Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 100.0%

Time: 13.7s

Alternatives: 23

Speedup: 1.8×

• 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: 100.0% 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 85.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. 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 100.0%

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

Alternative 2: 87.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := 2 \cdot \sinh \ell\\ \mathbf{if}\;t\_0 \leq -0.88:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (* 2.0 (sinh l))))
   (if (<= t_0 -0.88)
     (fma (* (cos (* K 0.5)) (* 2.0 l)) J U)
     (if (<= t_0 -0.02)
       (fma (* t_1 (fma K (* K -0.125) 1.0)) J U)
       (fma t_1 J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = 2.0 * sinh(l);
	double tmp;
	if (t_0 <= -0.88) {
		tmp = fma((cos((K * 0.5)) * (2.0 * l)), J, U);
	} else if (t_0 <= -0.02) {
		tmp = fma((t_1 * fma(K, (K * -0.125), 1.0)), J, U);
	} else {
		tmp = fma(t_1, J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(2.0 * sinh(l))
	tmp = 0.0
	if (t_0 <= -0.88)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(2.0 * l)), J, U);
	elseif (t_0 <= -0.02)
		tmp = fma(Float64(t_1 * fma(K, Float64(K * -0.125), 1.0)), J, U);
	else
		tmp = fma(t_1, J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.88], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(t$95$1 * N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(t$95$1 * J + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.880000000000000004

   1. Initial program 67.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   if -0.880000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 91.2%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 77.9

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

   7. Applied rewrites 77.9%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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

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

   7. Applied rewrites 95.6%

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

      1. lift-sinh.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.6

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

   9. Applied rewrites 95.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.88:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \sinh \ell\right) \cdot \mathsf{fma}\left(K, K \cdot -0.125, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.87:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.87)
     (fma (* (cos (* K 0.5)) (* 2.0 l)) J U)
     (if (<= t_0 -0.02)
       (fma
        (fma K (* K -0.125) 1.0)
        (*
         l
         (*
          J
          (fma
           l
           (*
            l
            (fma
             (* l l)
             (fma l (* l 0.0003968253968253968) 0.016666666666666666)
             0.3333333333333333))
           2.0)))
        U)
       (fma (* 2.0 (sinh l)) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.87) {
		tmp = fma((cos((K * 0.5)) * (2.0 * l)), J, U);
	} else if (t_0 <= -0.02) {
		tmp = fma(fma(K, (K * -0.125), 1.0), (l * (J * fma(l, (l * fma((l * l), fma(l, (l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333)), 2.0))), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.87)
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(2.0 * l)), J, U);
	elseif (t_0 <= -0.02)
		tmp = fma(fma(K, Float64(K * -0.125), 1.0), Float64(l * Float64(J * fma(l, Float64(l * fma(Float64(l * l), fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333)), 2.0))), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.87], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(J * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.869999999999999996

   1. Initial program 72.1%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 80.1

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

   7. Applied rewrites 80.1%

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

   if -0.869999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 90.8%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + \ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + \ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + \ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2} + 1\right) \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + U \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

   8. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), U\right)} \]

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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

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

   7. Applied rewrites 95.6%

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

      1. lift-sinh.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.6

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

   9. Applied rewrites 95.6%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.87:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.87:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(2 \cdot \ell\right), U\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.87)
     (fma (cos (* K 0.5)) (* J (* 2.0 l)) U)
     (if (<= t_0 -0.02)
       (fma
        (fma K (* K -0.125) 1.0)
        (*
         l
         (*
          J
          (fma
           l
           (*
            l
            (fma
             (* l l)
             (fma l (* l 0.0003968253968253968) 0.016666666666666666)
             0.3333333333333333))
           2.0)))
        U)
       (fma (* 2.0 (sinh l)) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.87) {
		tmp = fma(cos((K * 0.5)), (J * (2.0 * l)), U);
	} else if (t_0 <= -0.02) {
		tmp = fma(fma(K, (K * -0.125), 1.0), (l * (J * fma(l, (l * fma((l * l), fma(l, (l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333)), 2.0))), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.87)
		tmp = fma(cos(Float64(K * 0.5)), Float64(J * Float64(2.0 * l)), U);
	elseif (t_0 <= -0.02)
		tmp = fma(fma(K, Float64(K * -0.125), 1.0), Float64(l * Float64(J * fma(l, Float64(l * fma(Float64(l * l), fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333)), 2.0))), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.87], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(J * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.869999999999999996

