Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 87.0% → 99.9%

Time: 12.1s

Alternatives: 24

Speedup: 2.1×

• 50.7% 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: 87.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

Alternative 2: 70.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot 2\right), J, U\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \ell, 4, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U)))
   (if (<= t_0 (- INFINITY))
     (fma (* (* l l) (* l 2.0)) J U)
     (if (<= t_0 1e+291) (fma (+ l l) J U) (fma (* l l) 4.0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((l * l) * (l * 2.0)), J, U);
	} else if (t_0 <= 1e+291) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = fma((l * l), 4.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(Float64(l * l) * Float64(l * 2.0)), J, U);
	elseif (t_0 <= 1e+291)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = fma(Float64(l * l), 4.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(l * l), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[t$95$0, 1e+291], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * 4.0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 36.6

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

   5. Applied rewrites 36.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 28.9%

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

   10. Applied rewrites 28.9%

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

   12. Applied rewrites 53.5%

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

   if -inf.0 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 9.9999999999999996e290

   1. Initial program 72.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 83.9%

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

   10. Applied rewrites 83.9%

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

   if 9.9999999999999996e290 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U)

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 30.5

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

   5. Applied rewrites 30.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 20.9%

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

   10. Applied rewrites 20.9%

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

   12. Applied rewrites 64.9%

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

Alternative 3: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, \mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right) \cdot \ell, K \cdot K, \ell\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U) 1e+306)
   (fma (* 1.0 J) (* (fma 0.3333333333333333 (* l l) 2.0) l) U)
   (fma
    (* l 2.0)
    (fma (* (fma (* K K) 0.0026041666666666665 -0.125) l) (* K K) l)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U) <= 1e+306) {
		tmp = fma((1.0 * J), (fma(0.3333333333333333, (l * l), 2.0) * l), U);
	} else {
		tmp = fma((l * 2.0), fma((fma((K * K), 0.0026041666666666665, -0.125) * l), (K * K), l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U) <= 1e+306)
		tmp = fma(Float64(1.0 * J), Float64(fma(0.3333333333333333, Float64(l * l), 2.0) * l), U);
	else
		tmp = fma(Float64(l * 2.0), fma(Float64(fma(Float64(K * K), 0.0026041666666666665, -0.125) * l), Float64(K * K), l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[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], 1e+306], N[(N[(1.0 * J), $MachinePrecision] * N[(N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * 2.0), $MachinePrecision] * N[(N[(N[(N[(K * K), $MachinePrecision] * 0.0026041666666666665 + -0.125), $MachinePrecision] * l), $MachinePrecision] * N[(K * K), $MachinePrecision] + l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 1.00000000000000002e306

   1. Initial program 80.8%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-/.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 74.6%

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

   if 1.00000000000000002e306 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U)

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg-rev N/A

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

      4. lower-cos.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 29.4

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

   7. Applied rewrites 29.4%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 36.0%

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

   12. Applied rewrites 82.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot 2, \mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right) \cdot \ell, K \cdot K, \ell\right), U\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \leq 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \ell, 4, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U) 1e+291)
   (fma (+ l l) J U)
   (fma (* l l) 4.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U) <= 1e+291) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = fma((l * l), 4.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U) <= 1e+291)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = fma(Float64(l * l), 4.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[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], 1e+291], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * 4.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 9.9999999999999996e290

   1. Initial program 80.7%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 66.9%

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

   10. Applied rewrites 66.9%

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

   if 9.9999999999999996e290 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U)

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 30.5

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

   5. Applied rewrites 30.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 20.9%

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

   10. Applied rewrites 20.9%

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

   12. Applied rewrites 64.9%

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

Alternative 5: 55.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot J, 4, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U) 1e+306)
   (fma (+ l l) J U)
   (fma (* J J) 4.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U) <= 1e+306) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = fma((J * J), 4.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U) <= 1e+306)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = fma(Float64(J * J), 4.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[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], 1e+306], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(J * J), $MachinePrecision] * 4.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 1.00000000000000002e306

   1. Initial program 80.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 67.1%

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

   10. Applied rewrites 67.1%

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

   if 1.00000000000000002e306 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U)

