Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 99.9%

Time: 9.2s

Alternatives: 21

Speedup: 1.4×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.4% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. 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. Add Preprocessing

Alternative 2: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+290}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+236}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot 2\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0)))))
   (if (<= t_0 -1e+290)
     (fma (* (* (sinh l) 1.0) 2.0) J U)
     (if (<= t_0 2e+236)
       (fma (* (cos (* 0.5 K)) (* J 2.0)) l U)
       (fma (* (* (sinh l) (fma (* K K) -0.125 1.0)) 2.0) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double tmp;
	if (t_0 <= -1e+290) {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	} else if (t_0 <= 2e+236) {
		tmp = fma((cos((0.5 * K)) * (J * 2.0)), l, U);
	} else {
		tmp = fma(((sinh(l) * fma((K * K), -0.125, 1.0)) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0)))
	tmp = 0.0
	if (t_0 <= -1e+290)
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	elseif (t_0 <= 2e+236)
		tmp = fma(Float64(cos(Float64(0.5 * K)) * Float64(J * 2.0)), l, U);
	else
		tmp = fma(Float64(Float64(sinh(l) * fma(Float64(K * K), -0.125, 1.0)) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+290], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[t$95$0, 2e+236], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -1.00000000000000006e290

   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 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 80.8%

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

   if -1.00000000000000006e290 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 2.00000000000000011e236

   1. Initial program 75.8%

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

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

   if 2.00000000000000011e236 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 99.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. 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 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 81.9

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

   7. Applied rewrites 81.9%

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

Alternative 3: 58.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+236}:\\ \;\;\;\;\left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)\right) \cdot J\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \left(4 \cdot \left(J \cdot J\right)\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))))
   (if (<= t_0 -1e+236)
     (* (* l (fma -0.25 (* K K) 2.0)) J)
     (if (<= t_0 2e+287)
       (fma (+ l l) J U)
       (fma (* (* J 2.0) (* 4.0 (* J J))) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double tmp;
	if (t_0 <= -1e+236) {
		tmp = (l * fma(-0.25, (K * K), 2.0)) * J;
	} else if (t_0 <= 2e+287) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = fma(((J * 2.0) * (4.0 * (J * J))), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if (t_0 <= -1e+236)
		tmp = Float64(Float64(l * fma(-0.25, Float64(K * K), 2.0)) * J);
	elseif (t_0 <= 2e+287)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = fma(Float64(Float64(J * 2.0) * Float64(4.0 * Float64(J * J))), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+236], N[(N[(l * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[t$95$0, 2e+287], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * 2.0), $MachinePrecision] * N[(4.0 * N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -1.00000000000000005e236

   1. Initial program 99.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 20.2

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

   5. Applied rewrites 20.2%

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

   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 17.9%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 29.7%

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

   12. Taylor expanded in J around inf

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

   14. Applied rewrites 34.9%

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

   if -1.00000000000000005e236 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 2.0000000000000002e287

   1. Initial program 75.9%

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

   3. 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 99.7

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

   5. Applied rewrites 99.7%

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

   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 89.6%

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

   10. Applied rewrites 89.6%

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

   if 2.0000000000000002e287 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 30.9

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

   5. Applied rewrites 30.9%

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

   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 23.8%

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

   10. Applied rewrites 23.8%

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

   12. Applied rewrites 36.3%

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

Alternative 4: 58.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+236}:\\ \;\;\;\;\left(\ell \cdot \mathsf{fma}\left(-0.25, K \cdot K, 2\right)\right) \cdot J\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \left|J\right|, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))))
   (if (<= t_0 -1e+236)
     (* (* l (fma -0.25 (* K K) 2.0)) J)
     (if (<= t_0 2e+287) (fma (+ l l) J U) (fma (* (* J 2.0) (fabs J)) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double tmp;
	if (t_0 <= -1e+236) {
		tmp = (l * fma(-0.25, (K * K), 2.0)) * J;
	} else if (t_0 <= 2e+287) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = fma(((J * 2.0) * fabs(J)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if (t_0 <= -1e+236)
		tmp = Float64(Float64(l * fma(-0.25, Float64(K * K), 2.0)) * J);
	elseif (t_0 <= 2e+287)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = fma(Float64(Float64(J * 2.0) * abs(J)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+236], N[(N[(l * N[(-0.25 * N[(K * K), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[t$95$0, 2e+287], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * 2.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -1.00000000000000005e236

