Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 100.0%

Time: 9.8s

Alternatives: 14

Speedup: 1.5×

• 49.5% 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: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 89.4

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 96.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.75:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.75)
     (+
      (*
       (*
        (*
         (fma
          (fma
           (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
           (* l l)
           0.3333333333333333)
          (* l l)
          2.0)
         l)
        J)
       t_0)
      U)
     (fma (* 2.0 (* J (sinh l))) 1.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.75) {
		tmp = (((fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l) * J) * t_0) + U;
	} else {
		tmp = fma((2.0 * (J * sinh(l))), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.75)
		tmp = Float64(Float64(Float64(Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l) * J) * t_0) + U);
	else
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), 1.0, 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, 0.75], N[(N[(N[(N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.75

   1. Initial program 83.4%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if 0.75 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 93.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 93.4

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 98.7%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.75:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.75:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.75)
     (+
      (*
       (*
        (*
         (fma
          (fma 0.016666666666666666 (* l l) 0.3333333333333333)
          (* l l)
          2.0)
         l)
        J)
       t_0)
      U)
     (fma (* 2.0 (* J (sinh l))) 1.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.75) {
		tmp = (((fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l) * J) * t_0) + U;
	} else {
		tmp = fma((2.0 * (J * sinh(l))), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.75)
		tmp = Float64(Float64(Float64(Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l) * J) * t_0) + U);
	else
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), 1.0, 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, 0.75], N[(N[(N[(N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.75

   1. Initial program 83.4%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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 93.3

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

   5. Applied rewrites 93.3%

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

   if 0.75 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 93.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 93.4

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 98.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.75:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.6:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.6)
     (+ (* (* (* (fma (* l l) 0.3333333333333333 2.0) l) J) t_0) U)
     (fma (* 2.0 (* J (sinh l))) 1.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.6) {
		tmp = (((fma((l * l), 0.3333333333333333, 2.0) * l) * J) * t_0) + U;
	} else {
		tmp = fma((2.0 * (J * sinh(l))), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.6)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J) * t_0) + U);
	else
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), 1.0, 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, 0.6], N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.599999999999999978

   1. Initial program 83.4%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if 0.599999999999999978 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 93.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 93.2

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 98.5%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.6:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.335:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.335)
   (fma (* (* (fma (* l l) 0.3333333333333333 2.0) J) (cos (* 0.5 K))) l U)
   (fma (* 2.0 (* J (sinh l))) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.335) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * J) * cos((0.5 * K))), l, U);
	} else {
		tmp = fma((2.0 * (J * sinh(l))), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.335)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J) * cos(Float64(0.5 * K))), l, U);
	else
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.335], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.33500000000000002

   1. Initial program 86.4%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 84.4%

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

   if 0.33500000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 90.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 90.9

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 96.2%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.335:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(J \cdot \sinh \ell\right)\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (* J (sinh l)))))
   (if (<= (cos (/ K 2.0)) -0.002)
     (fma t_0 (fma (* K K) -0.125 1.0) U)
     (fma t_0 1.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (J * sinh(l));
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma(t_0, fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(t_0, 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * Float64(J * sinh(l)))
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(t_0, fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(t_0, 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(t$95$0 * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(t$95$0 * 1.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 87.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 87.6

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 90.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lower-fma.f64 90.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 94.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma
    (* (* (fma (* l l) 0.3333333333333333 2.0) J) l)
    (fma (* K K) -0.125 1.0)
    U)
   (fma (* 2.0 (* J (sinh l))) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * J) * l), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma((2.0 * (J * sinh(l))), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J) * l), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(2.0 * Float64(J * sinh(l))), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 87.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 87.6

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 70.6

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

   10. Applied rewrites 70.6%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 90.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lower-fma.f64 90.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 94.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(J \cdot \sinh \ell\right), 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right) \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma
    (* (* (fma (* l l) 0.3333333333333333 2.0) J) l)
    (fma (* K K) -0.125 1.0)
    U)
   (fma
    (*
     (*
      (fma
       (* (fma 0.008333333333333333 (* l l) 0.16666666666666666) J)
       (* l l)
       J)
      l)
     2.0)
    1.0
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * J) * l), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(((fma((fma(0.008333333333333333, (l * l), 0.16666666666666666) * J), (l * l), J) * l) * 2.0), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J) * l), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(l * l), 0.16666666666666666) * J), Float64(l * l), J) * l) * 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], -0.002], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.008333333333333333 * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * J), $MachinePrecision] * N[(l * l), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 87.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 87.6

