Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 100.0%

Time: 7.4s

Alternatives: 20

Speedup: 1.5×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. lift-+.f64 N/A

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

   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. lower-fma.f64 87.3

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 56.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * J + 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)))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      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 30.3

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

   5. Applied rewrites 30.3%

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

   6. Taylor expanded in J around inf

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

   8. Applied rewrites 30.3%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 26.6%

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

   if -inf.0 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 82.9%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-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. 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 66.4%

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

4. Final simplification 56.1%

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

Alternative 3: 96.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.97:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.97)
     (+
      (*
       (*
        (*
         (fma
          (fma 0.016666666666666666 (* l l) 0.3333333333333333)
          (* l l)
          2.0)
         l)
        J)
       t_0)
      U)
     (fma (* J (sinh l)) 2.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.97) {
		tmp = (((fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l) * J) * t_0) + U;
	} else {
		tmp = fma((J * sinh(l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.97)
		tmp = Float64(Float64(Float64(Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l) * J) * t_0) + U);
	else
		tmp = fma(Float64(J * sinh(l)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.97], N[(N[(N[(N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.96999999999999997

   1. Initial program 82.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

   3. Taylor expanded in 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 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.97:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.97)
     (+ (* (* (* (fma (* l l) 0.3333333333333333 2.0) l) J) t_0) U)
     (fma (* J (sinh l)) 2.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.97) {
		tmp = (((fma((l * l), 0.3333333333333333, 2.0) * l) * J) * t_0) + U;
	} else {
		tmp = fma((J * sinh(l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.97)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J) * t_0) + U);
	else
		tmp = fma(Float64(J * sinh(l)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.97], N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.96999999999999997

   1. Initial program 82.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

   3. Taylor expanded in 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 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.97:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.97)
     (+ (* (* (* (fma (* l l) 0.3333333333333333 2.0) J) l) t_0) U)
     (fma (* J (sinh l)) 2.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.97) {
		tmp = (((fma((l * l), 0.3333333333333333, 2.0) * J) * l) * t_0) + U;
	} else {
		tmp = fma((J * sinh(l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.97)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J) * l) * t_0) + U);
	else
		tmp = fma(Float64(J * sinh(l)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.97], N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.96999999999999997

   1. Initial program 82.3%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 86.5

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

   5. Applied rewrites 86.5%

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

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

   3. Taylor expanded in 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 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.97)
   (fma (* (cos (* 0.5 K)) (* (fma (* l l) 0.3333333333333333 2.0) J)) l U)
   (fma (* J (sinh l)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.97) {
		tmp = fma((cos((0.5 * K)) * (fma((l * l), 0.3333333333333333, 2.0) * J)), l, U);
	} else {
		tmp = fma((J * sinh(l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.97)
		tmp = fma(Float64(cos(Float64(0.5 * K)) * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J)), l, U);
	else
		tmp = fma(Float64(J * sinh(l)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.97], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.96999999999999997

   1. Initial program 82.3%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 86.5%

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

   if 0.96999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.3%

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

   3. Taylor expanded in 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 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(2.0 * l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.7%

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

   3. Taylor expanded in 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 70.2

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

   5. Applied rewrites 70.2%

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

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

   1. Initial program 88.9%

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

   3. Taylor expanded in 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 88.9

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 95.7%

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

Alternative 8: 87.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (fma
    (*
     (*
      (fma
       (fma
        (fma (* l l) 0.0003968253968253968 0.016666666666666666)
        (* l l)
        0.3333333333333333)
       (* l l)
       2.0)
      l)
     (fma (* K K) -0.125 1.0))
    J
    U)
   (fma (* J (sinh l)) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma(((fma(fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l) * fma((K * K), -0.125, 1.0)), J, U);
	} else {
		tmp = fma((J * sinh(l)), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l) * fma(Float64(K * K), -0.125, 1.0)), J, U);
	else
		tmp = fma(Float64(J * sinh(l)), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.7%

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

   3. Taylor expanded in 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 91.7

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

   5. Applied rewrites 91.7%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 91.6%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 55.1

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

   10. Applied rewrites 55.1%

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

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

   1. Initial program 88.9%

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

   3. Taylor expanded in 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 88.9

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 95.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \sinh \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 100:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \sinh \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 100.0)
   (fma (* (* J (sinh l)) 2.0) (fma (* K K) -0.125 1.0) U)
   (+
    (*
     (cos (/ K 2.0))
     (*
      (*
       (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 ((K / 2.0) <= 100.0) {
		tmp = fma(((J * sinh(l)) * 2.0), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = (cos((K / 2.0)) * ((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 (Float64(K / 2.0) <= 100.0)
		tmp = fma(Float64(Float64(J * sinh(l)) * 2.0), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = Float64(Float64(cos(Float64(K / 2.0)) * Float64(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[(K / 2.0), $MachinePrecision], 100.0], N[(N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $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), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 100