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 19.6%

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

   10. Applied rewrites 19.6%

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

   12. Applied rewrites 27.0%

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

Alternative 6: 55.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U) 1e+306)
   (fma (+ l l) J U)
   (fma (+ J J) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U) <= 1e+306) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = fma((J + J), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U) <= 1e+306)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = fma(Float64(J + J), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[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], 1e+306], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U) < 1.00000000000000002e306

   1. Initial program 80.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 67.1%

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

   10. Applied rewrites 67.1%

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

   if 1.00000000000000002e306 < (+.f64 (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) U)

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 19.6%

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

   10. Applied rewrites 19.6%

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

   12. Applied rewrites 27.0%

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

Alternative 7: 39.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\ell, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) 0.0)
   (fma l 2.0 U)
   (fma (+ J J) J U)))

C

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

Julia

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

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(l * 2.0 + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 80.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 80.0

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 67.8%

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

   10. Applied rewrites 67.8%

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

   12. Applied rewrites 45.3%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 30.4

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

   5. Applied rewrites 30.4%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 20.5%

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

   10. Applied rewrites 20.5%

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

   12. Applied rewrites 25.9%

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

Alternative 8: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.91:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell\right) \cdot 2, J, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(\left|0.3333333333333333 \cdot \ell\right|, \ell, 2\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.91)
     (fma (* (* (fma (* K K) -0.125 1.0) l) 2.0) J U)
     (if (<= t_0 -0.02)
       (fma (* 1.0 J) (* (fma (fabs (* 0.3333333333333333 l)) l 2.0) l) U)
       (fma
        (* 1.0 J)
        (*
         (fma
          (fma 0.016666666666666666 (* l l) 0.3333333333333333)
          (* l l)
          2.0)
         l)
        U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.91) {
		tmp = fma(((fma((K * K), -0.125, 1.0) * l) * 2.0), J, U);
	} else if (t_0 <= -0.02) {
		tmp = fma((1.0 * J), (fma(fabs((0.3333333333333333 * l)), l, 2.0) * l), U);
	} else {
		tmp = fma((1.0 * J), (fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.91)
		tmp = fma(Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * l) * 2.0), J, U);
	elseif (t_0 <= -0.02)
		tmp = fma(Float64(1.0 * J), Float64(fma(abs(Float64(0.3333333333333333 * l)), l, 2.0) * l), U);
	else
		tmp = fma(Float64(1.0 * J), Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.91], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(1.0 * J), $MachinePrecision] * N[(N[(N[Abs[N[(0.3333333333333333 * l), $MachinePrecision]], $MachinePrecision] * l + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(1.0 * J), $MachinePrecision] * N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.910000000000000031

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg-rev N/A

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

      4. lower-cos.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 56.5

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

   7. Applied rewrites 56.5%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 56.8%

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

   if -0.910000000000000031 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 89.7%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-/.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

   7. Applied rewrites 90.0%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 49.4%

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

   12. Applied rewrites 68.7%

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

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

   1. Initial program 84.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-/.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 82.1%

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

   11. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 85.7

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

   13. Applied rewrites 85.7%

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

Alternative 9: 76.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.91:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell\right) \cdot 2, J, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(\left|0.3333333333333333 \cdot \ell\right|, \ell, 2\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.91)
     (fma (* (* (fma (* K K) -0.125 1.0) l) 2.0) J U)
     (if (<= t_0 -0.02)
       (fma (* 1.0 J) (* (fma (fabs (* 0.3333333333333333 l)) l 2.0) l) U)
       (fma (* 1.0 J) (* (fma 0.3333333333333333 (* l l) 2.0) l) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.91) {
		tmp = fma(((fma((K * K), -0.125, 1.0) * l) * 2.0), J, U);
	} else if (t_0 <= -0.02) {
		tmp = fma((1.0 * J), (fma(fabs((0.3333333333333333 * l)), l, 2.0) * l), U);
	} else {
		tmp = fma((1.0 * J), (fma(0.3333333333333333, (l * l), 2.0) * l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.91)
		tmp = fma(Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * l) * 2.0), J, U);
	elseif (t_0 <= -0.02)
		tmp = fma(Float64(1.0 * J), Float64(fma(abs(Float64(0.3333333333333333 * l)), l, 2.0) * l), U);
	else
		tmp = fma(Float64(1.0 * J), Float64(fma(0.3333333333333333, Float64(l * l), 2.0) * l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.91], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(1.0 * J), $MachinePrecision] * N[(N[(N[Abs[N[(0.3333333333333333 * l), $MachinePrecision]], $MachinePrecision] * l + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[(1.0 * J), $MachinePrecision] * N[(N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.910000000000000031

