Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.9% → 99.9%

Time: 3.9s

Alternatives: 11

Speedup: 1.2×

• 49.8% 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: 85.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

Alternative 2: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left(1 + -0.125 \cdot {K}^{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+ (* (* J (- (exp l) (exp (- l)))) (+ 1.0 (* -0.125 (pow K 2.0)))) U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = ((J * (exp(l) - exp(-l))) * (1.0 + (-0.125 * pow(K, 2.0)))) + U;
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * Float64(1.0 + Float64(-0.125 * (K ^ 2.0)))) + U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 85.9%

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

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.2

         \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right) + U \]

   4. Applied rewrites 64.2%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. lift-sinh.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 80.5

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

   6. Applied rewrites 80.5%

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

Alternative 3: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right), J + J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma (* (sinh l) (fma (* K K) -0.125 1.0)) (+ J J) U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma((sinh(l) * fma((K * K), -0.125, 1.0)), (J + J), U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(Float64(sinh(l) * fma(Float64(K * K), -0.125, 1.0)), Float64(J + J), U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[Sinh[l], $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(J + J), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 85.9%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 68.7

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

   6. Applied rewrites 68.7%

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} \cdot J\right) \cdot \sinh \ell, 2, U\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

   8. Applied rewrites 69.2%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. lift-sinh.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 80.5

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

   6. Applied rewrites 80.5%

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

Alternative 4: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{e^{\ell}}, \ell \cdot \left(2 + 2 \cdot \ell\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma (/ J (exp l)) (* l (+ 2.0 (* 2.0 l))) U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma((J / exp(l)), (l * (2.0 + (2.0 * l))), U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(Float64(J / exp(l)), Float64(l * Float64(2.0 + Float64(2.0 * l))), U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(J / N[Exp[l], $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 + N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. lift-sinh.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 80.5

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

   6. Applied rewrites 80.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-sinh.f64 N/A

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

      7. sinh-undef N/A

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

      8. lift-exp.f64 N/A

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

      9. exp-neg N/A

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

      10. lift-exp.f64 N/A

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

      11. sub-to-fraction N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. exp-sum N/A

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

      15. lift-+.f64 N/A

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

      16. lift-expm1.f64 N/A

          \[\leadsto J \cdot \frac{\mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}} + U \]

      17. associate-/l* N/A

          \[\leadsto \frac{J \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}} + U \]

      18. *-commutative N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{e^{\ell}} + U \]

      19. associate-/l* N/A

          \[\leadsto \mathsf{expm1}\left(\ell + \ell\right) \cdot \frac{J}{e^{\ell}} + U \]

      20. *-commutative N/A

          \[\leadsto \frac{J}{e^{\ell}} \cdot \mathsf{expm1}\left(\ell + \ell\right) + U \]

   8. Applied rewrites 62.6%

      \[\leadsto \mathsf{fma}\left(\frac{J}{e^{\ell}}, \color{blue}{\mathsf{expm1}\left(\ell + \ell\right)}, U\right) \]

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 49.3

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

   11. Applied rewrites 49.3%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. lift-sinh.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 80.5

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

   6. Applied rewrites 80.5%

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

Alternative 5: 86.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(J + -1 \cdot \left(J \cdot \ell\right), \mathsf{expm1}\left(\ell + \ell\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (fma (+ J (* -1.0 (* J l))) (expm1 (+ l l)) U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = fma((J + (-1.0 * (J * l))), expm1((l + l)), U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = fma(Float64(J + Float64(-1.0 * Float64(J * l))), expm1(Float64(l + l)), U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(J + N[(-1.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Exp[N[(l + l), $MachinePrecision]] - 1), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. lift-sinh.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 80.5

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

   6. Applied rewrites 80.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-sinh.f64 N/A

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

      7. sinh-undef N/A

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

      8. lift-exp.f64 N/A

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

      9. exp-neg N/A

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

      10. lift-exp.f64 N/A

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

      11. sub-to-fraction N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. exp-sum N/A

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

      15. lift-+.f64 N/A

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

      16. lift-expm1.f64 N/A

          \[\leadsto J \cdot \frac{\mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}} + U \]

