Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 99.9%

Time: 5.3s

Alternatives: 13

Speedup: 1.2×

• 49.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

3. Applied rewrites 99.9%

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

Alternative 2: 88.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.816:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{elif}\;t\_0 \leq 0.39:\\ \;\;\;\;-U \cdot \mathsf{fma}\left(J \cdot \frac{\cos \left(K \cdot 0.5\right) \cdot \ell}{U}, -2, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell, J + J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.816)
     (+ (* (* J (- (exp l) (exp (- l)))) (fma (* K K) -0.125 1.0)) U)
     (if (<= t_0 0.39)
       (- (* U (fma (* J (/ (* (cos (* K 0.5)) l) U)) -2.0 -1.0)))
       (fma (sinh l) (+ J J) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.816) {
		tmp = ((J * (exp(l) - exp(-l))) * fma((K * K), -0.125, 1.0)) + U;
	} else if (t_0 <= 0.39) {
		tmp = -(U * fma((J * ((cos((K * 0.5)) * l) / U)), -2.0, -1.0));
	} else {
		tmp = fma(sinh(l), (J + J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.816)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * fma(Float64(K * K), -0.125, 1.0)) + U);
	elseif (t_0 <= 0.39)
		tmp = Float64(-Float64(U * fma(Float64(J * Float64(Float64(cos(Float64(K * 0.5)) * l) / U)), -2.0, -1.0)));
	else
		tmp = fma(sinh(l), Float64(J + J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.816], N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 0.39], (-N[(U * N[(N[(J * N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.815999999999999948

   1. Initial program 86.4%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left({K}^{2} \cdot \frac{-1}{8} + 1\right) + U \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   4. Applied rewrites 64.5%

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

   if -0.815999999999999948 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.39000000000000001

   1. Initial program 86.4%

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

   2. Taylor expanded in U around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}{U} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}{U} - 1\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}{U} - 1\right) \]

      4. sub-negate-rev N/A

         \[\leadsto -U \cdot \left(\mathsf{neg}\left(\left(1 - -1 \cdot \frac{J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}{U}\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto -U \cdot \left(\mathsf{neg}\left(\left(1 - \left(\mathsf{neg}\left(\frac{J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}{U}\right)\right)\right)\right)\right) \]

      6. add-flip N/A

         \[\leadsto -U \cdot \left(\mathsf{neg}\left(\left(1 + \frac{J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}{U}\right)\right)\right) \]

      7. add-negate N/A

         \[\leadsto -U \cdot \left(\left(\mathsf{neg}\left(1\right)\right) - \frac{J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}{U}\right) \]

      8. metadata-eval N/A

         \[\leadsto -U \cdot \left(-1 - \frac{J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}{U}\right) \]

   4. Applied rewrites 98.0%

      \[\leadsto \color{blue}{-U \cdot \left(-1 - \frac{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot J}{U}\right)} \]

   5. Taylor expanded in l around 0

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

      1. sub-flip N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 72.8

          \[\leadsto -U \cdot \mathsf{fma}\left(J \cdot \frac{\cos \left(K \cdot 0.5\right) \cdot \ell}{U}, -2, -1\right) \]

   7. Applied rewrites 72.8%

      \[\leadsto -U \cdot \mathsf{fma}\left(J \cdot \frac{\cos \left(K \cdot 0.5\right) \cdot \ell}{U}, -2, -1\right) \]

   if 0.39000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef-rev N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.2

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

   6. Applied rewrites 80.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sinh.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 80.2

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

   8. Applied rewrites 80.2%

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

Alternative 3: 87.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.75:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{elif}\;t\_0 \leq 0.39:\\ \;\;\;\;\mathsf{fma}\left(J + J, \cos \left(0.5 \cdot K\right) \cdot \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell, J + J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.75)
     (+ (* (* J (- (exp l) (exp (- l)))) (fma (* K K) -0.125 1.0)) U)
     (if (<= t_0 0.39)
       (fma (+ J J) (* (cos (* 0.5 K)) l) U)
       (fma (sinh l) (+ J J) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.75) {
		tmp = ((J * (exp(l) - exp(-l))) * fma((K * K), -0.125, 1.0)) + U;
	} else if (t_0 <= 0.39) {
		tmp = fma((J + J), (cos((0.5 * K)) * l), U);
	} else {
		tmp = fma(sinh(l), (J + J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.75)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * fma(Float64(K * K), -0.125, 1.0)) + U);
	elseif (t_0 <= 0.39)
		tmp = fma(Float64(J + J), Float64(cos(Float64(0.5 * K)) * l), U);
	else
		tmp = fma(sinh(l), Float64(J + J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.75], N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 0.39], N[(N[(J + J), $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] + U), $MachinePrecision], N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 86.4%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left({K}^{2} \cdot \frac{-1}{8} + 1\right) + U \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   4. Applied rewrites 64.5%

