Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.9%

Time: 11.0s

Alternatives: 20

Speedup: 1.5×

• 49.6% 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: 86.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 85.2

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 86.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.99999:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (fma (* 2.0 (* J (sinh l))) (fma (* K K) -0.125 1.0) U)))
   (if (<= t_0 -0.05)
     t_1
     (if (<= t_0 0.99999)
       (fma
        (*
         (fma
          (fma
           (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
           (* l l)
           0.3333333333333333)
          (* l l)
          2.0)
         l)
        J
        U)
       t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = fma((2.0 * (J * sinh(l))), fma((K * K), -0.125, 1.0), U);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_1;
	} else if (t_0 <= 0.99999) {
		tmp = fma((fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l), J, U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = fma(Float64(2.0 * Float64(J * sinh(l))), fma(Float64(K * K), -0.125, 1.0), U)
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = t_1;
	elseif (t_0 <= 0.99999)
		tmp = fma(Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l), J, U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], t$95$1, If[LessEqual[t$95$0, 0.99999], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003 or 0.999990000000000046 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 84.9

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 88.6

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

   7. Applied rewrites 88.6%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.999990000000000046

   1. Initial program 86.3%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 82.2%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.99999:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.82:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.82)
     (fma (* (* 2.0 l) J) (cos (* 0.5 K)) U)
     (if (<= t_0 -0.05)
       (fma
        (* (* (fma (* l l) 0.3333333333333333 2.0) J) l)
        (fma (* K K) -0.125 1.0)
        U)
       (fma
        (*
         (fma
          (fma
           (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
           (* l l)
           0.3333333333333333)
          (* l l)
          2.0)
         l)
        J
        U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.82) {
		tmp = fma(((2.0 * l) * J), cos((0.5 * K)), U);
	} else if (t_0 <= -0.05) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * J) * l), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.82)
		tmp = fma(Float64(Float64(2.0 * l) * J), cos(Float64(0.5 * K)), U);
	elseif (t_0 <= -0.05)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J) * l), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.82], N[(N[(N[(2.0 * l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.819999999999999951

   1. Initial program 79.9%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if -0.819999999999999951 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 86.4

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 79.2

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.5

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

   10. Applied rewrites 68.5%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 91.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.82:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.99999:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J, \cos \left(\frac{K}{-2}\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.99999)
   (fma
    (*
     (*
      (fma (fma 0.016666666666666666 (* l l) 0.3333333333333333) (* l l) 2.0)
      l)
     J)
    (cos (/ K -2.0))
    U)
   (fma (* 2.0 (* J (sinh l))) (fma (* K K) -0.125 1.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.99999) {
		tmp = fma(((fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l) * J), cos((K / -2.0)), U);
	} else {
		tmp = fma((2.0 * (J * sinh(l))), fma((K * K), -0.125, 1.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.99999)
		tmp = fma(Float64(Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l) * J), cos(Float64(K / -2.0)), U);
	else
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), fma(Float64(K * K), -0.125, 1.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.99999], N[(N[(N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(K / -2.0), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.999990000000000046

   1. Initial program 85.1%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 95.2

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

   5. Applied rewrites 95.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 95.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.2

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

      7. lift-cos.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. cos-neg N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval 95.2

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

   7. Applied rewrites 95.2%

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

   if 0.999990000000000046 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 85.3

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot J\right) \cdot 2, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.99999:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J, \cos \left(\frac{K}{-2}\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 6: 84.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{-2}\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 0.0002)
   (fma (* 2.0 (* J (sinh l))) (fma (* K K) -0.125 1.0) U)
   (fma
    (*
     (*
      (fma
       (fma
        (fma (* l l) 0.0003968253968253968 0.016666666666666666)
        (* l l)
        0.3333333333333333)
       (* l l)
       2.0)
      l)
     (cos (/ K -2.0)))
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 0.0002) {
		tmp = fma((2.0 * (J * sinh(l))), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(((fma(fma(fma((l * l), 0.0003968253968253968, 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l) * cos((K / -2.0))), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 0.0002)
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(Float64(fma(fma(fma(Float64(l * l), 0.0003968253968253968, 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l) * cos(Float64(K / -2.0))), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 0.0002], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(K / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 2.0000000000000001e-4