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg-rev N/A

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

      4. lower-cos.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 56.5

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

   7. Applied rewrites 56.5%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 56.8%

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

   if -0.910000000000000031 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 89.7%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-/.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

   7. Applied rewrites 90.0%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 49.4%

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

   12. Applied rewrites 68.7%

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

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

   1. Initial program 84.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-/.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 82.1%

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

Alternative 10: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 \cdot \sinh \ell\right) \cdot J, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* (cos (* -0.5 K)) (* J (fma (* l l) 0.3333333333333333 2.0))) l U)
   (fma (* (* 1.0 (sinh l)) J) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma((cos((-0.5 * K)) * (J * fma((l * l), 0.3333333333333333, 2.0))), l, U);
	} else {
		tmp = fma(((1.0 * sinh(l)) * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(cos(Float64(-0.5 * K)) * Float64(J * fma(Float64(l * l), 0.3333333333333333, 2.0))), l, U);
	else
		tmp = fma(Float64(Float64(1.0 * sinh(l)) * J), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(1.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.3%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 84.9%

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

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

   1. Initial program 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 93.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 93.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 93.9

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

   9. Applied rewrites 93.9%

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

Alternative 11: 87.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 \cdot \sinh \ell\right) \cdot J, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* (+ J J) l) (cos (* -0.5 K)) U)
   (fma (* (* 1.0 (sinh l)) J) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma(((J + J) * l), cos((-0.5 * K)), U);
	} else {
		tmp = fma(((1.0 * sinh(l)) * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(Float64(J + J) * l), cos(Float64(-0.5 * K)), U);
	else
		tmp = fma(Float64(Float64(1.0 * sinh(l)) * J), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(1.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 68.3

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

   5. Applied rewrites 68.3%

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

   7. Applied rewrites 68.3%

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

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

   1. Initial program 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 93.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 93.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 93.9

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

   9. Applied rewrites 93.9%

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

Alternative 12: 86.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    (* (fma (* K K) -0.125 1.0) J)
    (* (fma 0.3333333333333333 (* l l) 2.0) l)
    U)
   (fma (* (* 1.0 (sinh l)) J) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma((fma((K * K), -0.125, 1.0) * J), (fma(0.3333333333333333, (l * l), 2.0) * l), U);
	} else {
		tmp = fma(((1.0 * sinh(l)) * J), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * J), Float64(fma(0.3333333333333333, Float64(l * l), 2.0) * l), U);
	else
		tmp = fma(Float64(Float64(1.0 * sinh(l)) * J), 2.0, U);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.3%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-/.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.2

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

   10. Applied rewrites 62.2%

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

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

   1. Initial program 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 93.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 93.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 93.9

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

   9. Applied rewrites 93.9%

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

Alternative 13: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in K around inf

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

   1. cos-neg-rev N/A

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

   2. lower-cos.f64 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

Alternative 14: 88.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 \cdot \sinh \ell\right) \cdot J, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 5e-30)
   (fma (* (* 1.0 (sinh l)) J) 2.0 U)
   (+
    (*
     (*
      J
      (*
       (fma
        (fma
         (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
         (* l l)
         0.3333333333333333)
        (* l l)
        2.0)
       l))
     (cos (/ K 2.0)))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 5e-30) {
		tmp = fma(((1.0 * sinh(l)) * J), 2.0, U);
	} else {
		tmp = ((J * (fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 5e-30)
		tmp = fma(Float64(Float64(1.0 * sinh(l)) * J), 2.0, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 5e-30], N[(N[(N[(1.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 4.99999999999999972e-30

   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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 84.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 84.5

