinh(l)) * fma(-0.125, Float64(K * K), 1.0)), J, U);
	else
		tmp = Float64(U + Float64(Float64(J * Float64(l * fma(Float64(l * l), fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0))) * cos(Float64(K / 2.0))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e-115], N[(N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(U + N[(N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 1.0000000000000001e-115

   1. Initial program 85.9%

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

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.2

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

   7. Applied rewrites 77.2%

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

   if 1.0000000000000001e-115 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 85.4%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) + 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)} + \frac{1}{3}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), \frac{1}{3}\right)}, 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      10. lower-*.f64 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      11. +-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{2} + \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      12. *-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{2520}} + \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{2520}, \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      14. unpow2 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{2520}, \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      15. lower-*.f64 96.8

          \[\leadsto \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied rewrites 96.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e-115)
   (fma (* (* 2.0 (sinh l)) (fma -0.125 (* K K) 1.0)) J U)
   (fma
    (*
     (cos (* K 0.5))
     (*
      l
      (fma
       (* l l)
       (fma
        l
        (* l (fma l (* l 0.0003968253968253968) 0.016666666666666666))
        0.3333333333333333)
       2.0)))
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e-115) {
		tmp = fma(((2.0 * sinh(l)) * fma(-0.125, (K * K), 1.0)), J, U);
	} else {
		tmp = fma((cos((K * 0.5)) * (l * fma((l * l), fma(l, (l * fma(l, (l * 0.0003968253968253968), 0.016666666666666666)), 0.3333333333333333), 2.0))), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e-115)
		tmp = fma(Float64(Float64(2.0 * sinh(l)) * fma(-0.125, Float64(K * K), 1.0)), J, U);
	else
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(l * fma(Float64(l * l), fma(l, Float64(l * fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666)), 0.3333333333333333), 2.0))), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e-115], N[(N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 1.0000000000000001e-115

   1. Initial program 85.9%

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

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.2

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

   7. Applied rewrites 77.2%

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

   if 1.0000000000000001e-115 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 85.4%

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

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) + 2\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)}\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), 2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}}, 2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}, 2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)} + \frac{1}{3}, 2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), \frac{1}{3}\right)}, 2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}, \frac{1}{3}\right), 2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\frac{1}{2520} \cdot {\ell}^{2} + \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{1}{2520}} + \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{2520} + \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{2520}\right)} + \frac{1}{60}\right), \frac{1}{3}\right), 2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \frac{1}{2520}, \frac{1}{60}\right)}, \frac{1}{3}\right), 2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right), J, U\right) \]

      16. lower-*.f64 96.8

          \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot 0.0003968253968253968}, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right) \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]

   7. Applied rewrites 96.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)} \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e-13)
   (fma (* (* 2.0 (sinh l)) (fma -0.125 (* K K) 1.0)) J U)
   (+
    U
    (*
     (cos (/ K 2.0))
     (*
      J
      (*
       l
       (fma
        (* l l)
        (fma (* l l) 0.016666666666666666 0.3333333333333333)
        2.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e-13) {
		tmp = fma(((2.0 * sinh(l)) * fma(-0.125, (K * K), 1.0)), J, U);
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e-13)
		tmp = fma(Float64(Float64(2.0 * sinh(l)) * fma(-0.125, Float64(K * K), 1.0)), J, U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), 2.0)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e-13], N[(N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 1e-13

   1. Initial program 86.4%

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

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 80.2

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

   7. Applied rewrites 80.2%

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

   if 1e-13 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 83.8%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 94.3

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

   5. Applied rewrites 94.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K, K \cdot \left(J \cdot -0.125\right), J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* l (fma K (* K (* J -0.125)) J)) 2.0 U)
   (fma
    J
    (*
     l
     (fma
      l
      (*
       l
       (fma
        (* l l)
        (fma (* l l) 0.0003968253968253968 0.016666666666666666)
        0.3333333333333333))
      2.0))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((l * fma(K, (K * (J * -0.125)), J)), 2.0, U);
	} else {
		tmp = fma(J, (l * fma(l, (l * fma((l * l), fma((l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333)), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(l * fma(K, Float64(K * Float64(J * -0.125)), J)), 2.0, U);
	else
		tmp = fma(J, Float64(l * fma(l, Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333)), 2.0)), 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[(l * N[(K * N[(K * N[(J * -0.125), $MachinePrecision]), $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(J * N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $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 74.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 48.7%