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 70.6

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

   10. Applied rewrites 70.6%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 90.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lower-fma.f64 90.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 83.1

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \color{blue}{\ell \cdot \ell}, J\right) \cdot \ell\right) \cdot 2, 1, U\right) \]

   10. Applied rewrites 83.1%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, J\right) \cdot \ell\right)} \cdot 2, 1, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right) \cdot J, \ell \cdot \ell, J\right) \cdot \ell\right) \cdot 2, 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666, \left(\ell \cdot \ell\right) \cdot J, J\right) \cdot \ell\right) \cdot 2, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma
    (* (* (fma (* l l) 0.3333333333333333 2.0) J) l)
    (fma (* K K) -0.125 1.0)
    U)
   (fma (* (* (fma 0.16666666666666666 (* (* l l) J) J) l) 2.0) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma(((fma((l * l), 0.3333333333333333, 2.0) * J) * l), fma((K * K), -0.125, 1.0), U);
	} else {
		tmp = fma(((fma(0.16666666666666666, ((l * l) * J), J) * l) * 2.0), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * J) * l), fma(Float64(K * K), -0.125, 1.0), U);
	else
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(Float64(l * l) * J), J) * l) * 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], -0.002], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 87.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 87.6

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 70.6

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

   10. Applied rewrites 70.6%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 90.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lower-fma.f64 90.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 77.4

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{\left(\ell \cdot \ell\right)} \cdot J, J\right) \cdot \ell\right) \cdot 2, 1, U\right) \]

   10. Applied rewrites 77.4%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, \left(\ell \cdot \ell\right) \cdot J, J\right) \cdot \ell\right)} \cdot 2, 1, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot J\right) \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666, \left(\ell \cdot \ell\right) \cdot J, J\right) \cdot \ell\right) \cdot 2, 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666, \left(\ell \cdot \ell\right) \cdot J, J\right) \cdot \ell\right) \cdot 2, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (fma (* (* (fma 0.16666666666666666 (* (* l l) J) J) l) 2.0) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma(((fma(0.16666666666666666, ((l * l) * J), J) * l) * 2.0), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(Float64(l * l) * J), J) * l) * 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], -0.002], N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * N[(N[(l * l), $MachinePrecision] * J), $MachinePrecision] + J), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 63.6%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 90.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lower-fma.f64 90.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 77.4

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{\left(\ell \cdot \ell\right)} \cdot J, J\right) \cdot \ell\right) \cdot 2, 1, U\right) \]

   10. Applied rewrites 77.4%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, \left(\ell \cdot \ell\right) \cdot J, J\right) \cdot \ell\right)} \cdot 2, 1, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666, \left(\ell \cdot \ell\right) \cdot J, J\right) \cdot \ell\right) \cdot 2, 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (fma (* (* (fma 0.3333333333333333 (* l l) 2.0) J) l) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma(((fma(0.3333333333333333, (l * l), 2.0) * J) * l), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = fma(Float64(Float64(fma(0.3333333333333333, Float64(l * l), 2.0) * J) * l), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 63.6%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 90.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. lower-fma.f64 90.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 77.4

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

   10. Applied rewrites 77.4%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, 2\right) \cdot J\right) \cdot \ell, 1, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (fma (* J l) (fma (* K K) -0.25 2.0) U)
   (fma (* J l) 2.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.002) {
		tmp = fma((J * l), fma((K * K), -0.25, 2.0), U);
	} else {
		tmp = fma((J * l), 2.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.002)
		tmp = fma(Float64(J * l), fma(Float64(K * K), -0.25, 2.0), U);
	else
		tmp = fma(Float64(J * l), 2.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.002], N[(N[(J * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.25 + 2.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(J * l), $MachinePrecision] * 2.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -2e-3

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 63.6%

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

   if -2e-3 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 90.0%

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 67.9

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 62.7%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, \mathsf{fma}\left(K \cdot K, -0.25, 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot \ell, 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.8% accurate, 27.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 63.8

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

5. Applied rewrites 63.8%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 51.9%

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

9. Final simplification 51.9%

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

Alternative 14: 19.4% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = (2.0d0 * l) * j
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.4%

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

3. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 63.8

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

5. Applied rewrites 63.8%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 51.9%

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

9. Taylor expanded in U around 0

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

11. Applied rewrites 18.7%

    \[\leadsto \left(\ell \cdot J\right) \cdot 2 \]
12. Step-by-step derivation

13. Applied rewrites 18.7%

    \[\leadsto \left(2 \cdot \ell\right) \cdot J \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))