   1. Initial program 72.1%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-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 80.0

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

   5. Applied rewrites 80.0%

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

   if -0.869999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 90.8%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + \ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + \ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + \ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2} + 1\right) \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + U \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

   8. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), U\right)} \]

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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

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

   7. Applied rewrites 95.6%

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

      1. lift-sinh.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.6

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

   9. Applied rewrites 95.6%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.87:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right), J \cdot \left(2 \cdot \ell\right), U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.984:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.984)
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (fma
          (* l l)
          (fma (* l l) 0.016666666666666666 0.3333333333333333)
          2.0)))))
     (fma (* 2.0 (sinh l)) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.984) {
		tmp = U + (t_0 * (J * (l * fma((l * l), fma((l * l), 0.016666666666666666, 0.3333333333333333), 2.0))));
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.984)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(Float64(l * l), fma(Float64(l * l), 0.016666666666666666, 0.3333333333333333), 2.0)))));
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.984], N[(U + N[(t$95$0 * 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], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.98399999999999999

   1. Initial program 84.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 94.0

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

   5. Applied rewrites 94.0%

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

   if 0.98399999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 99.1%

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

      1. lift-sinh.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.1

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

   9. Applied rewrites 99.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.984:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right), J, U\right)\\ \mathbf{elif}\;t\_0 \leq 0.999999998:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \left(\ell \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.02)
     (fma (* l (* -0.25 (* K K))) J U)
     (if (<= t_0 0.999999998)
       (fma J (* l (* l (* l (* (* l l) 0.016666666666666666)))) U)
       (fma J (* l (fma l (* l 0.3333333333333333) 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 <= -0.02) {
		tmp = fma((l * (-0.25 * (K * K))), J, U);
	} else if (t_0 <= 0.999999998) {
		tmp = fma(J, (l * (l * (l * ((l * l) * 0.016666666666666666)))), U);
	} else {
		tmp = fma(J, (l * fma(l, (l * 0.3333333333333333), 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 <= -0.02)
		tmp = fma(Float64(l * Float64(-0.25 * Float64(K * K))), J, U);
	elseif (t_0 <= 0.999999998)
		tmp = fma(J, Float64(l * Float64(l * Float64(l * Float64(Float64(l * l) * 0.016666666666666666)))), 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(l * N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[t$95$0, 0.999999998], N[(J * N[(l * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 85.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{4}} + 2\right), J, U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{4}, 2\right)}, J, U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{4}, 2\right), J, U\right) \]

      7. lower-*.f64 60.8

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

   10. Applied rewrites 60.8%

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

   11. Taylor expanded in K around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}, J, U\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(K \cdot K\right)}\right), J, U\right) \]

       3. lower-*.f64 60.8

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

   13. Applied rewrites 60.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.999999997999999946

   1. Initial program 85.2%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 82.3%

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

   9. Taylor expanded in l around inf

      \[\leadsto \mathsf{fma}\left(J, \ell \cdot \color{blue}{\left(\frac{1}{60} \cdot {\ell}^{4}\right)}, U\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 79.1

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

   11. Applied rewrites 79.1%

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

   if 0.999999997999999946 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.2%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 94.1%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 88.8

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

   11. Applied rewrites 88.8%

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

Alternative 7: 93.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.625:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.625)
     (+ U (* t_0 (* J (* l (fma l (* l 0.3333333333333333) 2.0)))))
     (fma (* 2.0 (sinh l)) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.625) {
		tmp = U + (t_0 * (J * (l * fma(l, (l * 0.3333333333333333), 2.0))));
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.625)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * fma(l, Float64(l * 0.3333333333333333), 2.0)))));
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.625], 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[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.625

   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. 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 90.3

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

   5. Applied rewrites 90.3%

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

   if 0.625 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 96.7%

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

      1. lift-sinh.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 96.7

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

   9. Applied rewrites 96.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.625:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.625:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.625)
   (fma l (* (fma l (* l 0.3333333333333333) 2.0) (* (cos (* K 0.5)) J)) U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.625) {
		tmp = fma(l, (fma(l, (l * 0.3333333333333333), 2.0) * (cos((K * 0.5)) * J)), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.625)
		tmp = fma(l, Float64(fma(l, Float64(l * 0.3333333333333333), 2.0) * Float64(cos(Float64(K * 0.5)) * J)), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.625], 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[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.625

