Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 87.1% → 99.9%

Time: 10.9s

Alternatives: 21

Speedup: 2.1×

• 48.8% 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: 87.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

Alternative 2: 58.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \leq -5 \cdot 10^{+289}:\\ \;\;\;\;\left(\ell \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, -0.25, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot J, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) -5e+289)
   (* (* l J) (fma (* K K) -0.25 2.0))
   (fma (* 2.0 J) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (((J * (exp(l) - exp(-l))) * cos((K / 2.0))) <= -5e+289) {
		tmp = (l * J) * fma((K * K), -0.25, 2.0);
	} else {
		tmp = fma((2.0 * J), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) <= -5e+289)
		tmp = Float64(Float64(l * J) * fma(Float64(K * K), -0.25, 2.0));
	else
		tmp = fma(Float64(2.0 * J), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e+289], N[(N[(l * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * J), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -5.00000000000000031e289

   1. Initial program 99.0%

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 28.9

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

   5. Applied rewrites 28.9%

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

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{\left(2 \cdot \left(J \cdot \ell\right) + {K}^{2} \cdot \left(\frac{-1}{4} \cdot \left(J \cdot \ell\right) + \frac{1}{192} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 10.7%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, 0.005208333333333333, -0.25\right), \color{blue}{K \cdot K}, \mathsf{fma}\left(2 \cdot J, \ell, U\right)\right) \]

   9. Taylor expanded in J around inf

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

   11. Applied rewrites 29.2%

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

   12. Taylor expanded in K around 0

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

   14. Applied rewrites 43.4%

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

   if -5.00000000000000031e289 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 79.7%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 67.0%

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

Alternative 3: 87.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    (* (* (fma -0.125 (* K K) 1.0) (fma 0.3333333333333333 (* l l) 2.0)) J)
    l
    U)
   (fma (* (sinh 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(((fma(-0.125, (K * K), 1.0) * fma(0.3333333333333333, (l * l), 2.0)) * J), l, U);
	} else {
		tmp = fma((sinh(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(Float64(fma(-0.125, Float64(K * K), 1.0) * fma(0.3333333333333333, Float64(l * l), 2.0)) * J), l, U);
	else
		tmp = fma(Float64(sinh(l) * 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[(N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $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 83.1%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 52.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right), \ell, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 55.5%

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

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

   1. Initial program 84.4%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 84.3

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

   5. Applied rewrites 84.3%

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

   7. Applied rewrites 94.8%

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

Alternative 4: 89.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-46)
   (fma (* (sinh l) 2.0) J U)
   (+
    (*
     (*
      J
      (*
       (fma
        (fma
         (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
         (* l l)
         0.3333333333333333)
        (* l l)
        2.0)
       l))
     (cos (/ K 2.0)))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-46) {
		tmp = fma((sinh(l) * 2.0), J, U);
	} else {
		tmp = ((J * (fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-46)
		tmp = fma(Float64(sinh(l) * 2.0), J, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-46], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 4.99999999999999992e-46

   1. Initial program 86.0%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 77.6

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 87.7%

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

   if 4.99999999999999992e-46 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 79.4%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 93.6

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

   5. Applied rewrites 93.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-46)
   (fma (* (sinh l) 2.0) J U)
   (+
    (*
     (*
      J
      (*
       (fma (fma 0.016666666666666666 (* l l) 0.3333333333333333) (* l l) 2.0)
       l))
     (cos (/ K 2.0)))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-46) {
		tmp = fma((sinh(l) * 2.0), J, U);
	} else {
		tmp = ((J * (fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-46)
		tmp = fma(Float64(sinh(l) * 2.0), J, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-46], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 4.99999999999999992e-46

   1. Initial program 86.0%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 77.6

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 87.7%

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

   if 4.99999999999999992e-46 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 79.4%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.0% accurate, 2.0× speedup

Math

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

FPCore

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.1%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 52.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right), \ell, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 55.5%

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

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

   1. Initial program 84.4%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 84.3

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

   5. Applied rewrites 84.3%

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 89.2%

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

Alternative 7: 87.7% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5.0)
   (fma (* (sinh l) 2.0) J U)
   (+ (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) (cos (/ K 2.0))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5.0) {
		tmp = fma((sinh(l) * 2.0), J, U);
	} else {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5.0)
		tmp = fma(Float64(sinh(l) * 2.0), J, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5.0], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 5