   1. Initial program 99.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 20.2

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

   5. Applied rewrites 20.2%

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

   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 17.9%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 29.7%

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

   12. Taylor expanded in J around inf

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

   14. Applied rewrites 34.9%

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

   if -1.00000000000000005e236 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 2.0000000000000002e287

   1. Initial program 75.9%

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

   3. 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 99.7

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

   5. Applied rewrites 99.7%

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

   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 89.6%

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

   10. Applied rewrites 89.6%

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

   if 2.0000000000000002e287 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 30.9

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

   5. Applied rewrites 30.9%

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

   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 23.8%

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

   10. Applied rewrites 23.8%

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

   12. Applied rewrites 34.7%

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

Alternative 5: 57.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \left|J\right|, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U) 5e+302)
   (fma (+ l l) J U)
   (fma (* (* J 2.0) (fabs 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) <= 5e+302) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = fma(((J * 2.0) * fabs(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) <= 5e+302)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = fma(Float64(Float64(J * 2.0) * abs(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], 5e+302], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * 2.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $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) < 5e302

   1. Initial program 82.2%

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

   3. Taylor expanded in 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 80.4

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

   5. Applied rewrites 80.4%

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

   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 70.4%

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

   10. Applied rewrites 70.4%

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

   if 5e302 < (+.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 23.5

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

   5. Applied rewrites 23.5%

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

   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 23.0%

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

   10. Applied rewrites 23.0%

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

   12. Applied rewrites 37.0%

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

Alternative 6: 39.0% 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 82.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 79.9

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

   5. Applied rewrites 79.9%

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

   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 70.2%

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

   10. Applied rewrites 70.2%

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

   12. Applied rewrites 49.6%

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

   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 99.2%

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

   3. Taylor expanded in 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 29.9

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

   5. Applied rewrites 29.9%

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

   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 28.0%

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

   10. Applied rewrites 28.0%

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

   12. Applied rewrites 28.8%

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

Alternative 7: 96.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.775:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.775)
     (+
      (*
       (*
        J
        (*
         (fma
          (fma
           (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
           (* l l)
           0.3333333333333333)
          (* l l)
          2.0)
         l))
       t_0)
      U)
     (fma (* (* (sinh l) 1.0) 2.0) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.775) {
		tmp = ((J * (fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * t_0) + U;
	} else {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.775)
		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)) * t_0) + U);
	else
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.775], 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] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.775000000000000022

   1. Initial program 82.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 94.6

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

   5. Applied rewrites 94.6%

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

   if 0.775000000000000022 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 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 99.8%

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

Alternative 8: 95.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.775:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.775)
     (+
      (*
       (*
        J
        (*
         (fma
          (fma 0.016666666666666666 (* l l) 0.3333333333333333)
          (* l l)
          2.0)
         l))
       t_0)
      U)
     (fma (* (* (sinh l) 1.0) 2.0) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.775) {
		tmp = ((J * (fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * t_0) + U;
	} else {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.775)
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * t_0) + U);
	else
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.775], N[(N[(N[(J * N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.775000000000000022

   1. Initial program 82.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if 0.775000000000000022 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 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 99.8%

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

Alternative 9: 93.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.66:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.66)
     (+ (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) t_0) U)
     (fma (* (* (sinh l) 1.0) 2.0) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.66) {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * t_0) + U;
	} else {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.66)
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * t_0) + U);
	else
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.66], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.660000000000000031

   1. Initial program 82.3%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if 0.660000000000000031 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 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 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 99.3%

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

Alternative 10: 92.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.66:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.66)
   (fma (* (cos (* -0.5 K)) (* J (fma (* l l) 0.3333333333333333 2.0))) l U)
   (fma (* (* (sinh l) 1.0) 2.0) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.66) {
		tmp = fma((cos((-0.5 * K)) * (J * fma((l * l), 0.3333333333333333, 2.0))), l, U);
	} else {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.66)
		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(sinh(l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.66], 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[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.660000000000000031

   1. Initial program 82.3%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. 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 87.2%

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

   if 0.660000000000000031 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 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 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 99.3%