      17. associate-/l* N/A

          \[\leadsto \frac{J \cdot \mathsf{expm1}\left(\ell + \ell\right)}{e^{\ell}} + U \]

      18. *-commutative N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{e^{\ell}} + U \]

      19. associate-/l* N/A

          \[\leadsto \mathsf{expm1}\left(\ell + \ell\right) \cdot \frac{J}{e^{\ell}} + U \]

      20. *-commutative N/A

          \[\leadsto \frac{J}{e^{\ell}} \cdot \mathsf{expm1}\left(\ell + \ell\right) + U \]

   8. Applied rewrites 62.6%

      \[\leadsto \mathsf{fma}\left(\frac{J}{e^{\ell}}, \color{blue}{\mathsf{expm1}\left(\ell + \ell\right)}, U\right) \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(J + -1 \cdot \left(J \cdot \ell\right), \mathsf{expm1}\left(\color{blue}{\ell + \ell}\right), U\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(J + -1 \cdot \left(J \cdot \ell\right), \mathsf{expm1}\left(\ell + \color{blue}{\ell}\right), U\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(J + -1 \cdot \left(J \cdot \ell\right), \mathsf{expm1}\left(\ell + \ell\right), U\right) \]

       3. lower-*.f64 55.8

          \[\leadsto \mathsf{fma}\left(J + -1 \cdot \left(J \cdot \ell\right), \mathsf{expm1}\left(\ell + \ell\right), U\right) \]

   11. Applied rewrites 55.8%

       \[\leadsto \mathsf{fma}\left(J + -1 \cdot \left(J \cdot \ell\right), \mathsf{expm1}\left(\color{blue}{\ell + \ell}\right), U\right) \]

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. lift-sinh.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 80.5

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

   6. Applied rewrites 80.5%

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

Alternative 6: 86.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\left(\ell \cdot K\right) \cdot K\right)\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+ (* 2.0 (* J (+ l (* -0.125 (* (* l K) K))))) U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = (2.0 * (J * (l + (-0.125 * ((l * K) * K))))) + U;
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(Float64(2.0 * Float64(J * Float64(l + Float64(-0.125 * Float64(Float64(l * K) * K))))) + U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(2.0 * N[(J * N[(l + N[(-0.125 * N[(N[(l * K), $MachinePrecision] * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 64.7

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U \]

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 49.9

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left({K}^{2} \cdot \ell\right)\right)\right) + U \]

   7. Applied rewrites 49.9%

      \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \color{blue}{-0.125 \cdot \left({K}^{2} \cdot \ell\right)}\right)\right) + U \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 51.2

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\left(\ell \cdot K\right) \cdot K\right)\right)\right) + U \]

   9. Applied rewrites 51.2%

      \[\leadsto 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\left(\ell \cdot K\right) \cdot K\right)\right)\right) + U \]

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. lift-sinh.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 80.5

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

   6. Applied rewrites 80.5%

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

Alternative 7: 85.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;\left(J + J\right) \cdot \mathsf{fma}\left(\left(K \cdot K\right) \cdot \ell, -0.125, \ell\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+ (* (+ J J) (fma (* (* K K) l) -0.125 l)) U)
   (fma (+ J J) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = ((J + J) * fma(((K * K) * l), -0.125, l)) + U;
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(Float64(Float64(J + J) * fma(Float64(Float64(K * K) * l), -0.125, l)) + U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(N[(N[(J + J), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * l), $MachinePrecision] * -0.125 + l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 64.7

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U \]

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 49.9

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left({K}^{2} \cdot \ell\right)\right)\right) + U \]

   7. Applied rewrites 49.9%

      \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \color{blue}{-0.125 \cdot \left({K}^{2} \cdot \ell\right)}\right)\right) + U \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. count-2 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 49.8

         \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(\ell + -0.125 \cdot \left({K}^{2} \cdot \ell\right)\right)} + U \]

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 49.8

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 49.8

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

   9. Applied rewrites 49.8%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. lift-sinh.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 80.5