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 84.5

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

   9. Applied rewrites 84.5%

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

   if 4.99999999999999972e-30 < K

   1. Initial program 87.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

Alternative 15: 82.8% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    (* (fma (* K K) -0.125 1.0) J)
    (* (fma 0.3333333333333333 (* l l) 2.0) l)
    U)
   (fma
    (* 1.0 J)
    (*
     (fma
      (fma
       (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
       (* l l)
       0.3333333333333333)
      (* l l)
      2.0)
     l)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma((fma((K * K), -0.125, 1.0) * J), (fma(0.3333333333333333, (l * l), 2.0) * l), U);
	} else {
		tmp = fma((1.0 * J), (fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * J), Float64(fma(0.3333333333333333, Float64(l * l), 2.0) * l), U);
	else
		tmp = fma(Float64(1.0 * J), Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l), U);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.3%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-/.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.2

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

   10. Applied rewrites 62.2%

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

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

   1. Initial program 84.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-/.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 82.1%

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

   11. Taylor expanded in l around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 87.7

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

   13. Applied rewrites 87.7%

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

Alternative 16: 88.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 \cdot \sinh \ell\right) \cdot J, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 5.2e-6)
   (fma (* (* 1.0 (sinh l)) J) 2.0 U)
   (+
    (*
     (*
      J
      (*
       (fma (fma 0.016666666666666666 (* l l) 0.3333333333333333) (* l l) 2.0)
       l))
     (cos (/ K 2.0)))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 5.2e-6) {
		tmp = fma(((1.0 * sinh(l)) * J), 2.0, U);
	} else {
		tmp = ((J * (fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 5.2e-6)
		tmp = fma(Float64(Float64(1.0 * sinh(l)) * J), 2.0, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 5.2e-6], N[(N[(N[(1.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 5.20000000000000019e-6

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 84.8

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 84.8

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

   9. Applied rewrites 84.8%

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

   if 5.20000000000000019e-6 < K

   1. Initial program 86.1%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 90.2

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

   5. Applied rewrites 90.2%

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

Alternative 17: 81.3% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma
    (* (fma (* K K) -0.125 1.0) J)
    (* (fma 0.3333333333333333 (* l l) 2.0) l)
    U)
   (fma
    (* 1.0 J)
    (*
     (fma (fma 0.016666666666666666 (* l l) 0.3333333333333333) (* l l) 2.0)
     l)
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma((fma((K * K), -0.125, 1.0) * J), (fma(0.3333333333333333, (l * l), 2.0) * l), U);
	} else {
		tmp = fma((1.0 * J), (fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * J), Float64(fma(0.3333333333333333, Float64(l * l), 2.0) * l), U);
	else
		tmp = fma(Float64(1.0 * J), Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l), U);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.3%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-/.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.2

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

   10. Applied rewrites 62.2%

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

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

   1. Initial program 84.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3}, 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. lower-*.f64 87.7

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied rewrites 87.7%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right)} + U \]

      4. lift-*.f64 N/A

         \[\leadsto \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right)} + U \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)} + U \]

      6. lift-cos.f64 N/A

         \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\frac{K}{2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      8. metadata-eval N/A

         \[\leadsto \left(\cos \left(\frac{K}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\mathsf{neg}\left(\frac{K}{-2}\right)\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{-2}}\right)\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      11. cos-neg-rev N/A

          \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{-2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{-2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      13. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{-2}\right) \cdot J\right)} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

   7. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{K}{2}\right) \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right)} \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot J, \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right) \cdot \ell, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 82.1%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right) \]