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

Alternative 11: 87.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.1)
   (fma (* (+ J J) l) (cos (* -0.5 K)) U)
   (fma (* (* (sinh l) 1.0) 2.0) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.1) {
		tmp = fma(((J + J) * l), cos((-0.5 * K)), U);
	} else {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.1)
		tmp = fma(Float64(Float64(J + J) * l), cos(Float64(-0.5 * K)), U);
	else
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.1], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.10000000000000001

   1. Initial program 89.4%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

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

   if 0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 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 95.8%

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

Alternative 12: 83.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{4 \cdot \left(\ell \cdot \ell\right)}, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.1)
   (fma (sqrt (* 4.0 (* l l))) J U)
   (fma (* (* (sinh l) 1.0) 2.0) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.1) {
		tmp = fma(sqrt((4.0 * (l * l))), J, U);
	} else {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.1)
		tmp = fma(sqrt(Float64(4.0 * Float64(l * l))), J, U);
	else
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.1], N[(N[Sqrt[N[(4.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.10000000000000001

   1. Initial program 89.4%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 71.5

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

   5. Applied rewrites 71.5%

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

   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 52.5%

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

   10. Applied rewrites 61.7%

       \[\leadsto \mathsf{fma}\left(\sqrt{4 \cdot \left(\ell \cdot \ell\right)}, J, U\right) \]

   if 0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 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 95.8%

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

Alternative 13: 77.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{4 \cdot \left(\ell \cdot \ell\right)}, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right)\right) \cdot \ell + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.1)
   (fma (sqrt (* 4.0 (* l l))) J U)
   (+
    (*
     (*
      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)) <= 0.1) {
		tmp = fma(sqrt((4.0 * (l * l))), J, U);
	} else {
		tmp = ((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)) <= 0.1)
		tmp = fma(sqrt(Float64(4.0 * Float64(l * l))), J, U);
	else
		tmp = Float64(Float64(Float64(J * 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], 0.1], N[(N[Sqrt[N[(4.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.10000000000000001

   1. Initial program 89.4%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 71.5

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

   5. Applied rewrites 71.5%

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

   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 52.5%

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

   10. Applied rewrites 61.7%

       \[\leadsto \mathsf{fma}\left(\sqrt{4 \cdot \left(\ell \cdot \ell\right)}, J, U\right) \]

   if 0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.5%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 88.7%

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

Alternative 14: 73.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{4 \cdot \left(\ell \cdot \ell\right)}, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot J, 0.3333333333333333, 2 \cdot J\right) \cdot \ell + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.1)
   (fma (sqrt (* 4.0 (* l l))) J U)
   (+ (* (fma (* (* l l) J) 0.3333333333333333 (* 2.0 J)) l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.1) {
		tmp = fma(sqrt((4.0 * (l * l))), J, U);
	} else {
		tmp = (fma(((l * l) * J), 0.3333333333333333, (2.0 * J)) * l) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.1)
		tmp = fma(sqrt(Float64(4.0 * Float64(l * l))), J, U);
	else
		tmp = Float64(Float64(fma(Float64(Float64(l * l) * J), 0.3333333333333333, Float64(2.0 * J)) * l) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.1], N[(N[Sqrt[N[(4.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision] * 0.3333333333333333 + N[(2.0 * J), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.10000000000000001

   1. Initial program 89.4%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 71.5

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

   5. Applied rewrites 71.5%

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

   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 52.5%

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

   10. Applied rewrites 61.7%

       \[\leadsto \mathsf{fma}\left(\sqrt{4 \cdot \left(\ell \cdot \ell\right)}, J, U\right) \]

   if 0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.5%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 88.7%

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

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 80.9%

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

Alternative 15: 73.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{4 \cdot \left(\ell \cdot \ell\right)}, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333 \cdot J, \ell \cdot \ell, J\right) + J\right) \cdot \ell + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.1)
   (fma (sqrt (* 4.0 (* l l))) J U)
   (+ (* (+ (fma (* 0.3333333333333333 J) (* l l) J) J) l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.1) {
		tmp = fma(sqrt((4.0 * (l * l))), J, U);
	} else {
		tmp = ((fma((0.3333333333333333 * J), (l * l), J) + J) * l) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.1)
		tmp = fma(sqrt(Float64(4.0 * Float64(l * l))), J, U);
	else
		tmp = Float64(Float64(Float64(fma(Float64(0.3333333333333333 * J), Float64(l * l), J) + J) * l) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.1], N[(N[Sqrt[N[(4.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 * J), $MachinePrecision] * N[(l * l), $MachinePrecision] + J), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.10000000000000001