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

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

   1. Initial program 88.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 58.7%

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

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(J, \ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}, U\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 92.4%

       \[\leadsto \mathsf{fma}\left(J, \ell \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)}, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K, K \cdot \left(J \cdot -0.125\right), J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e-13)
   (fma (* (* 2.0 (sinh l)) (fma -0.125 (* K K) 1.0)) J U)
   (fma
    (*
     (cos (* K 0.5))
     (*
      l
      (fma l (* l (fma (* l l) 0.016666666666666666 0.3333333333333333)) 2.0)))
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e-13) {
		tmp = fma(((2.0 * sinh(l)) * fma(-0.125, (K * K), 1.0)), J, U);
	} else {
		tmp = fma((cos((K * 0.5)) * (l * fma(l, (l * fma((l * l), 0.016666666666666666, 0.3333333333333333)), 2.0))), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e-13)
		tmp = fma(Float64(Float64(2.0 * sinh(l)) * fma(-0.125, Float64(K * K), 1.0)), J, U);
	else
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(l * fma(l, Float64(l * fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333)), 2.0))), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e-13], N[(N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 1e-13

   1. Initial program 86.4%

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

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 80.2

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

   7. Applied rewrites 80.2%

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

   if 1e-13 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 83.8%

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

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 94.2

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

   7. Applied rewrites 94.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)} \cdot \cos \left(K \cdot 0.5\right), J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K, K \cdot \left(J \cdot -0.125\right), J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* l (fma K (* K (* J -0.125)) J)) 2.0 U)
   (fma
    J
    (*
     l
     (fma l (* l (fma (* l l) 0.016666666666666666 0.3333333333333333)) 2.0))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((l * fma(K, (K * (J * -0.125)), J)), 2.0, U);
	} else {
		tmp = fma(J, (l * fma(l, (l * fma((l * l), 0.016666666666666666, 0.3333333333333333)), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(l * fma(K, Float64(K * Float64(J * -0.125)), J)), 2.0, U);
	else
		tmp = fma(J, Float64(l * fma(l, Float64(l * fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333)), 2.0)), 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[(l * N[(K * N[(K * N[(J * -0.125), $MachinePrecision]), $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(J * N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $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 74.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 48.7%

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

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

   1. Initial program 88.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 89.7%

      \[\leadsto \mathsf{fma}\left(J, \ell \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)}, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K, K \cdot \left(J \cdot -0.125\right), J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K, K \cdot \left(J \cdot -0.125\right), J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right), 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* l (fma K (* K (* J -0.125)) J)) 2.0 U)
   (fma J (* l (fma l (* l (* (* l l) 0.016666666666666666)) 2.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((l * fma(K, (K * (J * -0.125)), J)), 2.0, U);
	} else {
		tmp = fma(J, (l * fma(l, (l * ((l * l) * 0.016666666666666666)), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(l * fma(K, Float64(K * Float64(J * -0.125)), J)), 2.0, U);
	else
		tmp = fma(J, Float64(l * fma(l, Float64(l * Float64(Float64(l * l) * 0.016666666666666666)), 2.0)), 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[(l * N[(K * N[(K * N[(J * -0.125), $MachinePrecision]), $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(J * N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $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 74.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 48.7%

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

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

   1. Initial program 88.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 89.7%

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

   9. Taylor expanded in l around inf

      \[\leadsto \mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\frac{1}{60} \cdot {\ell}^{\color{blue}{2}}\right), 2\right), U\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 89.6%

       \[\leadsto \mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right), 2\right), U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(K, K \cdot \left(J \cdot -0.125\right), J\right), 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right), 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(l * fma(K, Float64(K * Float64(J * -0.125)), J)), 2.0, U);
	else
		tmp = fma(J, Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)), 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[(l * N[(K * N[(K * N[(J * -0.125), $MachinePrecision]), $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(J * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $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 74.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 48.7%

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

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

   1. Initial program 88.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 83.1%

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

4. Final simplification 76.5%

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

Alternative 15: 76.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* l J) (fma -0.25 (* K K) 2.0) U)
   (fma J (* l (fma l (* l 0.3333333333333333) 2.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((l * J), fma(-0.25, (K * K), 2.0), U);
	} else {
		tmp = fma(J, (l * fma(l, (l * 0.3333333333333333), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(l * J), fma(-0.25, Float64(K * K), 2.0), U);
	else
		tmp = fma(J, Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)), 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[(l * J), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $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 74.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   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 48.6%