   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. 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 89.1%

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

   if 0.625 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 96.7%

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

      1. lift-sinh.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 96.7

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

   9. Applied rewrites 96.7%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.625:\\ \;\;\;\;\mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot J\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(\ell, 2 \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (- (exp l) (exp (- l))) 0.004)
   (fma l (* 2.0 J) U)
   (* (* l J) (fma -0.25 (* K K) 2.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((exp(l) - exp(-l)) <= 0.004) {
		tmp = fma(l, (2.0 * J), U);
	} else {
		tmp = (l * J) * fma(-0.25, (K * K), 2.0);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(exp(l) - exp(Float64(-l))) <= 0.004)
		tmp = fma(l, Float64(2.0 * J), U);
	else
		tmp = Float64(Float64(l * J) * fma(-0.25, Float64(K * K), 2.0));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], 0.004], N[(l * N[(2.0 * J), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * J), $MachinePrecision] * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0040000000000000001

   1. Initial program 81.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-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 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 65.0

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

   8. Applied rewrites 65.0%

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

   if 0.0040000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 21.1

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

   7. Applied rewrites 21.1%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{4}} + 2\right), J, U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{4}, 2\right)}, J, U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{4}, 2\right), J, U\right) \]

      7. lower-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in l around inf

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \frac{-1}{4} \cdot {K}^{2}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \frac{-1}{4} \cdot {K}^{2}\right)} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\ell \cdot J\right)} \cdot \left(2 + \frac{-1}{4} \cdot {K}^{2}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\ell \cdot J\right)} \cdot \left(2 + \frac{-1}{4} \cdot {K}^{2}\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\ell \cdot J\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2} + 2\right)} \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\ell \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {K}^{2}, 2\right)} \]

       7. unpow2 N/A

          \[\leadsto \left(\ell \cdot J\right) \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{K \cdot K}, 2\right) \]

       8. lower-*.f64 30.5

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

   13. Applied rewrites 30.5%

       \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 54.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(\ell, 2 \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.25 \cdot \left(K \cdot K\right)\right) \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (- (exp l) (exp (- l))) 0.004)
   (fma l (* 2.0 J) U)
   (* (* -0.25 (* K K)) (* l J))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((exp(l) - exp(-l)) <= 0.004) {
		tmp = fma(l, (2.0 * J), U);
	} else {
		tmp = (-0.25 * (K * K)) * (l * J);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(exp(l) - exp(Float64(-l))) <= 0.004)
		tmp = fma(l, Float64(2.0 * J), U);
	else
		tmp = Float64(Float64(-0.25 * Float64(K * K)) * Float64(l * J));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], 0.004], N[(l * N[(2.0 * J), $MachinePrecision] + U), $MachinePrecision], N[(N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision] * N[(l * J), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0040000000000000001

   1. Initial program 81.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-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 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 65.0

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

   8. Applied rewrites 65.0%

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

   if 0.0040000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 21.1

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

   7. Applied rewrites 21.1%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{4}} + 2\right), J, U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{4}, 2\right)}, J, U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{4}, 2\right), J, U\right) \]

      7. lower-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in K around inf

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

       1. *-commutative N/A

          \[\leadsto \frac{-1}{4} \cdot \left(J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\left(J \cdot \ell\right) \cdot {K}^{2}\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot \ell\right)\right) \cdot {K}^{2}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(J \cdot \ell\right) \cdot \frac{-1}{4}\right)} \cdot {K}^{2} \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right)} \]

       7. *-commutative N/A

          \[\leadsto \color{blue}{\left(\ell \cdot J\right)} \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\ell \cdot J\right)} \cdot \left(\frac{-1}{4} \cdot {K}^{2}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\ell \cdot J\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)} \]

       10. unpow2 N/A

           \[\leadsto \left(\ell \cdot J\right) \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(K \cdot K\right)}\right) \]

       11. lower-*.f64 21.0

           \[\leadsto \left(\ell \cdot J\right) \cdot \left(-0.25 \cdot \color{blue}{\left(K \cdot K\right)}\right) \]

