Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.9%

Time: 8.5s

Alternatives: 15

Speedup: 2.1×

• 50.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return ((cos((K * -0.5)) * (sinh(l) * 2.0)) * J) + 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 = ((cos((k * (-0.5d0))) * (sinh(l) * 2.0d0)) * j) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((Math.cos((K * -0.5)) * (Math.sinh(l) * 2.0)) * J) + U;
}

Python

def code(J, l, K, U):
	return ((math.cos((K * -0.5)) * (math.sinh(l) * 2.0)) * J) + U

Julia

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

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((cos((K * -0.5)) * (sinh(l) * 2.0)) * J) + U;
end

Wolfram

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

Derivation

1. Initial program 89.4%

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

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 100.0%

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

Alternative 2: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+162} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+148}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\sinh \ell \cdot 2\right)\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \ell\right) \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0)))))
   (if (or (<= t_0 -1e+162) (not (<= t_0 2e+148)))
     (+ (* (* (fma (* K K) -0.125 1.0) (* (sinh l) 2.0)) J) U)
     (+ (* (* (* 2.0 l) (cos (* -0.5 K))) J) U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double tmp;
	if ((t_0 <= -1e+162) || !(t_0 <= 2e+148)) {
		tmp = ((fma((K * K), -0.125, 1.0) * (sinh(l) * 2.0)) * J) + U;
	} else {
		tmp = (((2.0 * l) * cos((-0.5 * K))) * J) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0)))
	tmp = 0.0
	if ((t_0 <= -1e+162) || !(t_0 <= 2e+148))
		tmp = Float64(Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(sinh(l) * 2.0)) * J) + U);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * l) * cos(Float64(-0.5 * K))) * J) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+162], N[Not[LessEqual[t$95$0, 2e+148]], $MachinePrecision]], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(2.0 * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + 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)))) < -9.9999999999999994e161 or 2.0000000000000001e148 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 99.5%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 78.8

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

   7. Applied rewrites 78.8%

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

   if -9.9999999999999994e161 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 2.0000000000000001e148

   1. Initial program 75.9%

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

      1. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \leq -1 \cdot 10^{+162} \lor \neg \left(\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \leq 2 \cdot 10^{+148}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(\sinh \ell \cdot 2\right)\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \ell\right) \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \ell + U\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0)))
     (+ (* (* (* (* l l) J) 0.3333333333333333) l) U)
     (+ (* (* 2.0 l) J) U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.0)) {
		tmp = ((((l * l) * J) * 0.3333333333333333) * l) + U;
	} else {
		tmp = ((2.0 * l) * J) + U;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.0)) {
		tmp = ((((l * l) * J) * 0.3333333333333333) * l) + U;
	} else {
		tmp = ((2.0 * l) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 0.0):
		tmp = ((((l * l) * J) * 0.3333333333333333) * l) + U
	else:
		tmp = ((2.0 * l) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) * J) * 0.3333333333333333) * l) + U);
	else
		tmp = Float64(Float64(Float64(2.0 * l) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 0.0)))
		tmp = ((((l * l) * J) * 0.3333333333333333) * l) + U;
	else
		tmp = ((2.0 * l) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] * J), $MachinePrecision] + 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 or 0.0 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 68.7

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

   5. Applied rewrites 68.7%

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

   7. Applied rewrites 68.7%

      \[\leadsto \left(\sinh \ell \cdot 2\right) \cdot J + U \]

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 49.5%

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

   11. Taylor expanded in l around inf

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

   13. Applied rewrites 49.5%

       \[\leadsto \left(\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \ell + U \]

   if -inf.0 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 0.0

   1. Initial program 75.5%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 75.5

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

   5. Applied rewrites 75.5%

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

   7. Applied rewrites 83.2%

      \[\leadsto \left(\sinh \ell \cdot 2\right) \cdot J + U \]