   1. Initial program 85.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 85.6

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if 2.0000000000000001e-4 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 83.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 96.7

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

   5. Applied rewrites 96.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 96.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.0003968253968253968, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \cos \left(\frac{K}{-2}\right), J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.170138888888889 \cdot 10^{-5}, K \cdot K, 0.0026041666666666665\right), K \cdot K, -0.125\right), K \cdot K, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma
    (* (* (fma (* l l) 0.3333333333333333 2.0) J) l)
    (fma
     (fma
      (fma -2.170138888888889e-5 (* K K) 0.0026041666666666665)
      (* K K)
      -0.125)
     (* K K)
     1.0)
    U)
   (fma
    (*
     (fma
      (fma
       (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
       (* l l)
       0.3333333333333333)
      (* l l)
      2.0)
     l)
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * J) * l), fma(fma(fma(-2.170138888888889e-5, (K * K), 0.0026041666666666665), (K * K), -0.125), (K * K), 1.0), U);
	} else {
		tmp = fma((fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J) * l), fma(fma(fma(-2.170138888888889e-5, Float64(K * K), 0.0026041666666666665), Float64(K * K), -0.125), Float64(K * K), 1.0), U);
	else
		tmp = fma(Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(-2.170138888888889e-5 * N[(K * K), $MachinePrecision] + 0.0026041666666666665), $MachinePrecision] * N[(K * K), $MachinePrecision] + -0.125), $MachinePrecision] * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 84.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 84.1

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \color{blue}{1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right)}, U\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \color{blue}{{K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) + 1}, U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}\right) \cdot {K}^{2}} + 1, U\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \color{blue}{\mathsf{fma}\left({K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) - \frac{1}{8}, {K}^{2}, 1\right)}, U\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\color{blue}{{K}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, {K}^{2}, 1\right), U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\color{blue}{\left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) \cdot {K}^{2}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), {K}^{2}, 1\right), U\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}\right) \cdot {K}^{2} + \color{blue}{\frac{-1}{8}}, {K}^{2}, 1\right), U\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{384} + \frac{-1}{46080} \cdot {K}^{2}, {K}^{2}, \frac{-1}{8}\right)}, {K}^{2}, 1\right), U\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{46080} \cdot {K}^{2} + \frac{1}{384}}, {K}^{2}, \frac{-1}{8}\right), {K}^{2}, 1\right), U\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{46080}, {K}^{2}, \frac{1}{384}\right)}, {K}^{2}, \frac{-1}{8}\right), {K}^{2}, 1\right), U\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{46080}, \color{blue}{K \cdot K}, \frac{1}{384}\right), {K}^{2}, \frac{-1}{8}\right), {K}^{2}, 1\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{46080}, \color{blue}{K \cdot K}, \frac{1}{384}\right), {K}^{2}, \frac{-1}{8}\right), {K}^{2}, 1\right), U\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{46080}, K \cdot K, \frac{1}{384}\right), \color{blue}{K \cdot K}, \frac{-1}{8}\right), {K}^{2}, 1\right), U\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{46080}, K \cdot K, \frac{1}{384}\right), \color{blue}{K \cdot K}, \frac{-1}{8}\right), {K}^{2}, 1\right), U\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{46080}, K \cdot K, \frac{1}{384}\right), K \cdot K, \frac{-1}{8}\right), \color{blue}{K \cdot K}, 1\right), U\right) \]

      15. lower-*.f64 61.6

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.170138888888889 \cdot 10^{-5}, K \cdot K, 0.0026041666666666665\right), K \cdot K, -0.125\right), \color{blue}{K \cdot K}, 1\right), U\right) \]