   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 K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 87.9%

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

   if 5 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 78.5%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.5

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

   5. Applied rewrites 86.5%

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

4. Final simplification 87.5%

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

Alternative 8: 81.5% accurate, 2.1× speedup

Math

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

FPCore

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.1%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 52.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right), \ell, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 55.5%

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

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

   1. Initial program 84.4%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 84.3

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

   5. Applied rewrites 84.3%

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 88.7%

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

Alternative 9: 82.5% accurate, 2.1× speedup

Math

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

FPCore

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.1%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 52.8%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 53.8%

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

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

   1. Initial program 84.4%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 84.3

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

   5. Applied rewrites 84.3%

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 88.7%

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

Alternative 10: 78.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.22:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(2 \cdot J\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.22)
   (fma (* (fma (* K K) -0.125 1.0) (* 2.0 J)) l U)
   (fma
    (*
     (fma (fma 0.016666666666666666 (* l l) 0.3333333333333333) (* l l) 2.0)
     l)
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.22) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (2.0 * J)), l, U);
	} else {
		tmp = fma((fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.22)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(2.0 * J)), l, U);
	else
		tmp = fma(Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.22], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(2.0 * J), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.220000000000000001

   1. Initial program 82.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 52.9%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 49.7%

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

   if -0.220000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.5%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 92.5%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 86.6%

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

Alternative 11: 86.9% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5.0)
   (fma (* (sinh l) 2.0) J U)
   (fma (* (cos (* 0.5 K)) (* J (fma (* l l) 0.3333333333333333 2.0))) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5.0) {
		tmp = fma((sinh(l) * 2.0), J, U);
	} else {
		tmp = fma((cos((0.5 * K)) * (J * fma((l * l), 0.3333333333333333, 2.0))), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5.0)
		tmp = fma(Float64(sinh(l) * 2.0), J, U);
	else
		tmp = fma(Float64(cos(Float64(0.5 * K)) * Float64(J * fma(Float64(l * l), 0.3333333333333333, 2.0))), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5.0], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 5

   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 K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 87.9%

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

   if 5 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 78.5%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

Alternative 12: 75.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.22:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(2 \cdot J\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.22)
   (fma (* (fma (* K K) -0.125 1.0) (* 2.0 J)) l U)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.22) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (2.0 * J)), l, U);
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.22)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(2.0 * J)), l, U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.22], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(2.0 * J), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.220000000000000001

   1. Initial program 82.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 52.9%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 49.7%

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

   if -0.220000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.5%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 83.0%

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

Alternative 13: 86.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot J\right), \ell, U\right)\\ \mathbf{elif}\;\ell \leq -460 \lor \neg \left(\ell \leq 4.8 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \cos \left(-0.5 \cdot K\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.3e+96)
   (fma (* (fma (* K K) -0.125 1.0) (* (* (* l l) 0.3333333333333333) J)) l U)
   (if (or (<= l -460.0) (not (<= l 4.8e-11)))
     (fma (* (sinh l) 2.0) J U)
     (fma (* (* 2.0 l) (cos (* -0.5 K))) J U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.3e+96) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (((l * l) * 0.3333333333333333) * J)), l, U);
	} else if ((l <= -460.0) || !(l <= 4.8e-11)) {
		tmp = fma((sinh(l) * 2.0), J, U);
	} else {
		tmp = fma(((2.0 * l) * cos((-0.5 * K))), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.3e+96)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(Float64(Float64(l * l) * 0.3333333333333333) * J)), l, U);
	elseif ((l <= -460.0) || !(l <= 4.8e-11))
		tmp = fma(Float64(sinh(l) * 2.0), J, U);
	else
		tmp = fma(Float64(Float64(2.0 * l) * cos(Float64(-0.5 * K))), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.3e+96], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], If[Or[LessEqual[l, -460.0], N[Not[LessEqual[l, 4.8e-11]], $MachinePrecision]], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.29999999999999984e96

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 88.2%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 88.2%