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 82.7%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \leq -\infty \lor \neg \left(\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \leq 0\right):\\ \;\;\;\;\left(\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \ell + U\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.2 \lor \neg \left(t\_0 \leq 0.996\right):\\ \;\;\;\;\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \mathsf{fma}\left(0.3333333333333333 \cdot \ell, \ell, 2\right)\right) \cdot \ell + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (or (<= t_0 -0.2) (not (<= t_0 0.996)))
     (+
      (*
       (* (fma (* K K) -0.125 1.0) J)
       (*
        (fma
         (fma
          (fma (* l l) 0.0003968253968253968 0.016666666666666666)
          (* l l)
          0.3333333333333333)
         (* l l)
         2.0)
        l))
      U)
     (+ (* (* J (fma (* 0.3333333333333333 l) l 2.0)) l) U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 0.996)) {
		tmp = ((fma((K * K), -0.125, 1.0) * J) * (fma(fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) + U;
	} else {
		tmp = ((J * fma((0.3333333333333333 * l), l, 2.0)) * l) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if ((t_0 <= -0.2) || !(t_0 <= 0.996))
		tmp = Float64(Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * J) * Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) + U);
	else
		tmp = Float64(Float64(Float64(J * fma(Float64(0.3333333333333333 * l), l, 2.0)) * l) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.2], N[Not[LessEqual[t$95$0, 0.996]], $MachinePrecision]], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * J), $MachinePrecision] * 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] + U), $MachinePrecision], N[(N[(N[(J * N[(N[(0.3333333333333333 * l), $MachinePrecision] * l + 2.0), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.20000000000000001 or 0.996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 92.3%

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

   3. 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 95.9

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

   5. Applied rewrites 95.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

   7. Applied rewrites 96.0%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.9

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

   10. Applied rewrites 82.9%

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

   if -0.20000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.996

   1. Initial program 80.8%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 84.0%

      \[\leadsto \left(\sinh \ell \cdot 2\right) \cdot J + U \]

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 70.8%

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

   12. Applied rewrites 70.8%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.2 \lor \neg \left(\cos \left(\frac{K}{2}\right) \leq 0.996\right):\\ \;\;\;\;\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \mathsf{fma}\left(0.3333333333333333 \cdot \ell, \ell, 2\right)\right) \cdot \ell + U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.95:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\left(\sinh \ell \cdot 2\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.95)
     (+
      (*
       (*
        J
        (*
         (fma
          (fma 0.016666666666666666 (* l l) 0.3333333333333333)
          (* l l)
          2.0)
         l))
       t_0)
      U)
     (+ (* (* (sinh l) 2.0) J) U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.95) {
		tmp = ((J * (fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * t_0) + U;
	} else {
		tmp = ((sinh(l) * 2.0) * J) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.95)
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * t_0) + U);
	else
		tmp = Float64(Float64(Float64(sinh(l) * 2.0) * J) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.95], N[(N[(N[(J * N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.94999999999999996

   1. Initial program 85.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

   if 0.94999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 93.9%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 93.9

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

   5. Applied rewrites 93.9%

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

   7. Applied rewrites 99.4%

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

Alternative 6: 95.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.95:\\ \;\;\;\;\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right)\right) \cdot \ell + U\\ \mathbf{else}:\\ \;\;\;\;\left(\sinh \ell \cdot 2\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.95)
   (+
    (*
     (*
      (* (cos (* 0.5 K)) J)
      (fma (fma 0.016666666666666666 (* l l) 0.3333333333333333) (* l l) 2.0))
     l)
    U)
   (+ (* (* (sinh l) 2.0) J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.95) {
		tmp = (((cos((0.5 * K)) * J) * fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0)) * l) + U;
	} else {
		tmp = ((sinh(l) * 2.0) * J) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.95)
		tmp = Float64(Float64(Float64(Float64(cos(Float64(0.5 * K)) * J) * fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0)) * l) + U);
	else
		tmp = Float64(Float64(Float64(sinh(l) * 2.0) * J) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.95], N[(N[(N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.94999999999999996

   1. Initial program 85.3%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.3%

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

   if 0.94999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 93.9%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 93.9