   10. Applied rewrites 61.6%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.170138888888889 \cdot 10^{-5}, K \cdot K, 0.0026041666666666665\right), K \cdot K, -0.125\right), K \cdot K, 1\right)}, U\right) \]

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 91.6%

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

Alternative 8: 83.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 6.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right) \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \cos \left(-0.5 \cdot K\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 6.5e-33)
   (fma (* 2.0 (* J (sinh l))) (fma (* K K) -0.125 1.0) U)
   (fma
    (*
     (*
      (fma
       (* (fma 0.008333333333333333 (* l l) 0.16666666666666666) J)
       (* l l)
       J)
      l)
     2.0)
    (cos (* -0.5 K))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 6.5e-33) {
		tmp = fma((2.0 * (J * sinh(l))), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(((fma((fma(0.008333333333333333, (l * l), 0.16666666666666666) * J), (l * l), J) * l) * 2.0), cos((-0.5 * K)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 6.5e-33)
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(l * l), 0.16666666666666666) * J), Float64(l * l), J) * l) * 2.0), cos(Float64(-0.5 * K)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 6.5e-33], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.008333333333333333 * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * J), $MachinePrecision] * N[(l * l), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 6.4999999999999993e-33

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 85.4

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 78.8

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

   7. Applied rewrites 78.8%

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

   if 6.4999999999999993e-33 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 84.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 84.8

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{120} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{6} \cdot J\right)\right) \cdot \ell\right)} \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(J + {\ell}^{2} \cdot \left(\frac{1}{120} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{6} \cdot J\right)\right) \cdot \ell\right)} \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{120} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{6} \cdot J\right) + J\right)} \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(\frac{1}{120} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{6} \cdot J\right) \cdot {\ell}^{2}} + J\right) \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(J \cdot {\ell}^{2}\right) + \frac{1}{6} \cdot J, {\ell}^{2}, J\right)} \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \color{blue}{\left({\ell}^{2} \cdot J\right)} + \frac{1}{6} \cdot J, {\ell}^{2}, J\right) \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} \cdot {\ell}^{2}\right) \cdot J} + \frac{1}{6} \cdot J, {\ell}^{2}, J\right) \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\color{blue}{J \cdot \left(\frac{1}{120} \cdot {\ell}^{2} + \frac{1}{6}\right)}, {\ell}^{2}, J\right) \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(J \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)}, {\ell}^{2}, J\right) \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\color{blue}{J \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)}, {\ell}^{2}, J\right) \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(J \cdot \color{blue}{\left(\frac{1}{120} \cdot {\ell}^{2} + \frac{1}{6}\right)}, {\ell}^{2}, J\right) \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(J \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {\ell}^{2}, \frac{1}{6}\right)}, {\ell}^{2}, J\right) \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, J\right) \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{\ell \cdot \ell}, \frac{1}{6}\right), {\ell}^{2}, J\right) \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{120}, \ell \cdot \ell, \frac{1}{6}\right), \color{blue}{\ell \cdot \ell}, J\right) \cdot \ell\right) \cdot 2, \cos \left(K \cdot \frac{-1}{2}\right), U\right) \]

      16. lower-*.f64 90.3

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \color{blue}{\ell \cdot \ell}, J\right) \cdot \ell\right) \cdot 2, \cos \left(K \cdot -0.5\right), U\right) \]

   7. Applied rewrites 90.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, J\right) \cdot \ell\right)} \cdot 2, \cos \left(K \cdot -0.5\right), U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 6.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right) \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \cos \left(-0.5 \cdot K\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma
    (* (* (fma (* l l) 0.3333333333333333 2.0) J) l)
    (fma (* K K) -0.125 1.0)
    U)
   (fma
    (*
     (fma
      (fma
       (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
       (* l l)
       0.3333333333333333)
      (* l l)
      2.0)
     l)
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * J) * l), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J) * l), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 84.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 84.1