   11. Taylor expanded in l around 0

       \[\leadsto \mathsf{fma}\left(1 \cdot J, \color{blue}{\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}, U\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 \cdot J, \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) \cdot \ell}, U\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(1 \cdot J, \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) \cdot \ell}, U\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 \cdot J, \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right) + 2\right)} \cdot \ell, U\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 \cdot J, \left(\color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right) \cdot {\ell}^{2}} + 2\right) \cdot \ell, U\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(1 \cdot J, \color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}, {\ell}^{2}, 2\right)} \cdot \ell, U\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(\color{blue}{\frac{1}{60} \cdot {\ell}^{2} + \frac{1}{3}}, {\ell}^{2}, 2\right) \cdot \ell, U\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60}, {\ell}^{2}, \frac{1}{3}\right)}, {\ell}^{2}, 2\right) \cdot \ell, U\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \color{blue}{\ell \cdot \ell}, \frac{1}{3}\right), {\ell}^{2}, 2\right) \cdot \ell, U\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \color{blue}{\ell \cdot \ell}, \frac{1}{3}\right), {\ell}^{2}, 2\right) \cdot \ell, U\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \ell \cdot \ell, \frac{1}{3}\right), \color{blue}{\ell \cdot \ell}, 2\right) \cdot \ell, U\right) \]

       11. lower-*.f64 85.7

           \[\leadsto \mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \color{blue}{\ell \cdot \ell}, 2\right) \cdot \ell, U\right) \]

   13. Applied rewrites 85.7%

       \[\leadsto \mathsf{fma}\left(1 \cdot J, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell}, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 81.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 20:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 20.0)
   (fma (* (* (sinh l) (fma (* K K) -0.125 1.0)) 2.0) J U)
   (+ (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) (cos (/ K 2.0))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 20.0) {
		tmp = fma(((sinh(l) * fma((K * K), -0.125, 1.0)) * 2.0), J, U);
	} else {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 20.0)
		tmp = fma(Float64(Float64(sinh(l) * fma(Float64(K * K), -0.125, 1.0)) * 2.0), J, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 20.0], N[(N[(N[(N[Sinh[l], $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 20

   1. Initial program 85.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}\right) \cdot 2, J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2} + 1\right)}\right) \cdot 2, J, U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{8}} + 1\right)\right) \cdot 2, J, U\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{8}, 1\right)}\right) \cdot 2, J, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{8}, 1\right)\right) \cdot 2, J, U\right) \]

      5. lower-*.f64 74.4

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right)\right) \cdot 2, J, U\right) \]

   7. Applied rewrites 74.4%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}\right) \cdot 2, J, U\right) \]

   if 20 < K

   1. Initial program 87.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3}, 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. lower-*.f64 82.8

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied rewrites 82.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 77.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* (* (fma (* K K) -0.125 1.0) l) 2.0) J U)
   (fma (* 1.0 J) (* (fma 0.3333333333333333 (* l l) 2.0) l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma(((fma((K * K), -0.125, 1.0) * l) * 2.0), J, U);
	} else {
		tmp = fma((1.0 * J), (fma(0.3333333333333333, (l * l), 2.0) * l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * l) * 2.0), J, U);
	else
		tmp = fma(Float64(1.0 * J), Float64(fma(0.3333333333333333, Float64(l * l), 2.0) * l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(1.0 * J), $MachinePrecision] * N[(N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 88.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)} \cdot 2, J, U\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \ell\right)} \cdot 2, J, U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \ell\right)} \cdot 2, J, U\right) \]

      3. cos-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot \ell\right) \cdot 2, J, U\right) \]

      7. lower-*.f64 68.2

         \[\leadsto \mathsf{fma}\left(\left(\cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

   7. Applied rewrites 68.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)} \cdot 2, J, U\right) \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\ell + \color{blue}{\frac{-1}{8} \cdot \left({K}^{2} \cdot \ell\right)}\right) \cdot 2, J, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 58.1%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \color{blue}{\ell}\right) \cdot 2, J, U\right) \]

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.6%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3}, 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. lower-*.f64 87.7

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied rewrites 87.7%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right)} + U \]

      4. lift-*.f64 N/A

         \[\leadsto \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right)} + U \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)} + U \]

      6. lift-cos.f64 N/A

         \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\frac{K}{2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      8. metadata-eval N/A

         \[\leadsto \left(\cos \left(\frac{K}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\mathsf{neg}\left(\frac{K}{-2}\right)\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{-2}}\right)\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      11. cos-neg-rev N/A

          \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{-2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{-2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      13. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{-2}\right) \cdot J\right)} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