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

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

   1. Initial program 88.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 83.1%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma (* l J) (fma -0.25 (* K K) 2.0) U)
   (fma l (* J (fma l (* l 0.3333333333333333) 2.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((l * J), fma(-0.25, (K * K), 2.0), U);
	} else {
		tmp = fma(l, (J * fma(l, (l * 0.3333333333333333), 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(l * J), fma(-0.25, Float64(K * K), 2.0), U);
	else
		tmp = fma(l, Float64(J * fma(l, Float64(l * 0.3333333333333333), 2.0)), 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[(l * J), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(l * N[(J * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $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 74.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   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 48.6%

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

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

   1. Initial program 88.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 81.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, \mathsf{fma}\left(-0.25, K \cdot K, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 69.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -34000:\\ \;\;\;\;\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 44000000000000:\\ \;\;\;\;\mathsf{fma}\left(J, 2 \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -34000.0)
   (* l (* 0.3333333333333333 (* J (* l l))))
   (if (<= l 44000000000000.0)
     (fma J (* 2.0 l) U)
     (* l (* J (fma 0.3333333333333333 (* l l) 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -34000.0) {
		tmp = l * (0.3333333333333333 * (J * (l * l)));
	} else if (l <= 44000000000000.0) {
		tmp = fma(J, (2.0 * l), U);
	} else {
		tmp = l * (J * fma(0.3333333333333333, (l * l), 2.0));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -34000.0)
		tmp = Float64(l * Float64(0.3333333333333333 * Float64(J * Float64(l * l))));
	elseif (l <= 44000000000000.0)
		tmp = fma(J, Float64(2.0 * l), U);
	else
		tmp = Float64(l * Float64(J * fma(0.3333333333333333, Float64(l * l), 2.0)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -34000.0], N[(l * N[(0.3333333333333333 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 44000000000000.0], N[(J * N[(2.0 * l), $MachinePrecision] + U), $MachinePrecision], N[(l * N[(J * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -34000

   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 K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 57.1%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 57.4%

       \[\leadsto \ell \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(J \cdot \left(\ell \cdot \ell\right)\right)}\right) \]

   if -34000 < l < 4.4e13

   1. Initial program 72.7%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 82.2%

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

   if 4.4e13 < 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 K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 82.5

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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 67.3%

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

   9. Taylor expanded in J around inf

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

   11. Applied rewrites 67.3%

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

Alternative 18: 69.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{if}\;\ell \leq -34000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 44000000000000:\\ \;\;\;\;\mathsf{fma}\left(J, 2 \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* 0.3333333333333333 (* J (* l l))))))
   (if (<= l -34000.0)
     t_0
     (if (<= l 44000000000000.0) (fma J (* 2.0 l) U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (0.3333333333333333 * (J * (l * l)));
	double tmp;
	if (l <= -34000.0) {
		tmp = t_0;
	} else if (l <= 44000000000000.0) {
		tmp = fma(J, (2.0 * l), U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(0.3333333333333333 * Float64(J * Float64(l * l))))
	tmp = 0.0
	if (l <= -34000.0)
		tmp = t_0;
	elseif (l <= 44000000000000.0)
		tmp = fma(J, Float64(2.0 * l), U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(0.3333333333333333 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -34000.0], t$95$0, If[LessEqual[l, 44000000000000.0], N[(J * N[(2.0 * l), $MachinePrecision] + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -34000 or 4.4e13 < 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 K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 62.5%

       \[\leadsto \ell \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(J \cdot \left(\ell \cdot \ell\right)\right)}\right) \]

   if -34000 < l < 4.4e13

   1. Initial program 72.7%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 82.2%

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

Alternative 19: 69.9% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma l (* J (fma l (* l 0.3333333333333333) 2.0)) U))

C

double code(double J, double l, double K, double U) {
	return fma(l, (J * fma(l, (l * 0.3333333333333333), 2.0)), U);
}

Julia

function code(J, l, K, U)
	return fma(l, Float64(J * fma(l, Float64(l * 0.3333333333333333), 2.0)), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(l * N[(J * N[(l * N[(l * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 85.7%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 78.0

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

5. Applied rewrites 78.0%

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

6. Taylor expanded in l around 0

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

8. Applied rewrites 72.8%

   \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{J \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)}, U\right) \]
9. Add Preprocessing

Alternative 20: 54.1% accurate, 27.5× speedup

Math

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

FPCore

(FPCore (J l K U) :precision binary64 (fma J (* 2.0 l) U))

C

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

Julia

function code(J, l, K, U)
	return fma(J, Float64(2.0 * l), U)
end

Wolfram

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

Derivation

1. Initial program 85.7%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{U + J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 78.0

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

5. Applied rewrites 78.0%

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

6. Taylor expanded in l around 0

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

8. Applied rewrites 54.2%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))