   13. Applied rewrites 21.0%

       \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(\ell, 2 \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.25 \cdot \left(K \cdot K\right)\right) \cdot \left(\ell \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    (fma K (* K -0.125) 1.0)
    (*
     l
     (*
      J
      (fma
       l
       (*
        l
        (fma
         (* l l)
         (fma l (* l 0.0003968253968253968) 0.016666666666666666)
         0.3333333333333333))
       2.0)))
    U)
   (fma (* 2.0 (sinh l)) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma(fma(K, (K * -0.125), 1.0), (l * (J * fma(l, (l * fma((l * l), fma(l, (l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333)), 2.0))), U);
	} else {
		tmp = fma((2.0 * sinh(l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(fma(K, Float64(K * -0.125), 1.0), Float64(l * Float64(J * fma(l, Float64(l * fma(Float64(l * l), fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333)), 2.0))), U);
	else
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(J * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 85.6%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + \ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + \ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + \ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2} + 1\right) \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + U \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

   8. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), U\right)} \]

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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

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

   7. Applied rewrites 95.6%

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

      1. lift-sinh.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.6

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

   9. Applied rewrites 95.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(J \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), J \cdot 0.3333333333333333\right), 2 \cdot J\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e-8)
   (fma (* 2.0 (sinh l)) J U)
   (+
    U
    (*
     (*
      l
      (fma
       (* l l)
       (fma
        (* J (* l l))
        (fma (* l l) 0.0003968253968253968 0.016666666666666666)
        (* J 0.3333333333333333))
       (* 2.0 J)))
     (cos (/ K 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e-8) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = U + ((l * fma((l * l), fma((J * (l * l)), fma((l * l), 0.0003968253968253968, 0.016666666666666666), (J * 0.3333333333333333)), (2.0 * J))) * cos((K / 2.0)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e-8)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = Float64(U + Float64(Float64(l * fma(Float64(l * l), fma(Float64(J * Float64(l * l)), fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(J * 0.3333333333333333)), Float64(2.0 * J))) * cos(Float64(K / 2.0))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e-8], N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(U + N[(N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] + N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(2.0 * J), $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)) < 1e-8

   1. Initial program 84.7%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 85.1%

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

      1. lift-sinh.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 85.1

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

   9. Applied rewrites 85.1%

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

   if 1e-8 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 89.4%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 96.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(J \cdot \left(\ell \cdot \ell\right), \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), J \cdot 0.3333333333333333\right), 2 \cdot J\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e-8)
   (fma (* 2.0 (sinh l)) J U)
   (+
    U
    (*
     (cos (/ K 2.0))
     (*
      J
      (*
       l
       (fma
        (* l l)
        (fma
         l
         (* l (fma (* l l) 0.0003968253968253968 0.016666666666666666))
         0.3333333333333333)
        2.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e-8) {
		tmp = fma((2.0 * sinh(l)), J, U);
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * fma((l * l), fma(l, (l * fma((l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e-8)
		tmp = fma(Float64(2.0 * sinh(l)), J, U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * fma(Float64(l * l), fma(l, Float64(l * fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666)), 0.3333333333333333), 2.0)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e-8], N[(N[(2.0 * N[Sinh[l], $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[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 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-8

   1. Initial program 84.7%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 85.1%

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

      1. lift-sinh.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 85.1

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

   9. Applied rewrites 85.1%

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

   if 1e-8 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 89.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sinh \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 83.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    (fma K (* K -0.125) 1.0)
    (*
     l
     (*
      J
      (fma
       l
       (*
        l
        (fma
         (* l l)
         (fma l (* l 0.0003968253968253968) 0.016666666666666666)
         0.3333333333333333))
       2.0)))
    U)
   (fma
    (*
     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 (cos((K / 2.0)) <= -0.02) {
		tmp = fma(fma(K, (K * -0.125), 1.0), (l * (J * fma(l, (l * fma((l * l), fma(l, (l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333)), 2.0))), U);
	} else {
		tmp = fma((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 (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(fma(K, Float64(K * -0.125), 1.0), Float64(l * Float64(J * fma(l, Float64(l * fma(Float64(l * l), fma(l, Float64(l * 0.0003968253968253968), 0.016666666666666666), 0.3333333333333333)), 2.0))), U);
	else
		tmp = fma(Float64(l * fma(Float64(l * l), fma(Float64(l * l), fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * N[(J * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 85.6%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + \ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right) + \ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + \ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2} + 1\right) \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)} + U \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot \left(\ell \cdot \left(2 \cdot J + {\ell}^{2} \cdot \left(\frac{1}{3} \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + U \]

   8. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), \ell \cdot \left(J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right)\right), U\right)} \]

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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