   1. Initial program 87.1%

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

      1. lift-+.f64 N/A

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

      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. lower-fma.f64 87.1

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

   if 100 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 87.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 100:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \sinh \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J\right) + U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot t\_0, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma
            (fma (* l l) 0.0003968253968253968 0.016666666666666666)
            (* l l)
            0.3333333333333333)
           (* l l)
           2.0)
          l)))
   (if (<= (cos (/ K 2.0)) -0.01)
     (fma (* t_0 (fma (* K K) -0.125 1.0)) J U)
     (fma (* 1.0 t_0) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma(fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l;
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = fma((t_0 * fma((K * K), -0.125, 1.0)), J, U);
	} else {
		tmp = fma((1.0 * t_0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = fma(Float64(t_0 * fma(Float64(K * K), -0.125, 1.0)), J, U);
	else
		tmp = fma(Float64(1.0 * t_0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(N[(t$95$0 * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(1.0 * t$95$0), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.7%

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

   3. Taylor expanded in 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 91.7

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

   5. Applied rewrites 91.7%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 91.6%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 55.1

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

   10. Applied rewrites 55.1%

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

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

   1. Initial program 88.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 96.5

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

   5. Applied rewrites 96.5%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 92.3%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right), J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 100:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \sinh \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \cos \left(-0.5 \cdot K\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 100.0)
   (fma (* (* J (sinh l)) 2.0) (fma (* K K) -0.125 1.0) U)
   (fma
    (*
     (*
      (fma
       (fma
        (fma (* l l) 0.0003968253968253968 0.016666666666666666)
        (* l l)
        0.3333333333333333)
       (* l l)
       2.0)
      l)
     (cos (* -0.5 K)))
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 100.0) {
		tmp = fma(((J * sinh(l)) * 2.0), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(((fma(fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l) * cos((-0.5 * K))), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 100.0)
		tmp = fma(Float64(Float64(J * sinh(l)) * 2.0), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l) * cos(Float64(-0.5 * K))), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 100.0], N[(N[(N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 100

   1. Initial program 87.1%

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

      1. lift-+.f64 N/A

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

      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. lower-fma.f64 87.1

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

   if 100 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 87.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 98.3%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 100:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot \sinh \ell\right) \cdot 2, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \cos \left(-0.5 \cdot K\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.2)
   (fma
    (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)
    (fma (* K K) -0.125 1.0)
    U)
   (fma
    (*
     1.0
     (*
      (fma
       (fma
        (fma (* l l) 0.0003968253968253968 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)) <= -0.2) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * l) * J), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((1.0 * (fma(fma(fma((l * l), 0.0003968253968253968, 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)) <= -0.2)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(1.0 * Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 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], -0.2], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(1.0 * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.20000000000000001

   1. Initial program 81.5%

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

      1. lift-+.f64 N/A

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

      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. lower-fma.f64 81.5

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.9

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

   7. Applied rewrites 60.9%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 55.4

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

   10. Applied rewrites 55.4%

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

   if -0.20000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.8%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 96.1

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

   5. Applied rewrites 96.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 96.2%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 90.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right), J, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 79.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right) \cdot J, \ell \cdot \ell, J\right) \cdot \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.2)
   (fma
    (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)
    (fma (* K K) -0.125 1.0)
    U)
   (fma
    (*
     (fma
      (* (fma 0.008333333333333333 (* l l) 0.16666666666666666) J)
      (* l l)
      J)
     l)
    2.0
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.2) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * l) * J), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((fma((fma(0.008333333333333333, (l * l), 0.16666666666666666) * J), (l * l), J) * l), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.2)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(fma(Float64(fma(0.008333333333333333, Float64(l * l), 0.16666666666666666) * J), Float64(l * l), J) * l), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.2], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * J), $MachinePrecision] * N[(l * l), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.20000000000000001

   1. Initial program 81.5%

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

      1. lift-+.f64 N/A

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

      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. lower-fma.f64 81.5

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.9

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

   7. Applied rewrites 60.9%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 55.4

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

   10. Applied rewrites 55.4%

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

   if -0.20000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 87.3

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

   5. Applied rewrites 87.3%

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

   7. Applied rewrites 93.9%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 86.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, J\right) \cdot \ell, 2, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right) \cdot J, \ell \cdot \ell, J\right) \cdot \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right) \cdot J, \ell \cdot \ell, J\right) \cdot \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.3)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (fma
    (*
     (fma
      (* (fma 0.008333333333333333 (* l l) 0.16666666666666666) J)
      (* l l)
      J)
     l)
    2.0
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.3) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma((fma((fma(0.008333333333333333, (l * l), 0.16666666666666666) * J), (l * l), J) * l), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.3)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = fma(Float64(fma(Float64(fma(0.008333333333333333, Float64(l * l), 0.16666666666666666) * J), Float64(l * l), J) * l), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.3], N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * J), $MachinePrecision] * N[(l * l), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.299999999999999989