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.9

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

   10. Applied rewrites 60.9%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 91.6%

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

Alternative 10: 82.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma
    (* (* (fma (* l l) 0.3333333333333333 2.0) J) l)
    (fma (* K K) -0.125 1.0)
    U)
   (fma
    (*
     (fma (fma 0.016666666666666666 (* l l) 0.3333333333333333) (* l l) 2.0)
     l)
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * J) * l), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J) * l), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 84.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 84.1

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.9

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

   10. Applied rewrites 60.9%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 89.5%

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

Alternative 11: 80.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (fma
    (*
     (fma (fma 0.016666666666666666 (* l l) 0.3333333333333333) (* l l) 2.0)
     l)
    J
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma((fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = fma(Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 49.6%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 89.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 6.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 6.5e-33)
   (fma (* 2.0 (* J (sinh l))) (fma (* K K) -0.125 1.0) U)
   (fma (* (* (fma 0.3333333333333333 (* l l) 2.0) l) J) (cos (* -0.5 K)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 6.5e-33) {
		tmp = fma((2.0 * (J * sinh(l))), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(((fma(0.3333333333333333, (l * l), 2.0) * l) * J), cos((-0.5 * K)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 6.5e-33)
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(Float64(fma(0.3333333333333333, Float64(l * l), 2.0) * l) * J), cos(Float64(-0.5 * K)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 6.5e-33], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 6.4999999999999993e-33

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 85.4

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 78.8

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

   7. Applied rewrites 78.8%

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

   if 6.4999999999999993e-33 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 84.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 84.8

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 79.1

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

   7. Applied rewrites 79.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell}, \cos \left(K \cdot -0.5\right), U\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 85.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 6.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 0.0002)
   (fma (* 2.0 (* J (sinh l))) (fma (* K K) -0.125 1.0) U)
   (fma (* (* (fma (* l l) 0.3333333333333333 2.0) J) (cos (* 0.5 K))) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 0.0002) {
		tmp = fma((2.0 * (J * sinh(l))), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * J) * cos((0.5 * K))), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 0.0002)
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J) * cos(Float64(0.5 * K))), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 0.0002], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 2.0000000000000001e-4

   1. Initial program 85.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 85.6

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if 2.0000000000000001e-4 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 83.9%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 77.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666 \cdot \left(\ell \cdot \ell\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (fma (* (fma (* 0.016666666666666666 (* l l)) (* l l) 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma((fma((0.016666666666666666 * (l * l)), (l * l), 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = fma(Float64(fma(Float64(0.016666666666666666 * Float64(l * l)), Float64(l * l), 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 49.6%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 89.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666 \cdot \left(\ell \cdot \ell\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 49.6%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 83.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) J) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * J), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 49.6%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.6%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 78.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J, \ell, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.1% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{if}\;\ell \leq -1.85 \cdot 10^{+90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell, 0.5, 1\right), \ell, 1\right) - 1, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (* J l) (fma (* K K) -0.25 2.0) U)))
   (if (<= l -1.85e+90)
     t_0
     (if (<= l 1.05e-39)
       (fma (* 2.0 l) J U)
       (if (<= l 2.25e+117)
         t_0
         (fma (- (fma (fma l 0.5 1.0) l 1.0) 1.0) J U))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((J * l), fma((K * K), -0.25, 2.0), U);
	double tmp;
	if (l <= -1.85e+90) {
		tmp = t_0;
	} else if (l <= 1.05e-39) {
		tmp = fma((2.0 * l), J, U);
	} else if (l <= 2.25e+117) {
		tmp = t_0;
	} else {
		tmp = fma((fma(fma(l, 0.5, 1.0), l, 1.0) - 1.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U)
	tmp = 0.0
	if (l <= -1.85e+90)
		tmp = t_0;
	elseif (l <= 1.05e-39)
		tmp = fma(Float64(2.0 * l), J, U);
	elseif (l <= 2.25e+117)
		tmp = t_0;
	else
		tmp = fma(Float64(fma(fma(l, 0.5, 1.0), l, 1.0) - 1.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -1.85e+90], t$95$0, If[LessEqual[l, 1.05e-39], N[(N[(2.0 * l), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 2.25e+117], t$95$0, N[(N[(N[(N[(l * 0.5 + 1.0), $MachinePrecision] * l + 1.0), $MachinePrecision] - 1.0), $MachinePrecision] * J + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.85e90 or 1.04999999999999997e-39 < l < 2.25e117