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

   7. Applied rewrites 95.6%

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

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 90.6

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

   10. Applied rewrites 90.6%

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

Alternative 15: 83.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    (fma K (* K -0.125) 1.0)
    (*
     J
     (*
      l
      (fma (* l l) (fma l (* l 0.016666666666666666) 0.3333333333333333) 2.0)))
    U)
   (fma
    (*
     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 (cos((K / 2.0)) <= -0.02) {
		tmp = fma(fma(K, (K * -0.125), 1.0), (J * (l * fma((l * l), fma(l, (l * 0.016666666666666666), 0.3333333333333333), 2.0))), U);
	} else {
		tmp = fma((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 (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(fma(K, Float64(K * -0.125), 1.0), Float64(J * Float64(l * fma(Float64(l * l), fma(l, Float64(l * 0.016666666666666666), 0.3333333333333333), 2.0))), U);
	else
		tmp = fma(Float64(l * fma(Float64(l * l), fma(Float64(l * l), fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(K * N[(K * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 85.6%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{U + \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + \ell \cdot \left(2 \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right) + \ell \cdot \left(2 \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) + U} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)} + \ell \cdot \left(2 \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2} + 1\right) \cdot \left(\ell \cdot \left(2 \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot \left(\ell \cdot \left(2 \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {K}^{2}, \ell \cdot \left(2 \cdot J + J \cdot \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), U\right)} \]

   8. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(K, K \cdot -0.125, 1\right), J \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right)\right), U\right)} \]

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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

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

   7. Applied rewrites 95.6%

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

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 90.6

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

   10. Applied rewrites 90.6%

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

Alternative 16: 81.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* l (* -0.25 (* K K))) J U)
   (fma
    (*
     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 (cos((K / 2.0)) <= -0.02) {
		tmp = fma((l * (-0.25 * (K * K))), J, U);
	} else {
		tmp = fma((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 (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(l * Float64(-0.25 * Float64(K * K))), J, U);
	else
		tmp = fma(Float64(l * fma(Float64(l * l), fma(Float64(l * l), fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), 0.3333333333333333), 2.0)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(l * N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 85.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{4}} + 2\right), J, U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{4}, 2\right)}, J, U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{4}, 2\right), J, U\right) \]

      7. lower-*.f64 60.8

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

   10. Applied rewrites 60.8%

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

   11. Taylor expanded in K around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}, J, U\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(K \cdot K\right)}\right), J, U\right) \]

       3. lower-*.f64 60.8

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

   13. Applied rewrites 60.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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

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

   7. Applied rewrites 95.6%

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

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 90.6

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

   10. Applied rewrites 90.6%

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

Alternative 17: 81.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot \mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* l (* -0.25 (* K K))) J U)
   (fma
    l
    (*
     J
     (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)) <= -0.02) {
		tmp = fma((l * (-0.25 * (K * K))), J, U);
	} else {
		tmp = fma(l, (J * 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)) <= -0.02)
		tmp = fma(Float64(l * Float64(-0.25 * Float64(K * K))), J, U);
	else
		tmp = fma(l, Float64(J * fma(l, Float64(l * fma(Float64(l * l), fma(l, Float64(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], -0.02], N[(N[(l * N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(l * N[(J * N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.0003968253968253968), $MachinePrecision] + 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))) < -0.0200000000000000004

   1. Initial program 85.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{4}} + 2\right), J, U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{4}, 2\right)}, J, U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{4}, 2\right), J, U\right) \]

      7. lower-*.f64 60.8

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

   10. Applied rewrites 60.8%

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

   11. Taylor expanded in K around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}, J, U\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(K \cdot K\right)}\right), J, U\right) \]

       3. lower-*.f64 60.8

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

   13. Applied rewrites 60.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 90.1%

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

Alternative 18: 80.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\ell, \ell \cdot 0.016666666666666666, 0.3333333333333333\right), 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* l (* -0.25 (* K K))) J 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)) <= -0.02) {
		tmp = fma((l * (-0.25 * (K * K))), J, 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)) <= -0.02)
		tmp = fma(Float64(l * Float64(-0.25 * Float64(K * K))), J, U);
	else
		tmp = fma(J, Float64(l * fma(Float64(l * l), fma(l, Float64(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], -0.02], N[(N[(l * N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(l * N[(l * 0.016666666666666666), $MachinePrecision] + 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))) < -0.0200000000000000004