   1. Initial program 80.7%

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

   3. Taylor expanded in 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 73.1

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

   5. Applied rewrites 73.1%

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

   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 51.9%

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

   if -0.299999999999999989 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.9%

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

   3. Taylor expanded in 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 86.9

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

   5. Applied rewrites 86.9%

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 86.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, J\right) \cdot \ell, 2, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right) \cdot J, \ell \cdot \ell, J\right) \cdot \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right), \frac{J \cdot \ell}{U}, 1\right) \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.3)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (* (fma (fma (* l l) 0.3333333333333333 2.0) (/ (* J l) U) 1.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.3) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma(fma((l * l), 0.3333333333333333, 2.0), ((J * l) / U), 1.0) * U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.3)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = Float64(fma(fma(Float64(l * l), 0.3333333333333333, 2.0), Float64(Float64(J * l) / U), 1.0) * U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.3], 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] * N[(N[(J * l), $MachinePrecision] / U), $MachinePrecision] + 1.0), $MachinePrecision] * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.299999999999999989

   1. Initial program 80.7%

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

   3. Taylor expanded in 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 73.1

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

   5. Applied rewrites 73.1%

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

   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 51.9%

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

   if -0.299999999999999989 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.9%

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

   3. Taylor expanded in 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 86.9

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

   5. Applied rewrites 86.9%

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

   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 80.8%

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

   9. Taylor expanded in U around inf

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

   11. Applied rewrites 83.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right), \frac{\ell \cdot J}{U}, 1\right) \cdot U \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right), \frac{J \cdot \ell}{U}, 1\right) \cdot U\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right), \frac{\ell}{U} \cdot J, 1\right) \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.3)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (* (fma (fma (* l l) 0.3333333333333333 2.0) (* (/ l U) J) 1.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.3) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma(fma((l * l), 0.3333333333333333, 2.0), ((l / U) * J), 1.0) * U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.3)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = Float64(fma(fma(Float64(l * l), 0.3333333333333333, 2.0), Float64(Float64(l / U) * J), 1.0) * U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.3], 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] * N[(N[(l / U), $MachinePrecision] * J), $MachinePrecision] + 1.0), $MachinePrecision] * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.299999999999999989

   1. Initial program 80.7%

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

   3. Taylor expanded in 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 73.1

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

   5. Applied rewrites 73.1%

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

   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 51.9%

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

   if -0.299999999999999989 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.9%

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

   3. Taylor expanded in 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 86.9

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

   5. Applied rewrites 86.9%

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

   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 80.8%

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

   9. Taylor expanded in U around inf

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

   11. Applied rewrites 83.7%

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

   13. Applied rewrites 83.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right), J \cdot \frac{\ell}{U}, 1\right) \cdot U \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right), \frac{\ell}{U} \cdot J, 1\right) \cdot U\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 73.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.3)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) J) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.3) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * J), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.3)
		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) * J), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.3], 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] * J), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.299999999999999989

   1. Initial program 80.7%

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

   3. Taylor expanded in 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 73.1

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

   5. Applied rewrites 73.1%

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

   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 51.9%

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

   if -0.299999999999999989 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.9%

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

   3. Taylor expanded in 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 86.9

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

   5. Applied rewrites 86.9%

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

   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 80.8%

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

Alternative 18: 72.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.0135:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)))
   (if (<= l -1.7e+39) t_0 (if (<= l 0.0135) (fma (* 2.0 l) J U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (fma((l * l), 0.3333333333333333, 2.0) * l) * J;
	double tmp;
	if (l <= -1.7e+39) {
		tmp = t_0;
	} else if (l <= 0.0135) {
		tmp = fma((2.0 * l), J, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J)
	tmp = 0.0
	if (l <= -1.7e+39)
		tmp = t_0;
	elseif (l <= 0.0135)
		tmp = fma(Float64(2.0 * l), J, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -1.7e+39], t$95$0, If[LessEqual[l, 0.0135], N[(N[(2.0 * l), $MachinePrecision] * J + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.6999999999999999e39 or 0.0134999999999999998 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-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 81.4

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

   5. Applied rewrites 81.4%

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

   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 62.8%

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

   9. Taylor expanded in J around inf

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

   11. Applied rewrites 68.0%

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

   if -1.6999999999999999e39 < l < 0.0134999999999999998

   1. Initial program 75.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 72.4

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

   5. Applied rewrites 72.4%

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

   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 80.3%

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

Alternative 19: 54.6% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
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 76.8

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

5. Applied rewrites 76.8%

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

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 53.8%

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

Alternative 20: 19.9% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = (2.0d0 * j) * l
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
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 76.8

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

5. Applied rewrites 76.8%

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

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 53.8%

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

9. Taylor expanded in l around inf

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

11. Applied rewrites 43.2%

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

12. Taylor expanded in J around inf

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

14. Applied rewrites 18.8%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))