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

   if -3.29999999999999984e96 < l < -460 or 4.8000000000000002e-11 < l

   1. Initial program 99.3%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 77.2

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 77.9%

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

   if -460 < l < 4.8000000000000002e-11

   1. Initial program 70.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot J\right), \ell, U\right)\\ \mathbf{elif}\;\ell \leq -460 \lor \neg \left(\ell \leq 4.8 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \cos \left(-0.5 \cdot K\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 86.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot J\right), \ell, U\right)\\ \mathbf{elif}\;\ell \leq -460 \lor \neg \left(\ell \leq 4.8 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.3e+96)
   (fma (* (fma (* K K) -0.125 1.0) (* (* (* l l) 0.3333333333333333) J)) l U)
   (if (or (<= l -460.0) (not (<= l 4.8e-11)))
     (fma (* (sinh l) 2.0) J U)
     (fma (* (* 2.0 l) J) (cos (* 0.5 K)) U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.3e+96) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (((l * l) * 0.3333333333333333) * J)), l, U);
	} else if ((l <= -460.0) || !(l <= 4.8e-11)) {
		tmp = fma((sinh(l) * 2.0), J, U);
	} else {
		tmp = fma(((2.0 * l) * J), cos((0.5 * K)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.3e+96)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(Float64(Float64(l * l) * 0.3333333333333333) * J)), l, U);
	elseif ((l <= -460.0) || !(l <= 4.8e-11))
		tmp = fma(Float64(sinh(l) * 2.0), J, U);
	else
		tmp = fma(Float64(Float64(2.0 * l) * J), cos(Float64(0.5 * K)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.3e+96], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], If[Or[LessEqual[l, -460.0], N[Not[LessEqual[l, 4.8e-11]], $MachinePrecision]], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.29999999999999984e96

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 88.2%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 88.2%

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

   if -3.29999999999999984e96 < l < -460 or 4.8000000000000002e-11 < l

   1. Initial program 99.3%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 77.2

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 77.9%

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

   if -460 < l < 4.8000000000000002e-11

   1. Initial program 70.6%

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

   3. Taylor expanded in l around 0

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot J\right), \ell, U\right)\\ \mathbf{elif}\;\ell \leq -460 \lor \neg \left(\ell \leq 4.8 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 86.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot J\right), \ell, U\right)\\ \mathbf{elif}\;\ell \leq -460 \lor \neg \left(\ell \leq 4.8 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot 2\right), \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.3e+96)
   (fma (* (fma (* K K) -0.125 1.0) (* (* (* l l) 0.3333333333333333) J)) l U)
   (if (or (<= l -460.0) (not (<= l 4.8e-11)))
     (fma (* (sinh l) 2.0) J U)
     (fma (* (cos (* 0.5 K)) (* J 2.0)) l U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.3e+96) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (((l * l) * 0.3333333333333333) * J)), l, U);
	} else if ((l <= -460.0) || !(l <= 4.8e-11)) {
		tmp = fma((sinh(l) * 2.0), J, U);
	} else {
		tmp = fma((cos((0.5 * K)) * (J * 2.0)), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.3e+96)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(Float64(Float64(l * l) * 0.3333333333333333) * J)), l, U);
	elseif ((l <= -460.0) || !(l <= 4.8e-11))
		tmp = fma(Float64(sinh(l) * 2.0), J, U);
	else
		tmp = fma(Float64(cos(Float64(0.5 * K)) * Float64(J * 2.0)), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.3e+96], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], If[Or[LessEqual[l, -460.0], N[Not[LessEqual[l, 4.8e-11]], $MachinePrecision]], N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.29999999999999984e96

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 88.2%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 88.2%

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

   if -3.29999999999999984e96 < l < -460 or 4.8000000000000002e-11 < l

   1. Initial program 99.3%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 77.2

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 77.9%

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

   if -460 < l < 4.8000000000000002e-11

   1. Initial program 70.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 99.8%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right) \cdot J\right), \ell, U\right)\\ \mathbf{elif}\;\ell \leq -460 \lor \neg \left(\ell \leq 4.8 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot 2\right), \ell, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.22:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.22)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.22) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.22)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.22], N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.220000000000000001

   1. Initial program 82.6%

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

   3. Taylor expanded in l around 0

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 49.4%

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

   if -0.220000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.5%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 83.0%