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

   if -0.75 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.39000000000000001

   1. Initial program 86.4%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. mult-flip N/A

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

      11. lift-cos.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 64.3

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

   4. Applied rewrites 64.3%

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

   if 0.39000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef-rev N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.2

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

   6. Applied rewrites 80.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sinh.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 80.2

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

   8. Applied rewrites 80.2%

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

Alternative 4: 87.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell, J + J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (+ (* (* J (- (exp l) (exp (- l)))) (fma (* K K) -0.125 1.0)) U)
   (fma (sinh l) (+ J J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.1) {
		tmp = ((J * (exp(l) - exp(-l))) * fma((K * K), -0.125, 1.0)) + U;
	} else {
		tmp = fma(sinh(l), (J + J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.1)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * fma(Float64(K * K), -0.125, 1.0)) + U);
	else
		tmp = fma(sinh(l), Float64(J + J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.1], N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 86.4%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left({K}^{2} \cdot \frac{-1}{8} + 1\right) + U \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   4. Applied rewrites 64.5%

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

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

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef-rev N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.2

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

   6. Applied rewrites 80.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sinh.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 80.2

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

   8. Applied rewrites 80.2%

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

Alternative 5: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.74:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\sinh \ell \cdot 2\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell, J + J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.74)
   (fma J (* (* (sinh l) 2.0) (fma (* K K) -0.125 1.0)) U)
   (fma (sinh l) (+ J J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.74) {
		tmp = fma(J, ((sinh(l) * 2.0) * fma((K * K), -0.125, 1.0)), U);
	} else {
		tmp = fma(sinh(l), (J + J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.74)
		tmp = fma(J, Float64(Float64(sinh(l) * 2.0) * fma(Float64(K * K), -0.125, 1.0)), U);
	else
		tmp = fma(sinh(l), Float64(J + J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.74], N[(J * N[(N[(N[Sinh[l], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.73999999999999999

   1. Initial program 86.4%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left({K}^{2} \cdot \frac{-1}{8} + 1\right) + U \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   4. Applied rewrites 64.5%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)} + U \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sinh-undef-rev N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 69.2%

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

   if -0.73999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef-rev N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.2

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

   6. Applied rewrites 80.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sinh.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 80.2

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

   8. Applied rewrites 80.2%

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

Alternative 6: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.74:\\ \;\;\;\;\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot J\right) \cdot 0.3333333333333333\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell, J + J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.74)
   (+
    (* (* (* (* (* l l) l) J) 0.3333333333333333) (fma (* K K) -0.125 1.0))
    U)
   (fma (sinh l) (+ J J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.74) {
		tmp = (((((l * l) * l) * J) * 0.3333333333333333) * fma((K * K), -0.125, 1.0)) + U;
	} else {
		tmp = fma(sinh(l), (J + J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.74)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(l * l) * l) * J) * 0.3333333333333333) * fma(Float64(K * K), -0.125, 1.0)) + U);
	else
		tmp = fma(sinh(l), Float64(J + J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.74], N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.73999999999999999

   1. Initial program 86.4%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left({K}^{2} \cdot \frac{-1}{8} + 1\right) + U \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 60.8

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

   7. Applied rewrites 60.8%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 57.1

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

   10. Applied rewrites 57.1%

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

   if -0.73999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef-rev N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.2

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

   6. Applied rewrites 80.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sinh.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 80.2

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

   8. Applied rewrites 80.2%

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

Alternative 7: 82.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.74:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, \left(K \cdot K\right) \cdot -0.125, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell, J + J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.74)
   (fma (* (+ J J) l) (* (* K K) -0.125) U)
   (fma (sinh l) (+ J J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.74) {
		tmp = fma(((J + J) * l), ((K * K) * -0.125), U);
	} else {
		tmp = fma(sinh(l), (J + J), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.74)
		tmp = fma(Float64(Float64(J + J) * l), Float64(Float64(K * K) * -0.125), U);
	else
		tmp = fma(sinh(l), Float64(J + J), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.74], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] + U), $MachinePrecision], N[(N[Sinh[l], $MachinePrecision] * N[(J + J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.73999999999999999

   1. Initial program 86.4%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left({K}^{2} \cdot \frac{-1}{8} + 1\right) + U \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 48.9

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

   7. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\left(\left(J + J\right) \cdot \ell\right)} \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 48.9

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

   9. Applied rewrites 48.9%

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

   10. Taylor expanded in K around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 35.4

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

   12. Applied rewrites 35.4%

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

   if -0.73999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef-rev N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.2

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

   6. Applied rewrites 80.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sinh.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 80.2

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

   8. Applied rewrites 80.2%

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

Alternative 8: 73.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.74:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, \left(K \cdot K\right) \cdot -0.125, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.74)
   (fma (* (+ J J) l) (* (* K K) -0.125) U)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.74) {
		tmp = fma(((J + J) * l), ((K * K) * -0.125), U);
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.74)
		tmp = fma(Float64(Float64(J + J) * l), Float64(Float64(K * K) * -0.125), U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.74], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.73999999999999999

