Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.8% → 100.0%

Time: 11.1s

Alternatives: 19

Speedup: 2.5×

• 49.7% 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: 85.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

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 86.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
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 86.3

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

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. 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: 87.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := e^{\ell} - t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(1 - t\_0, J, U\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{\ell} - 1, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (- (exp l) t_0)))
   (if (<= t_1 -5e+140)
     (fma (- 1.0 t_0) J U)
     (if (<= t_1 5e-12)
       (fma (* (* 2.0 l) J) (cos (* 0.5 K)) U)
       (fma (- (exp l) 1.0) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = exp(l) - t_0;
	double tmp;
	if (t_1 <= -5e+140) {
		tmp = fma((1.0 - t_0), J, U);
	} else if (t_1 <= 5e-12) {
		tmp = fma(((2.0 * l) * J), cos((0.5 * K)), U);
	} else {
		tmp = fma((exp(l) - 1.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = Float64(exp(l) - t_0)
	tmp = 0.0
	if (t_1 <= -5e+140)
		tmp = fma(Float64(1.0 - t_0), J, U);
	elseif (t_1 <= 5e-12)
		tmp = fma(Float64(Float64(2.0 * l) * J), cos(Float64(0.5 * K)), U);
	else
		tmp = fma(Float64(exp(l) - 1.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+140], N[(N[(1.0 - t$95$0), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[t$95$1, 5e-12], N[(N[(N[(2.0 * l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(N[Exp[l], $MachinePrecision] - 1.0), $MachinePrecision] * J + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -5.00000000000000008e140

   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 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 74.6%

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

   if -5.00000000000000008e140 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 4.9999999999999997e-12

   1. Initial program 73.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   if 4.9999999999999997e-12 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   3. 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 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 83.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -5 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(1 - e^{-\ell}, J, U\right)\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot J, \cos \left(0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{\ell} - 1, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \mathsf{fma}\left(\left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) \cdot \ell, J, U\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l))))
        (t_1 (fma (* (* 0.3333333333333333 (* l l)) l) J U)))
   (if (<= t_0 -5e+140) t_1 (if (<= t_0 0.0) (fma (* J l) 2.0 U) t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = fma(((0.3333333333333333 * (l * l)) * l), J, U);
	double tmp;
	if (t_0 <= -5e+140) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = fma((J * l), 2.0, U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = fma(Float64(Float64(0.3333333333333333 * Float64(l * l)) * l), J, U)
	tmp = 0.0
	if (t_0 <= -5e+140)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = fma(Float64(J * l), 2.0, U);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+140], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(J * l), $MachinePrecision] * 2.0 + U), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -5.00000000000000008e140 or 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.6%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 64.7%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 64.7%

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

   if -5.00000000000000008e140 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 73.1%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 90.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -5 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) \cdot \ell, J, U\right)\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.87:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.87)
   (fma (* (* (fma (* l l) 0.3333333333333333 2.0) J) (cos (* 0.5 K))) l U)
   (fma
    (*
     (fma
      (fma
       (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
       (* l l)
       0.3333333333333333)
      (* l l)
      2.0)
     l)
    J
    U)))

C

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

Julia

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

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.87], 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], 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))) < 0.869999999999999996

   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. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 86.4%

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

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

   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 86.3

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

   5. Applied rewrites 86.3%

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

   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 93.4%

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

4. Final simplification 90.7%

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

Alternative 5: 95.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. 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 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 94.3

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

5. Applied rewrites 94.3%

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

6. Final simplification 94.3%

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

Alternative 6: 81.5% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (- (fma (fma 0.5 l 1.0) l 1.0) 1.0) J 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(fma(0.5, l, 1.0), l, 1.0) - 1.0), J, 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(fma(fma(0.5, l, 1.0), l, 1.0) - 1.0), J, 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[(0.5 * l + 1.0), $MachinePrecision] * l + 1.0), $MachinePrecision] - 1.0), $MachinePrecision] * J + 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 91.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 41.9

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

   5. Applied rewrites 41.9%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 42.8%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 62.4%

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

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

   1. Initial program 85.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 90.4%

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

Alternative 7: 92.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 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 92.8

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

5. Applied rewrites 92.8%

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

6. Final simplification 92.8%

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

Alternative 8: 79.9% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (- (fma (fma 0.5 l 1.0) l 1.0) 1.0) J 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(fma(0.5, l, 1.0), l, 1.0) - 1.0), J, 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(fma(fma(0.5, l, 1.0), l, 1.0) - 1.0), J, 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[(0.5 * l + 1.0), $MachinePrecision] * l + 1.0), $MachinePrecision] - 1.0), $MachinePrecision] * J + 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 91.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 41.9

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

   5. Applied rewrites 41.9%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 42.8%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 62.4%

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

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

   1. Initial program 85.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 88.9%

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

Alternative 9: 90.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right) \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \cos \left(-0.5 \cdot K\right), U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (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) {
	return fma(((fma((fma(0.008333333333333333, (l * l), 0.16666666666666666) * J), (l * l), J) * l) * 2.0), cos((-0.5 * K)), U);
}