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

   5. Applied rewrites 93.9%

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

   7. Applied rewrites 99.4%

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

Alternative 7: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.953:\\ \;\;\;\;\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, -1\right), \ell, 1\right)\right) \cdot J + U\\ \mathbf{elif}\;t\_0 \leq -0.2:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.25 + U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \mathsf{fma}\left(0.3333333333333333 \cdot \ell, \ell, 2\right)\right) \cdot \ell + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.953)
     (+ (* (- 1.0 (fma (fma 0.5 l -1.0) l 1.0)) J) U)
     (if (<= t_0 -0.2)
       (+ (* (* (* (* K K) l) J) -0.25) U)
       (+ (* (* J (fma (* 0.3333333333333333 l) l 2.0)) l) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.953) {
		tmp = ((1.0 - fma(fma(0.5, l, -1.0), l, 1.0)) * J) + U;
	} else if (t_0 <= -0.2) {
		tmp = ((((K * K) * l) * J) * -0.25) + U;
	} else {
		tmp = ((J * fma((0.3333333333333333 * l), l, 2.0)) * l) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.953)
		tmp = Float64(Float64(Float64(1.0 - fma(fma(0.5, l, -1.0), l, 1.0)) * J) + U);
	elseif (t_0 <= -0.2)
		tmp = Float64(Float64(Float64(Float64(Float64(K * K) * l) * J) * -0.25) + U);
	else
		tmp = Float64(Float64(Float64(J * fma(Float64(0.3333333333333333 * l), l, 2.0)) * l) + 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, -0.953], N[(N[(N[(1.0 - N[(N[(0.5 * l + -1.0), $MachinePrecision] * l + 1.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.2], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * -0.25), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J * N[(N[(0.3333333333333333 * l), $MachinePrecision] * l + 2.0), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.952999999999999958

   1. Initial program 81.0%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 41.0

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 41.4%

      \[\leadsto \left(1 - e^{-\ell}\right) \cdot J + U \]

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 68.2%

       \[\leadsto \left(1 - \mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, -1\right), \ell, 1\right)\right) \cdot J + U \]

   if -0.952999999999999958 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.20000000000000001

   1. Initial program 91.7%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 55.2%

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

   9. Taylor expanded in K around inf

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

   11. Applied rewrites 61.2%

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

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

   1. Initial program 89.3%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 89.3

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

   5. Applied rewrites 89.3%

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

   7. Applied rewrites 94.1%

      \[\leadsto \left(\sinh \ell \cdot 2\right) \cdot J + U \]

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 77.9%

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

   12. Applied rewrites 77.9%

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

Alternative 8: 65.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.953:\\ \;\;\;\;\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, -1\right), \ell, 1\right)\right) \cdot J + U\\ \mathbf{elif}\;t\_0 \leq -0.2:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.25 + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \ell + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.953)
     (+ (* (- 1.0 (fma (fma 0.5 l -1.0) l 1.0)) J) U)
     (if (<= t_0 -0.2)
       (+ (* (* (* (* K K) l) J) -0.25) U)
       (+ (* (* (* (* l l) J) 0.3333333333333333) l) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.953) {
		tmp = ((1.0 - fma(fma(0.5, l, -1.0), l, 1.0)) * J) + U;
	} else if (t_0 <= -0.2) {
		tmp = ((((K * K) * l) * J) * -0.25) + U;
	} else {
		tmp = ((((l * l) * J) * 0.3333333333333333) * l) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.953)
		tmp = Float64(Float64(Float64(1.0 - fma(fma(0.5, l, -1.0), l, 1.0)) * J) + U);
	elseif (t_0 <= -0.2)
		tmp = Float64(Float64(Float64(Float64(Float64(K * K) * l) * J) * -0.25) + U);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) * J) * 0.3333333333333333) * l) + 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, -0.953], N[(N[(N[(1.0 - N[(N[(0.5 * l + -1.0), $MachinePrecision] * l + 1.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.2], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * -0.25), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.952999999999999958