   1. Initial program 85.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{4}} + 2\right), J, U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{4}, 2\right)}, J, U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{4}, 2\right), J, U\right) \]

      7. lower-*.f64 60.8

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

   10. Applied rewrites 60.8%

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

   11. Taylor expanded in K around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}, J, U\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(K \cdot K\right)}\right), J, U\right) \]

       3. lower-*.f64 60.8

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

   13. Applied rewrites 60.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 89.6%

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

Alternative 19: 80.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \left(\ell \cdot \ell\right) \cdot 0.016666666666666666, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* l (* -0.25 (* K K))) J 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)) <= -0.02) {
		tmp = fma((l * (-0.25 * (K * K))), J, 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)) <= -0.02)
		tmp = fma(Float64(l * Float64(-0.25 * Float64(K * K))), J, U);
	else
		tmp = fma(J, Float64(l * fma(Float64(l * 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], -0.02], N[(N[(l * N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 85.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{4}} + 2\right), J, U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{4}, 2\right)}, J, U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{4}, 2\right), J, U\right) \]

      7. lower-*.f64 60.8

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

   10. Applied rewrites 60.8%

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

   11. Taylor expanded in K around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}, J, U\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(K \cdot K\right)}\right), J, U\right) \]

       3. lower-*.f64 60.8

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

   13. Applied rewrites 60.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 89.6

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

   11. Applied rewrites 89.6%

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

Alternative 20: 76.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot 0.3333333333333333, 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* l (* -0.25 (* K K))) J 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)) <= -0.02) {
		tmp = fma((l * (-0.25 * (K * K))), J, 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)) <= -0.02)
		tmp = fma(Float64(l * Float64(-0.25 * Float64(K * K))), J, 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], -0.02], N[(N[(l * N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + 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))) < -0.0200000000000000004

   1. Initial program 85.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{4}} + 2\right), J, U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{4}, 2\right)}, J, U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{4}, 2\right), J, U\right) \]

      7. lower-*.f64 60.8

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

   10. Applied rewrites 60.8%

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

   11. Taylor expanded in K around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}, J, U\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(K \cdot K\right)}\right), J, U\right) \]

       3. lower-*.f64 60.8

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

   13. Applied rewrites 60.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 82.5

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

   11. Applied rewrites 82.5%

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

Alternative 21: 59.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(-0.25 \cdot \left(K \cdot K\right)\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, 2 \cdot J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* l (* -0.25 (* K K))) J U)
   (fma l (* 2.0 J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma((l * (-0.25 * (K * K))), J, U);
	} else {
		tmp = fma(l, (2.0 * J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(l * Float64(-0.25 * Float64(K * K))), J, U);
	else
		tmp = fma(l, Float64(2.0 * J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(l * N[(-0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(l * N[(2.0 * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 85.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. 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 100.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right) \cdot \ell} + 2 \cdot \ell, J, U\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\ell \cdot \left(\frac{-1}{4} \cdot {K}^{2} + 2\right)}, J, U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{4}} + 2\right), J, U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{4}, 2\right)}, J, U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{4}, 2\right), J, U\right) \]

      7. lower-*.f64 60.8

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

   10. Applied rewrites 60.8%

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

   11. Taylor expanded in K around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \color{blue}{\left(\frac{-1}{4} \cdot {K}^{2}\right)}, J, U\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\ell \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(K \cdot K\right)}\right), J, U\right) \]

       3. lower-*.f64 60.8

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

   13. Applied rewrites 60.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 56.9

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

   8. Applied rewrites 56.9%

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

Alternative 22: 54.3% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
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 61.6

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

5. Applied rewrites 61.6%

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

6. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 52.2

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

8. Applied rewrites 52.2%

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

Alternative 23: 19.7% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

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 * (2.0d0 * l)
end function

Java

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

Python

def code(J, l, K, U):
	return J * (2.0 * l)

Julia

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

MATLAB

function tmp = code(J, l, K, U)
	tmp = J * (2.0 * l);
end

Wolfram

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

Derivation

1. Initial program 85.8%

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

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
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 61.6

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

5. Applied rewrites 61.6%

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

6. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 52.2

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

8. Applied rewrites 52.2%

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

9. Taylor expanded in l around inf

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 18.5

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

11. Applied rewrites 18.5%

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

12. Final simplification 18.5%

    \[\leadsto J \cdot \left(2 \cdot \ell\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))