Julia

function code(J, l, K, U)
	return 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

Wolfram

code[J_, l_, K_, U_] := 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. 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. 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 86.3

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

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 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 92.1

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

7. Applied rewrites 92.1%

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

8. Final simplification 92.1%

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

Alternative 10: 90.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(\left(0.008333333333333333 \cdot \left(\ell \cdot \ell\right)\right) \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \cos \left(-0.5 \cdot K\right), U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma
  (* (* (fma (* (* 0.008333333333333333 (* l l)) J) (* l l) J) l) 2.0)
  (cos (* -0.5 K))
  U))

C

double code(double J, double l, double K, double U) {
	return fma(((fma(((0.008333333333333333 * (l * l)) * J), (l * l), J) * l) * 2.0), cos((-0.5 * K)), U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(l * l)) * J), Float64(l * l), J) * l) * 2.0), cos(Float64(-0.5 * K)), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(N[(N[(N[(0.008333333333333333 * N[(l * l), $MachinePrecision]), $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. 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. 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 86.3

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

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 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 92.1

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

7. Applied rewrites 92.1%

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

8. Taylor expanded in l around inf

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

10. Applied rewrites 92.1%

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

11. Final simplification 92.1%

    \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\left(0.008333333333333333 \cdot \left(\ell \cdot \ell\right)\right) \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, \cos \left(-0.5 \cdot K\right), U\right) \]
12. Add Preprocessing

Alternative 11: 92.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -3:\\ \;\;\;\;\mathsf{fma}\left(1 - e^{-\ell}, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (fma
          (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)
          (cos (* -0.5 K))
          U)))
   (if (<= l -3.1e+98)
     t_0
     (if (<= l -3.0) (fma (- 1.0 (exp (- l))) J U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma(((fma((l * l), 0.3333333333333333, 2.0) * l) * J), cos((-0.5 * K)), U);
	double tmp;
	if (l <= -3.1e+98) {
		tmp = t_0;
	} else if (l <= -3.0) {
		tmp = fma((1.0 - exp(-l)), J, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J), cos(Float64(-0.5 * K)), U)
	tmp = 0.0
	if (l <= -3.1e+98)
		tmp = t_0;
	elseif (l <= -3.0)
		tmp = fma(Float64(1.0 - exp(Float64(-l))), J, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -3.1e+98], t$95$0, If[LessEqual[l, -3.0], N[(N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.10000000000000019e98 or -3 < l

   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. 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 85.1

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

   4. Applied rewrites 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 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 96.7

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in l around inf

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

   10. Applied rewrites 96.7%

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

   11. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-in N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 95.5

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

   13. Applied rewrites 95.5%

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

   if -3.10000000000000019e98 < l < -3

   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 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 77.3%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{elif}\;\ell \leq -3:\\ \;\;\;\;\mathsf{fma}\left(1 - e^{-\ell}, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J, \cos \left(-0.5 \cdot K\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, 1\right), \ell, 1\right) - 1, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (- (fma (fma 0.5 l 1.0) l 1.0) 1.0) J 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((fma(fma(0.5, l, 1.0), l, 1.0) - 1.0), J, 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(fma(fma(0.5, l, 1.0), l, 1.0) - 1.0), J, 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[(N[(N[(0.5 * l + 1.0), $MachinePrecision] * l + 1.0), $MachinePrecision] - 1.0), $MachinePrecision] * J + 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 91.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 41.9

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

   5. Applied rewrites 41.9%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 42.8%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 62.4%

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

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

   1. Initial program 85.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 87.5%

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

Alternative 13: 73.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, 1\right), \ell, 1\right) - 1, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (- (fma (fma 0.5 l 1.0) l 1.0) 1.0) J 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((fma(fma(0.5, l, 1.0), l, 1.0) - 1.0), J, 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(fma(fma(0.5, l, 1.0), l, 1.0) - 1.0), J, 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[(N[(N[(0.5 * l + 1.0), $MachinePrecision] * l + 1.0), $MachinePrecision] - 1.0), $MachinePrecision] * J + 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 91.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 41.9

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

   5. Applied rewrites 41.9%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 42.8%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 62.4%

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

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

   1. Initial program 85.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 83.2%

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

Alternative 14: 73.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \ell, 1\right), \ell, 1\right) - 1, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot \ell, \ell, 2\right) \cdot J, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (fma (- (fma (fma 0.5 l 1.0) l 1.0) 1.0) J U)
   (fma (* (fma (* 0.3333333333333333 l) l 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((fma(fma(0.5, l, 1.0), l, 1.0) - 1.0), J, U);
	} else {
		tmp = fma((fma((0.3333333333333333 * l), l, 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(fma(fma(0.5, l, 1.0), l, 1.0) - 1.0), J, U);
	else
		tmp = fma(Float64(fma(Float64(0.3333333333333333 * l), l, 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[(N[(N[(0.5 * l + 1.0), $MachinePrecision] * l + 1.0), $MachinePrecision] - 1.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * l), $MachinePrecision] * l + 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 91.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 41.9