   1. Initial program 89.4%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   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 71.5

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

   5. Applied rewrites 71.5%

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

   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 52.5%

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

   10. Applied rewrites 61.7%

       \[\leadsto \mathsf{fma}\left(\sqrt{4 \cdot \left(\ell \cdot \ell\right)}, J, U\right) \]

   if 0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.5%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 88.7%

      \[\leadsto \left(J \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right)\right) \cdot \ell + U \]
   9. Step-by-step derivation

   10. Applied rewrites 88.2%

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

   11. Taylor expanded in l around 0

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

   13. Applied rewrites 80.8%

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

Alternative 16: 60.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.35:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{4 \cdot \left(\ell \cdot \ell\right)}, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J \cdot \ell}{U}, 2, 1\right) \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.35)
   (fma (sqrt (* 4.0 (* l l))) J U)
   (* (fma (/ (* J l) U) 2.0 1.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.35) {
		tmp = fma(sqrt((4.0 * (l * l))), J, U);
	} else {
		tmp = fma(((J * l) / U), 2.0, 1.0) * U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.35)
		tmp = fma(sqrt(Float64(4.0 * Float64(l * l))), J, U);
	else
		tmp = Float64(fma(Float64(Float64(J * l) / U), 2.0, 1.0) * U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.35], N[(N[Sqrt[N[(4.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[(J * l), $MachinePrecision] / U), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision] * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.34999999999999998

   1. Initial program 89.5%

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

   3. Taylor expanded in 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 67.5

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

   5. Applied rewrites 67.5%

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

   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 50.1%

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

   10. Applied rewrites 60.2%

       \[\leadsto \mathsf{fma}\left(\sqrt{4 \cdot \left(\ell \cdot \ell\right)}, J, U\right) \]

   if 0.34999999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.2%

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

   3. Taylor expanded in 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 66.8

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

   5. Applied rewrites 66.8%

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

   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 62.8%

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

   9. Taylor expanded in U around inf

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

   11. Applied rewrites 66.5%

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

Alternative 17: 55.8% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|J \cdot \ell\right| \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 10.0) (fma (+ l l) J U) (fma (* (fabs (* J l)) 2.0) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 10.0) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = fma((fabs((J * l)) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 10.0)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = fma(Float64(abs(Float64(J * l)) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 10.0], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[Abs[N[(J * l), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 10

   1. Initial program 82.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 76.1

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

   5. Applied rewrites 76.1%

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

   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 68.3%

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

   10. Applied rewrites 68.3%

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

   if 10 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 34.0

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

   5. Applied rewrites 34.0%

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

   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 26.4%

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

   10. Applied rewrites 26.4%

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

   12. Applied rewrites 37.9%

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

Alternative 18: 54.7% accurate, 14.3× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 10.0) (fma (+ l l) J U) (fma (* (* J 2.0) J) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 10.0) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = fma(((J * 2.0) * J), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 10.0)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = fma(Float64(Float64(J * 2.0) * J), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 10.0], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * 2.0), $MachinePrecision] * J), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 10

   1. Initial program 82.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 76.1

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

   5. Applied rewrites 76.1%

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

   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 68.3%

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

   10. Applied rewrites 68.3%

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

   if 10 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 34.0

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

   5. Applied rewrites 34.0%

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

   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 26.4%

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

   10. Applied rewrites 26.4%

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

   12. Applied rewrites 34.4%

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

Alternative 19: 53.5% accurate, 33.0× speedup

Math

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

FPCore

(FPCore (J l K U) :precision binary64 (fma (+ l l) J U))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
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 67.0

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

5. Applied rewrites 67.0%

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

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 59.3%

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

10. Applied rewrites 59.3%

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

Alternative 20: 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 86.4%

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

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
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 67.0

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

5. Applied rewrites 67.0%

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

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 59.3%

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

10. Applied rewrites 59.3%

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

12. Applied rewrites 39.1%

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

Alternative 21: 27.3% 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 86.4%

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

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
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 67.0

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

5. Applied rewrites 67.0%

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

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 59.3%

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

10. Applied rewrites 59.3%

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

12. Applied rewrites 32.8%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))