Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.7% → 99.9%

Time: 5.1s

Alternatives: 12

Speedup: 1.2×

• 50.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.7% accurate, 1.0× speedup

Math

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

Alternative 2: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.42:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left(1 + -0.125 \cdot {K}^{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.42)
   (+ (* (* 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)) <= -0.42) {
		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)) <= -0.42)
		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], -0.42], 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))) < -0.419999999999999984

   1. Initial program 86.7%

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

   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 65.7%

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

   4. Applied rewrites 65.7%

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

   if -0.419999999999999984 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

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

      1. lift--.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. sub-to-fraction N/A

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

      7. lower-/.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-+.f64 74.9%

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

   3. Applied rewrites 74.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 62.0%

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

   6. Applied rewrites 62.0%

      \[\leadsto \color{blue}{U + \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}}} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   8. Applied rewrites 80.5%

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

Alternative 3: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.12:\\ \;\;\;\;\left(1 + \frac{\left(\left(J + J\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \ell}{U}\right) \cdot U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.12)
   (* (+ 1.0 (/ (* (* (+ J J) (cos (* 0.5 K))) l) U)) 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)) <= -0.12) {
		tmp = (1.0 + ((((J + J) * cos((0.5 * K))) * l) / U)) * 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)) <= -0.12)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(J + J) * cos(Float64(0.5 * K))) * l) / U)) * 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], -0.12], N[(N[(1.0 + N[(N[(N[(N[(J + J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / U), $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))) < -0.12

   1. Initial program 86.7%

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

   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 63.9%

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

   4. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 67.9%

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

   if -0.12 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

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

      1. lift--.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. sub-to-fraction N/A

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

      7. lower-/.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-+.f64 74.9%

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

   3. Applied rewrites 74.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 62.0%

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

   6. Applied rewrites 62.0%

      \[\leadsto \color{blue}{U + \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}}} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   8. Applied rewrites 80.5%

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

Alternative 4: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.516:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.516)
   (fma (* (+ J J) (sinh l)) (fma (* K K) -0.125 1.0) 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)) <= -0.516) {
		tmp = fma(((J + J) * sinh(l)), fma((K * K), -0.125, 1.0), 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)) <= -0.516)
		tmp = fma(Float64(Float64(J + J) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), 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], -0.516], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $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))) < -0.51600000000000001

   1. Initial program 86.7%

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

   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 65.7%

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

   4. Applied rewrites 65.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 65.7%

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

   6. Applied rewrites 70.6%

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

   if -0.51600000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

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

      1. lift--.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. sub-to-fraction N/A

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

      7. lower-/.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-+.f64 74.9%

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

   3. Applied rewrites 74.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 62.0%

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

   6. Applied rewrites 62.0%

      \[\leadsto \color{blue}{U + \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}}} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   8. Applied rewrites 80.5%

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

Alternative 5: 83.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.516:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell + \ell\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.516)
   (fma (* (+ l l) (fma -0.125 (* K K) 1.0)) 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)) <= -0.516) {
		tmp = fma(((l + l) * fma(-0.125, (K * K), 1.0)), 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)) <= -0.516)
		tmp = fma(Float64(Float64(l + l) * fma(-0.125, Float64(K * K), 1.0)), 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], -0.516], N[(N[(N[(l + l), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * J + 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))) < -0.51600000000000001

   1. Initial program 86.7%

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

   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 65.7%

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

   4. Applied rewrites 65.7%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 49.3%

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

   7. Applied rewrites 49.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 50.1%

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

   if -0.51600000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

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

      1. lift--.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. sub-to-fraction N/A

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

      7. lower-/.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-+.f64 74.9%

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

   3. Applied rewrites 74.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 62.0%

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

   6. Applied rewrites 62.0%

      \[\leadsto \color{blue}{U + \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{e^{\ell}}} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   8. Applied rewrites 80.5%

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

Alternative 6: 70.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell + \ell\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{1}\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (- (exp l) (exp (- l))) -1e-12)
   (fma (* (+ l l) (fma -0.125 (* K K) 1.0)) J U)
   (+ U (/ (* J (expm1 (* 2.0 l))) 1.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((exp(l) - exp(-l)) <= -1e-12) {
		tmp = fma(((l + l) * fma(-0.125, (K * K), 1.0)), J, U);
	} else {
		tmp = U + ((J * expm1((2.0 * l))) / 1.0);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(exp(l) - exp(Float64(-l))) <= -1e-12)
		tmp = fma(Float64(Float64(l + l) * fma(-0.125, Float64(K * K), 1.0)), J, U);
	else
		tmp = Float64(U + Float64(Float64(J * expm1(Float64(2.0 * l))) / 1.0));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], -1e-12], N[(N[(N[(l + l), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(U + N[(N[(J * N[(Exp[N[(2.0 * l), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -9.9999999999999998e-13

   1. Initial program 86.7%

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

   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 65.7%

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

   4. Applied rewrites 65.7%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 49.3%

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

   7. Applied rewrites 49.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 50.1%

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

   if -9.9999999999999998e-13 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 86.7%