   7. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{K}{2}\right) \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right)} \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot J, \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right) \cdot \ell, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 82.1%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 75.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 4, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -5e-310)
   (fma (* (* l l) 4.0) J U)
   (fma (* 1.0 J) (* (fma 0.3333333333333333 (* l l) 2.0) l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -5e-310) {
		tmp = fma(((l * l) * 4.0), J, U);
	} else {
		tmp = fma((1.0 * J), (fma(0.3333333333333333, (l * l), 2.0) * l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -5e-310)
		tmp = fma(Float64(Float64(l * l) * 4.0), J, U);
	else
		tmp = fma(Float64(1.0 * J), Float64(fma(0.3333333333333333, Float64(l * l), 2.0) * l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -5e-310], N[(N[(N[(l * l), $MachinePrecision] * 4.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(1.0 * J), $MachinePrecision] * N[(N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -4.999999999999985e-310

   1. Initial program 88.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 68.3

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.8%

      \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 39.8%

       \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 56.4%

       \[\leadsto \mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 4, J, U\right) \]

   if -4.999999999999985e-310 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.6%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3}, 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. lower-*.f64 87.7

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied rewrites 87.7%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right)} + U \]

      4. lift-*.f64 N/A

         \[\leadsto \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right)} + U \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)} + U \]

      6. lift-cos.f64 N/A

         \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\frac{K}{2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      8. metadata-eval N/A

         \[\leadsto \left(\cos \left(\frac{K}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\mathsf{neg}\left(\frac{K}{-2}\right)\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{-2}}\right)\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      11. cos-neg-rev N/A

          \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{-2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{-2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      13. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{-2}\right) \cdot J\right)} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

   7. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{K}{2}\right) \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right)} \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot J, \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right) \cdot \ell, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 82.1%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 81.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 20:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 20.0)
   (fma (* (* (sinh l) (fma (* K K) -0.125 1.0)) 2.0) J U)
   (fma (* (cos (* 0.5 K)) J) (* (fma 0.3333333333333333 (* l l) 2.0) l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 20.0) {
		tmp = fma(((sinh(l) * fma((K * K), -0.125, 1.0)) * 2.0), J, U);
	} else {
		tmp = fma((cos((0.5 * K)) * J), (fma(0.3333333333333333, (l * l), 2.0) * l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 20.0)
		tmp = fma(Float64(Float64(sinh(l) * fma(Float64(K * K), -0.125, 1.0)) * 2.0), J, U);
	else
		tmp = fma(Float64(cos(Float64(0.5 * K)) * J), Float64(fma(0.3333333333333333, Float64(l * l), 2.0) * l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 20.0], N[(N[(N[(N[Sinh[l], $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[(N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 20

   1. Initial program 85.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}\right) \cdot 2, J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2} + 1\right)}\right) \cdot 2, J, U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{8}} + 1\right)\right) \cdot 2, J, U\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{8}, 1\right)}\right) \cdot 2, J, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{8}, 1\right)\right) \cdot 2, J, U\right) \]

      5. lower-*.f64 74.4

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right)\right) \cdot 2, J, U\right) \]

   7. Applied rewrites 74.4%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}\right) \cdot 2, J, U\right) \]

   if 20 < K

   1. Initial program 87.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3}, 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. lower-*.f64 82.8

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied rewrites 82.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right)} + U \]

      4. lift-*.f64 N/A

         \[\leadsto \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right)} + U \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)} + U \]

      6. lift-cos.f64 N/A

         \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\frac{K}{2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      8. metadata-eval N/A

         \[\leadsto \left(\cos \left(\frac{K}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\mathsf{neg}\left(\frac{K}{-2}\right)\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{-2}}\right)\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      11. cos-neg-rev N/A

          \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{-2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{-2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      13. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{-2}\right) \cdot J\right)} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

   7. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{K}{2}\right) \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right)} \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot J, \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right) \cdot \ell, U\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 82.8

         \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(0.5 \cdot K\right)} \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right) \]