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

   6. Applied rewrites 80.5%

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

Alternative 8: 80.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.43:\\ \;\;\;\;\mathsf{fma}\left(1 - e^{-\ell}, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -0.43)
   (fma (- 1.0 (exp (- l))) J U)
   (+ U (/ (* (expm1 (+ l l)) J) 1.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -0.43) {
		tmp = fma((1.0 - exp(-l)), J, U);
	} else {
		tmp = U + ((expm1((l + l)) * J) / 1.0);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -0.43)
		tmp = fma(Float64(1.0 - exp(Float64(-l))), J, U);
	else
		tmp = Float64(U + Float64(Float64(expm1(Float64(l + l)) * J) / 1.0));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -0.43], N[(N[(1.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(U + N[(N[(N[(Exp[N[(l + l), $MachinePrecision]] - 1), $MachinePrecision] * J), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.429999999999999993

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.0

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

   4. Applied rewrites 73.0%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 55.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 55.7

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

   9. Applied rewrites 55.7%

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

   if -0.429999999999999993 < l

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. exp-neg N/A

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

      7. lift-exp.f64 N/A

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

      8. sub-to-fraction N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. exp-lft-sqr-rev N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-expm1.f64 62.6

          \[\leadsto U + \frac{\mathsf{expm1}\left(2 \cdot \ell\right) \cdot J}{e^{\ell}} \]

      18. lift-*.f64 N/A

          \[\leadsto U + \frac{\mathsf{expm1}\left(2 \cdot \ell\right) \cdot J}{e^{\ell}} \]

      19. count-2-rev N/A

          \[\leadsto U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{e^{\ell}} \]

      20. lower-+.f64 62.6

          \[\leadsto U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{e^{\ell}} \]

   6. Applied rewrites 62.6%

      \[\leadsto U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{\color{blue}{e^{\ell}}} \]

   7. Taylor expanded in l around 0

      \[\leadsto U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.9%

      \[\leadsto U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7600:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 7600.0) (fma (+ J J) l U) (+ U (/ (* (expm1 (+ l l)) J) 1.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 7600.0) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = U + ((expm1((l + l)) * J) / 1.0);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 7600.0)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = Float64(U + Float64(Float64(expm1(Float64(l + l)) * J) / 1.0));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 7600.0], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], N[(U + N[(N[(N[(Exp[N[(l + l), $MachinePrecision]] - 1), $MachinePrecision] * J), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7600

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. lift-sinh.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 80.5

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

   6. Applied rewrites 80.5%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 54.5%

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

   if 7600 < l

   1. Initial program 85.9%

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

   2. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. exp-neg N/A

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

      7. lift-exp.f64 N/A

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

      8. sub-to-fraction N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. exp-lft-sqr-rev N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-expm1.f64 62.6

          \[\leadsto U + \frac{\mathsf{expm1}\left(2 \cdot \ell\right) \cdot J}{e^{\ell}} \]

      18. lift-*.f64 N/A

          \[\leadsto U + \frac{\mathsf{expm1}\left(2 \cdot \ell\right) \cdot J}{e^{\ell}} \]

      19. count-2-rev N/A

          \[\leadsto U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{e^{\ell}} \]

      20. lower-+.f64 62.6

          \[\leadsto U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{e^{\ell}} \]

   6. Applied rewrites 62.6%

      \[\leadsto U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{\color{blue}{e^{\ell}}} \]

   7. Taylor expanded in l around 0

      \[\leadsto U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.9%

      \[\leadsto U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 54.5% 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 85.9%

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

2. Taylor expanded in K around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 73.0

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

4. Applied rewrites 73.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. sub-flip N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. lift-sinh.f64 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. remove-double-neg N/A

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

   15. lower-fma.f64 N/A

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

   16. count-2-rev N/A

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

   17. lower-+.f64 80.5

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

6. Applied rewrites 80.5%

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

7. Taylor expanded in l around 0

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

9. Applied rewrites 54.5%

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

Alternative 11: 36.6% 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 85.9%

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

2. Taylor expanded in J around 0

   \[\leadsto \color{blue}{U} \]
3. Step-by-step derivation

4. Applied rewrites 36.6%

   \[\leadsto \color{blue}{U} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))