   1. Initial program 97.9%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 36.6

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

   5. Applied rewrites 36.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 40.9%

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

   if -1.85e90 < l < 1.04999999999999997e-39

   1. Initial program 71.5%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 74.5%

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

   if 2.25e117 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 27.4%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 27.4%

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

   12. Taylor expanded in l around 0

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

   14. Applied rewrites 60.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell, 0.5, 1\right), \ell, 1\right) - 1, J, U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell, 0.5, 1\right), \ell, 1\right) - 1, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 60.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell, 0.5, 1\right), \ell, 1\right) - 1, J, U\right)\\ t_1 := \mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \mathbf{if}\;\ell \leq -4.9 \cdot 10^{+250}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -6.5 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (- (fma (fma l 0.5 1.0) l 1.0) 1.0) J U))
        (t_1 (fma (* 2.0 l) J U)))
   (if (<= l -4.9e+250) t_1 (if (<= l -6.5e+93) t_0 (if (<= l 2.0) t_1 t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((fma(fma(l, 0.5, 1.0), l, 1.0) - 1.0), J, U);
	double t_1 = fma((2.0 * l), J, U);
	double tmp;
	if (l <= -4.9e+250) {
		tmp = t_1;
	} else if (l <= -6.5e+93) {
		tmp = t_0;
	} else if (l <= 2.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(fma(fma(l, 0.5, 1.0), l, 1.0) - 1.0), J, U)
	t_1 = fma(Float64(2.0 * l), J, U)
	tmp = 0.0
	if (l <= -4.9e+250)
		tmp = t_1;
	elseif (l <= -6.5e+93)
		tmp = t_0;
	elseif (l <= 2.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[(l * 0.5 + 1.0), $MachinePrecision] * l + 1.0), $MachinePrecision] - 1.0), $MachinePrecision] * J + U), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * l), $MachinePrecision] * J + U), $MachinePrecision]}, If[LessEqual[l, -4.9e+250], t$95$1, If[LessEqual[l, -6.5e+93], t$95$0, If[LessEqual[l, 2.0], t$95$1, t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if l < -4.89999999999999993e250 or -6.4999999999999998e93 < l < 2

   1. Initial program 75.3%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 73.3%

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

   if -4.89999999999999993e250 < l < -6.4999999999999998e93 or 2 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 31.9%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 15.3%

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

   12. Taylor expanded in l around 0

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

   14. Applied rewrites 39.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\ell, 0.5, 1\right), \ell, 1\right) - 1, J, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 54.2% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in l around 0

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

8. Applied rewrites 49.9%

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

Alternative 20: 37.2% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1 - 1, J, U\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (fma (- 1.0 1.0) J U))

C

double code(double J, double l, double K, double U) {
	return fma((1.0 - 1.0), J, U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(1.0 - 1.0), J, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(1.0 - 1.0), $MachinePrecision] * J + U), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in K around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in l around 0

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

8. Applied rewrites 55.3%

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

9. Taylor expanded in l around 0

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

11. Applied rewrites 41.8%

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

12. Taylor expanded in l around 0

    \[\leadsto \mathsf{fma}\left(1 - 1, J, U\right) \]
13. Step-by-step derivation

14. Applied rewrites 32.6%

    \[\leadsto \mathsf{fma}\left(1 - 1, J, U\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024284 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))