   1. Initial program 81.0%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 41.0

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 41.4%

      \[\leadsto \left(1 - e^{-\ell}\right) \cdot J + U \]

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 68.2%

       \[\leadsto \left(1 - \mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, -1\right), \ell, 1\right)\right) \cdot J + U \]

   if -0.952999999999999958 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.20000000000000001

   1. Initial program 91.7%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 55.2%

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

   9. Taylor expanded in K around inf

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

   11. Applied rewrites 61.2%

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

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

   1. Initial program 89.3%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 89.3

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

   5. Applied rewrites 89.3%

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

   7. Applied rewrites 94.1%

      \[\leadsto \left(\sinh \ell \cdot 2\right) \cdot J + U \]

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 77.9%

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

   11. Taylor expanded in l around inf

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

   13. Applied rewrites 72.7%

       \[\leadsto \left(\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \ell + U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 94.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.95:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\left(\sinh \ell \cdot 2\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.95)
     (+ (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) t_0) U)
     (+ (* (* (sinh l) 2.0) J) U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.95) {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * t_0) + U;
	} else {
		tmp = ((sinh(l) * 2.0) * J) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.95)
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * t_0) + U);
	else
		tmp = Float64(Float64(Float64(sinh(l) * 2.0) * J) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.95], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.94999999999999996

   1. Initial program 85.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 88.2

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

   5. Applied rewrites 88.2%

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

   if 0.94999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 93.9%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 93.9

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

   5. Applied rewrites 93.9%

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

   7. Applied rewrites 99.4%

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

Alternative 10: 92.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.95:\\ \;\;\;\;\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right) \cdot \ell + U\\ \mathbf{else}:\\ \;\;\;\;\left(\sinh \ell \cdot 2\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.95)
   (+ (* (* (* (cos (* 0.5 K)) J) (fma (* l l) 0.3333333333333333 2.0)) l) U)
   (+ (* (* (sinh l) 2.0) J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.95) {
		tmp = (((cos((0.5 * K)) * J) * fma((l * l), 0.3333333333333333, 2.0)) * l) + U;
	} else {
		tmp = ((sinh(l) * 2.0) * J) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.95)
		tmp = Float64(Float64(Float64(Float64(cos(Float64(0.5 * K)) * J) * fma(Float64(l * l), 0.3333333333333333, 2.0)) * l) + U);
	else
		tmp = Float64(Float64(Float64(sinh(l) * 2.0) * J) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.95], N[(N[(N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.94999999999999996

   1. Initial program 85.3%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 85.4%

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

   if 0.94999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 93.9%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 93.9

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

   5. Applied rewrites 93.9%

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

   7. Applied rewrites 99.4%

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

Alternative 11: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.95:\\ \;\;\;\;\left(\left(J \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(\sinh \ell \cdot 2\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.95)
   (+ (* (* (* J l) (cos (* 0.5 K))) (fma (* l l) 0.3333333333333333 2.0)) U)
   (+ (* (* (sinh l) 2.0) J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.95) {
		tmp = (((J * l) * cos((0.5 * K))) * fma((l * l), 0.3333333333333333, 2.0)) + U;
	} else {
		tmp = ((sinh(l) * 2.0) * J) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.95)
		tmp = Float64(Float64(Float64(Float64(J * l) * cos(Float64(0.5 * K))) * fma(Float64(l * l), 0.3333333333333333, 2.0)) + U);
	else
		tmp = Float64(Float64(Float64(sinh(l) * 2.0) * J) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.95], N[(N[(N[(N[(J * l), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.94999999999999996

   1. Initial program 85.3%

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

   3. Taylor expanded in l around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   5. Applied rewrites 85.3%

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

   if 0.94999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 93.9%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 93.9