   1. Initial program 86.4%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left({K}^{2} \cdot \frac{-1}{8} + 1\right) + U \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 48.9

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

   7. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\left(\left(J + J\right) \cdot \ell\right)} \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right) + U \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 48.9

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

   9. Applied rewrites 48.9%

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

   10. Taylor expanded in K around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 35.4

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

   12. Applied rewrites 35.4%

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

   if -0.73999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef-rev N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.2

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. sinh-undef-rev N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 71.5

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

   9. Applied rewrites 71.5%

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

Alternative 9: 71.5% accurate, 4.1× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in K around 0

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

   1. +-commutative N/A

      \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

   2. *-commutative N/A

      \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

   3. rec-exp N/A

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

   4. mul-1-neg N/A

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

   5. mul-1-neg N/A

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

   6. sinh-undef-rev N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sinh.f64 80.2

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

6. Applied rewrites 80.2%

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

7. Taylor expanded in l around 0

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

   1. *-commutative N/A

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

   2. sinh-undef-rev N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 71.5

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

9. Applied rewrites 71.5%

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

Alternative 10: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\ell \cdot J}{U}, 2, 1\right) \cdot U\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, 2, \frac{U}{J}\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))))
   (if (<= t_0 (- INFINITY))
     (* (fma (/ (* l J) U) 2.0 1.0) U)
     (if (<= t_0 2e-61) (fma (+ J J) l U) (* (fma l 2.0 (/ U J)) J)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((l * J) / U), 2.0, 1.0) * U;
	} else if (t_0 <= 2e-61) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = fma(l, 2.0, (U / J)) * J;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(l * J) / U), 2.0, 1.0) * U);
	elseif (t_0 <= 2e-61)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = Float64(fma(l, 2.0, Float64(U / J)) * J);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(l * J), $MachinePrecision] / U), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision] * U), $MachinePrecision], If[LessEqual[t$95$0, 2e-61], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(l * 2.0 + N[(U / J), $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -inf.0

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef-rev N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.2

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 54.1

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

   9. Applied rewrites 54.1%

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

   10. Taylor expanded in U around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 57.7

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

   12. Applied rewrites 57.7%

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

   if -inf.0 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 2.0000000000000001e-61

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef-rev N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.2

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 54.1

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

   9. Applied rewrites 54.1%

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

   if 2.0000000000000001e-61 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef-rev N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.2

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 54.1

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

   9. Applied rewrites 54.1%

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

   10. Taylor expanded in J around inf

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

       1. *-commutative N/A

          \[\leadsto \left(2 \cdot \ell + \frac{U}{J}\right) \cdot J \]

       2. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \ell + \frac{U}{J}\right) \cdot J \]

       3. *-commutative N/A

          \[\leadsto \left(\ell \cdot 2 + \frac{U}{J}\right) \cdot J \]

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 50.7

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

   12. Applied rewrites 50.7%

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

Alternative 11: 55.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 0.0

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef-rev N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.2

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 54.1

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

   9. Applied rewrites 54.1%

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

   if 0.0 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

      3. rec-exp N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. sinh-undef-rev N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sinh.f64 80.2

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 54.1

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

   9. Applied rewrites 54.1%

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

   10. Taylor expanded in J around inf

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

       1. *-commutative N/A

          \[\leadsto \left(2 \cdot \ell + \frac{U}{J}\right) \cdot J \]

       2. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \ell + \frac{U}{J}\right) \cdot J \]

       3. *-commutative N/A

          \[\leadsto \left(\ell \cdot 2 + \frac{U}{J}\right) \cdot J \]

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 50.7

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

   12. Applied rewrites 50.7%

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

Alternative 12: 54.1% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in K around 0

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

   1. +-commutative N/A

      \[\leadsto J \cdot \left(e^{\ell} - \frac{1}{e^{\ell}}\right) + \color{blue}{U} \]

   2. *-commutative N/A

      \[\leadsto \left(e^{\ell} - \frac{1}{e^{\ell}}\right) \cdot J + U \]

   3. rec-exp N/A

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

   4. mul-1-neg N/A

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

   5. mul-1-neg N/A

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

   6. sinh-undef-rev N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sinh.f64 80.2

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

6. Applied rewrites 80.2%

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

7. Taylor expanded in l around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. count-2-rev N/A

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

   5. lift-+.f64 54.1

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

9. Applied rewrites 54.1%

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

Alternative 13: 37.1% accurate, 68.7× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 86.4%

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

2. Taylor expanded in J around 0

   \[\leadsto \color{blue}{U} \]
3. Step-by-step derivation

4. Applied rewrites 37.1%

   \[\leadsto \color{blue}{U} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025132 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))