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

   5. Applied rewrites 41.9%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 42.8%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 62.4%

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

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

   1. Initial program 85.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 83.2%

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

   10. Applied rewrites 82.7%

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

Alternative 15: 87.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.2:\\ \;\;\;\;\mathsf{fma}\left(1 - e^{-\ell}, J, U\right)\\ \mathbf{elif}\;\ell \leq 1.75:\\ \;\;\;\;\mathsf{fma}\left(\left(\cos \left(-0.5 \cdot K\right) \cdot \ell\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{\ell} - 1, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.2)
   (fma (- 1.0 (exp (- l))) J U)
   (if (<= l 1.75)
     (fma (* (* (cos (* -0.5 K)) l) 2.0) J U)
     (fma (- (exp l) 1.0) J U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.2) {
		tmp = fma((1.0 - exp(-l)), J, U);
	} else if (l <= 1.75) {
		tmp = fma(((cos((-0.5 * K)) * l) * 2.0), J, U);
	} else {
		tmp = fma((exp(l) - 1.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.2)
		tmp = fma(Float64(1.0 - exp(Float64(-l))), J, U);
	elseif (l <= 1.75)
		tmp = fma(Float64(Float64(cos(Float64(-0.5 * K)) * l) * 2.0), J, U);
	else
		tmp = fma(Float64(exp(l) - 1.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.2], N[(N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 1.75], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Exp[l], $MachinePrecision] - 1.0), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.2000000000000002

   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 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 74.6%

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

   if -2.2000000000000002 < l < 1.75

   1. Initial program 73.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 1.75 < 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 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 83.6%

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

Alternative 16: 71.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 9500000000000:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)))
   (if (<= l -8.5e+89)
     t_0
     (if (<= l 9500000000000.0) (fma (* J l) 2.0 U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (fma((l * l), 0.3333333333333333, 2.0) * l) * J;
	double tmp;
	if (l <= -8.5e+89) {
		tmp = t_0;
	} else if (l <= 9500000000000.0) {
		tmp = fma((J * l), 2.0, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J)
	tmp = 0.0
	if (l <= -8.5e+89)
		tmp = t_0;
	elseif (l <= 9500000000000.0)
		tmp = fma(Float64(J * l), 2.0, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -8.5e+89], t$95$0, If[LessEqual[l, 9500000000000.0], N[(N[(J * l), $MachinePrecision] * 2.0 + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -8.50000000000000045e89 or 9.5e12 < 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 79.2

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

   5. Applied rewrites 79.2%

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

   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 68.9%

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

   9. Taylor expanded in U around 0

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

   11. Applied rewrites 77.3%

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

   if -8.50000000000000045e89 < l < 9.5e12

   1. Initial program 77.4%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 77.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+89}:\\ \;\;\;\;\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \mathbf{elif}\;\ell \leq 9500000000000:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.9% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \ell\right) \cdot 2\\ \mathbf{if}\;\ell \leq -9.2 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 8800000000000:\\ \;\;\;\;\mathsf{fma}\left(1 - 1, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* J l) 2.0)))
   (if (<= l -9.2e-33)
     t_0
     (if (<= l 8800000000000.0) (fma (- 1.0 1.0) J U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (J * l) * 2.0;
	double tmp;
	if (l <= -9.2e-33) {
		tmp = t_0;
	} else if (l <= 8800000000000.0) {
		tmp = fma((1.0 - 1.0), J, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J * l) * 2.0)
	tmp = 0.0
	if (l <= -9.2e-33)
		tmp = t_0;
	elseif (l <= 8800000000000.0)
		tmp = fma(Float64(1.0 - 1.0), J, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J * l), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[l, -9.2e-33], t$95$0, If[LessEqual[l, 8800000000000.0], N[(N[(1.0 - 1.0), $MachinePrecision] * J + U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.19999999999999942e-33 or 8.8e12 < l

   1. Initial program 95.1%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 31.1%

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

   9. Taylor expanded in U around 0

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

   11. Applied rewrites 29.4%

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

   if -9.19999999999999942e-33 < l < 8.8e12

   1. Initial program 77.1%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in l around 0

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

   8. Applied rewrites 75.9%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 72.9%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.2 \cdot 10^{-33}:\\ \;\;\;\;\left(J \cdot \ell\right) \cdot 2\\ \mathbf{elif}\;\ell \leq 8800000000000:\\ \;\;\;\;\mathsf{fma}\left(1 - 1, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \ell\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 54.1% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 75.6

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

5. Applied rewrites 75.6%

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

6. Taylor expanded in l around 0

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

8. Applied rewrites 58.3%

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

9. Final simplification 58.3%

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

Alternative 19: 36.3% 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 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 75.6

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

5. Applied rewrites 75.6%

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

6. Taylor expanded in l around 0

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

8. Applied rewrites 55.6%

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

9. Taylor expanded in l around 0

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

11. Applied rewrites 37.5%

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

Reproduce

herbie shell --seed 2024241 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))