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

   5. Applied rewrites 93.9%

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

   7. Applied rewrites 99.4%

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

Alternative 12: 86.1% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.2)
   (+
    (*
     (* (fma (* K K) -0.125 1.0) J)
     (*
      (fma
       (fma
        (fma (* l l) 0.0003968253968253968 0.016666666666666666)
        (* l l)
        0.3333333333333333)
       (* l l)
       2.0)
      l))
    U)
   (+ (* (* (sinh l) 2.0) J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.2) {
		tmp = ((fma((K * K), -0.125, 1.0) * J) * (fma(fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) + U;
	} else {
		tmp = ((sinh(l) * 2.0) * J) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.2)
		tmp = Float64(Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * J) * Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) + U);
	else
		tmp = Float64(Float64(Float64(sinh(l) * 2.0) * J) + U);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 95.0

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

   5. Applied rewrites 95.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

   7. Applied rewrites 95.0%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 63.5

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

   10. Applied rewrites 63.5%

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

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

   1. Initial program 89.3%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 89.3

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

   5. Applied rewrites 89.3%

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

   7. Applied rewrites 94.1%

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

Alternative 13: 94.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  (*
   (* (cos (* -0.5 K)) J)
   (*
    (fma
     (fma
      (fma (* l l) 0.0003968253968253968 0.016666666666666666)
      (* l l)
      0.3333333333333333)
     (* l l)
     2.0)
    l))
  U))

C

double code(double J, double l, double K, double U) {
	return ((cos((-0.5 * K)) * J) * (fma(fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) + U;
}

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J) * Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) + U)
end

Wolfram

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

Derivation

1. Initial program 89.4%

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

3. Taylor expanded in l around 0

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

   1. *-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 96.2

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

5. Applied rewrites 96.2%

   \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

7. Applied rewrites 96.2%

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

Alternative 14: 66.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.2:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.25 + U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\ell \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \ell + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.2)
   (+ (* (* (* (* K K) l) J) -0.25) U)
   (+ (* (* (* (* l l) J) 0.3333333333333333) l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.2) {
		tmp = ((((K * K) * l) * J) * -0.25) + U;
	} else {
		tmp = ((((l * l) * J) * 0.3333333333333333) * l) + U;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.2d0)) then
        tmp = ((((k * k) * l) * j) * (-0.25d0)) + u
    else
        tmp = ((((l * l) * j) * 0.3333333333333333d0) * l) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.2) {
		tmp = ((((K * K) * l) * J) * -0.25) + U;
	} else {
		tmp = ((((l * l) * J) * 0.3333333333333333) * l) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.2:
		tmp = ((((K * K) * l) * J) * -0.25) + U
	else:
		tmp = ((((l * l) * J) * 0.3333333333333333) * l) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.2)
		tmp = Float64(Float64(Float64(Float64(Float64(K * K) * l) * J) * -0.25) + U);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) * J) * 0.3333333333333333) * l) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.2)
		tmp = ((((K * K) * l) * J) * -0.25) + U;
	else
		tmp = ((((l * l) * J) * 0.3333333333333333) * l) + U;
	end
	tmp_2 = 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[(K * K), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * -0.25), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.6%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 51.6%

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

   9. Taylor expanded in K around inf

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

   11. Applied rewrites 56.5%

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

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

   1. Initial program 89.3%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 89.3

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

   5. Applied rewrites 89.3%

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

   7. Applied rewrites 94.1%

      \[\leadsto \left(\sinh \ell \cdot 2\right) \cdot J + U \]

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 77.9%

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

   11. Taylor expanded in l around inf

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

   13. Applied rewrites 72.7%

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

Alternative 15: 53.6% accurate, 23.6× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return ((2.0 * l) * J) + 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 = ((2.0d0 * l) * j) + u
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.4%

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

3. Taylor expanded in K around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 71.6

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

5. Applied rewrites 71.6%

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

7. Applied rewrites 74.9%

   \[\leadsto \left(\sinh \ell \cdot 2\right) \cdot J + U \]

8. Taylor expanded in l around 0

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

10. Applied rewrites 47.6%

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

Reproduce

herbie shell --seed 2024298 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))