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

Alternative 17: 73.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.22:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \mathsf{fma}\left(0.3333333333333333 \cdot \ell, \ell, 2\right), \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.22)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (fma (* J (fma (* 0.3333333333333333 l) l 2.0)) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.22) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma((J * fma((0.3333333333333333 * l), l, 2.0)), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.22)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = fma(Float64(J * fma(Float64(0.3333333333333333 * l), l, 2.0)), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.22], N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(J * N[(N[(0.3333333333333333 * l), $MachinePrecision] * l + 2.0), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.220000000000000001

   1. Initial program 82.6%

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

   3. Taylor expanded in l around 0

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 49.4%

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

   if -0.220000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.5%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 62.9%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 80.6%

       \[\leadsto \mathsf{fma}\left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right), \color{blue}{\ell}, U\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 80.6%

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

Alternative 18: 86.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sinh \ell \cdot 2\\ \mathbf{if}\;\ell \leq -1040000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot t\_0, J, U\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \cos \left(-0.5 \cdot K\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (sinh l) 2.0)))
   (if (<= l -1040000000000.0)
     (fma (* (fma (* K K) -0.125 1.0) t_0) J U)
     (if (<= l 4.8e-11)
       (fma (* (* 2.0 l) (cos (* -0.5 K))) J U)
       (fma t_0 J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = sinh(l) * 2.0;
	double tmp;
	if (l <= -1040000000000.0) {
		tmp = fma((fma((K * K), -0.125, 1.0) * t_0), J, U);
	} else if (l <= 4.8e-11) {
		tmp = fma(((2.0 * l) * cos((-0.5 * K))), J, U);
	} else {
		tmp = fma(t_0, J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(sinh(l) * 2.0)
	tmp = 0.0
	if (l <= -1040000000000.0)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * t_0), J, U);
	elseif (l <= 4.8e-11)
		tmp = fma(Float64(Float64(2.0 * l) * cos(Float64(-0.5 * K))), J, U);
	else
		tmp = fma(t_0, J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[l, -1040000000000.0], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 4.8e-11], N[(N[(N[(2.0 * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(t$95$0 * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.04e12

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 86.5

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

   7. Applied rewrites 86.5%

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

   if -1.04e12 < l < 4.8000000000000002e-11

   1. Initial program 70.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 99.2

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

   7. Applied rewrites 99.2%

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

   if 4.8000000000000002e-11 < l

   1. Initial program 99.1%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 81.8%

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

Alternative 19: 72.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+42} \lor \neg \left(\ell \leq 7 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot J, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5.2e+42) (not (<= l 7e-5)))
   (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)
   (fma (* 2.0 J) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.2e+42) || !(l <= 7e-5)) {
		tmp = (fma((l * l), 0.3333333333333333, 2.0) * l) * J;
	} else {
		tmp = fma((2.0 * J), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5.2e+42) || !(l <= 7e-5))
		tmp = Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J);
	else
		tmp = fma(Float64(2.0 * J), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5.2e+42], N[Not[LessEqual[l, 7e-5]], $MachinePrecision]], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision], N[(N[(2.0 * J), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.1999999999999998e42 or 6.9999999999999994e-5 < l

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 24.0%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 55.5%

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

   12. Taylor expanded in J around inf

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

   14. Applied rewrites 59.7%

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

   if -5.1999999999999998e42 < l < 6.9999999999999994e-5

   1. Initial program 72.5%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 80.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+42} \lor \neg \left(\ell \leq 7 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot J, \ell, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 55.1% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 71.3

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

5. Applied rewrites 71.3%

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

6. Taylor expanded in l around 0

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

8. Applied rewrites 56.6%

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

Alternative 21: 19.7% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

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 = (2.0d0 * l) * j
end function

Java

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

Python

def code(J, l, K, U):
	return (2.0 * l) * J

Julia

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

MATLAB

function tmp = code(J, l, K, U)
	tmp = (2.0 * l) * J;
end

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 71.3

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

5. Applied rewrites 71.3%

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

6. Taylor expanded in l around 0

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

8. Applied rewrites 56.6%

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

9. Taylor expanded in J around inf

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

11. Applied rewrites 19.8%

    \[\leadsto \left(\ell \cdot J\right) \cdot 2 \]
12. Step-by-step derivation

13. Applied rewrites 19.8%

    \[\leadsto \left(2 \cdot \ell\right) \cdot J \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024315 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))