   10. Applied rewrites 82.8%

       \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(0.5 \cdot K\right)} \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 81.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 20:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 20.0)
   (fma (* (* (sinh l) (fma (* K K) -0.125 1.0)) 2.0) J U)
   (fma (* (* (fma (* l l) 0.3333333333333333 2.0) l) (cos (* 0.5 K))) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 20.0) {
		tmp = fma(((sinh(l) * fma((K * K), -0.125, 1.0)) * 2.0), J, U);
	} else {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * l) * cos((0.5 * K))), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 20.0)
		tmp = fma(Float64(Float64(sinh(l) * fma(Float64(K * K), -0.125, 1.0)) * 2.0), J, U);
	else
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * cos(Float64(0.5 * K))), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 20.0], N[(N[(N[(N[Sinh[l], $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 20

   1. Initial program 85.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}\right) \cdot 2, J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2} + 1\right)}\right) \cdot 2, J, U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{8}} + 1\right)\right) \cdot 2, J, U\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{8}, 1\right)}\right) \cdot 2, J, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{8}, 1\right)\right) \cdot 2, J, U\right) \]

      5. lower-*.f64 74.4

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right)\right) \cdot 2, J, U\right) \]

   7. Applied rewrites 74.4%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}\right) \cdot 2, J, U\right) \]

   if 20 < K

   1. Initial program 87.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3}, 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. lower-*.f64 82.8

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied rewrites 82.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right)} + U \]

      4. lift-*.f64 N/A

         \[\leadsto \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right)} + U \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)} + U \]

      6. lift-cos.f64 N/A

         \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\frac{K}{2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      8. metadata-eval N/A

         \[\leadsto \left(\cos \left(\frac{K}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\mathsf{neg}\left(\frac{K}{-2}\right)\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{-2}}\right)\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      11. cos-neg-rev N/A

          \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{-2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\color{blue}{\cos \left(\frac{K}{-2}\right)} \cdot J\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

      13. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{-2}\right) \cdot J\right)} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right) + U \]

   7. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{K}{2}\right) \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right)} \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot J, \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right) \cdot \ell, U\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 82.8

         \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(0.5 \cdot K\right)} \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right) \]

   10. Applied rewrites 82.8%

       \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(0.5 \cdot K\right)} \cdot J, \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot \ell, U\right) \]
   11. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right) \cdot \left(\mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right) \cdot \ell\right) + U} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right)} + U \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot J\right)} + U \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot J} + U \]

       5. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot \cos \left(\frac{1}{2} \cdot K\right), J, U\right)} \]

   12. Applied rewrites 82.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right), J, U\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 32.9% accurate, 47.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\ell, 2, U\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (fma l 2.0 U))

C

double code(double J, double l, double K, double U) {
	return fma(l, 2.0, U);
}

Julia

function code(J, l, K, U)
	return fma(l, 2.0, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(l * 2.0 + U), $MachinePrecision]

Derivation

1. Initial program 85.6%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in 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. *-commutative N/A

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

   3. associate-*r* N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

   6. associate-*r* N/A

      \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   12. cos-neg-rev N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

   16. metadata-eval 66.9

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

5. Applied rewrites 66.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

6. Taylor expanded in K around 0

   \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
7. Step-by-step derivation

8. Applied rewrites 55.2%

   \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
9. Step-by-step derivation

10. Applied rewrites 55.2%

    \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]
11. Step-by-step derivation

12. Applied rewrites 34.1%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, 2, U\right)} \]
13. Add Preprocessing

Alternative 24: 26.6% accurate, 47.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(J, 2, U\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (fma J 2.0 U))

C

double code(double J, double l, double K, double U) {
	return fma(J, 2.0, U);
}

Julia

function code(J, l, K, U)
	return fma(J, 2.0, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(J * 2.0 + U), $MachinePrecision]

Derivation

1. Initial program 85.6%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in 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. *-commutative N/A

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

   3. associate-*r* N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

   6. associate-*r* N/A

      \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   12. cos-neg-rev N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

   16. metadata-eval 66.9

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

5. Applied rewrites 66.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

6. Taylor expanded in K around 0

   \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
7. Step-by-step derivation

8. Applied rewrites 55.2%

   \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
9. Step-by-step derivation

10. Applied rewrites 55.2%

    \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]
11. Step-by-step derivation

12. Applied rewrites 24.6%

    \[\leadsto \mathsf{fma}\left(J, 2, U\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))