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

      1. lift--.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. sub-to-fraction N/A

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

      7. lower-/.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-+.f64 74.9%

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

   3. Applied rewrites 74.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 62.0%

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

   6. Applied rewrites 62.0%

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

   7. Taylor expanded in l around 0

      \[\leadsto U + \frac{J \cdot \mathsf{expm1}\left(2 \cdot \ell\right)}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.2%

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

Alternative 7: 60.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.516:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell + \ell\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\left(2 \cdot J\right) \cdot \ell}{U}\right) \cdot U\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.516)
   (fma (* (+ l l) (fma -0.125 (* K K) 1.0)) J U)
   (* (+ 1.0 (/ (* (* 2.0 J) l) U)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.516) {
		tmp = fma(((l + l) * fma(-0.125, (K * K), 1.0)), J, U);
	} else {
		tmp = (1.0 + (((2.0 * J) * l) / U)) * U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.516)
		tmp = fma(Float64(Float64(l + l) * fma(-0.125, Float64(K * K), 1.0)), J, U);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(2.0 * J) * l) / U)) * U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.516], N[(N[(N[(l + l), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(2.0 * J), $MachinePrecision] * l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.51600000000000001

   1. Initial program 86.7%

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

   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 65.7%

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

   4. Applied rewrites 65.7%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 49.3%

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

   7. Applied rewrites 49.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 50.1%

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

   if -0.51600000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

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

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.5%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 53.9%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. sum-to-mult N/A

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

       4. lower-unsound-*.f64 N/A

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

       5. lower-unsound-+.f64 N/A

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

       6. lower-unsound-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 57.2%

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

   11. Applied rewrites 57.2%

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

Alternative 8: 60.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.516:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell + \ell\right) \cdot J, \mathsf{fma}\left(-0.125, K \cdot K, 1\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\left(2 \cdot J\right) \cdot \ell}{U}\right) \cdot U\\ \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.516)
   (fma (* (+ l l) J) (fma -0.125 (* K K) 1.0) U)
   (* (+ 1.0 (/ (* (* 2.0 J) l) U)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.516) {
		tmp = fma(((l + l) * J), fma(-0.125, (K * K), 1.0), U);
	} else {
		tmp = (1.0 + (((2.0 * J) * l) / U)) * U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.516)
		tmp = fma(Float64(Float64(l + l) * J), fma(-0.125, Float64(K * K), 1.0), U);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(2.0 * J) * l) / U)) * U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.516], N[(N[(N[(l + l), $MachinePrecision] * J), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] + U), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(2.0 * J), $MachinePrecision] * l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.51600000000000001

   1. Initial program 86.7%

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

   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 65.7%

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

   4. Applied rewrites 65.7%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 49.3%

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

   7. Applied rewrites 49.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-fma.f64 49.3%

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

   9. Applied rewrites 49.3%

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

   if -0.51600000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

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

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sinh-undef N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 80.5%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 53.9%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. sum-to-mult N/A

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

       4. lower-unsound-*.f64 N/A

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

       5. lower-unsound-+.f64 N/A

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

       6. lower-unsound-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 57.2%

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

   11. Applied rewrites 57.2%

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

Alternative 9: 57.2% accurate, 4.3× speedup

Math

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

FPCore

(FPCore (J l K U) :precision binary64 (* (+ 1.0 (/ (* (* 2.0 J) l) U)) U))

C

double code(double J, double l, double K, double U) {
	return (1.0 + (((2.0 * J) * l) / U)) * 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 = (1.0d0 + (((2.0d0 * j) * l) / u)) * u
end function

Java

public static double code(double J, double l, double K, double U) {
	return (1.0 + (((2.0 * J) * l) / U)) * U;
}

Python

def code(J, l, K, U):
	return (1.0 + (((2.0 * J) * l) / U)) * U

Julia

function code(J, l, K, U)
	return Float64(Float64(1.0 + Float64(Float64(Float64(2.0 * J) * l) / U)) * U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = (1.0 + (((2.0 * J) * l) / U)) * U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(1.0 + N[(N[(N[(2.0 * J), $MachinePrecision] * l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]

Derivation

1. Initial program 86.7%

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

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in K around 0

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

6. Applied rewrites 80.5%

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

7. Taylor expanded in l around 0

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

9. Applied rewrites 53.9%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. sum-to-mult N/A

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

    4. lower-unsound-*.f64 N/A

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

    5. lower-unsound-+.f64 N/A

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

    6. lower-unsound-/.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. associate-*r* N/A

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

    10. lower-*.f64 N/A

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

    11. lower-*.f64 57.2%

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

11. Applied rewrites 57.2%

    \[\leadsto \color{blue}{\left(1 + \frac{\left(2 \cdot J\right) \cdot \ell}{U}\right) \cdot U} \]
12. Add Preprocessing

Alternative 10: 53.9% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in K around 0

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

6. Applied rewrites 80.5%

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

7. Taylor expanded in l around 0

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

9. Applied rewrites 53.9%

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

    1. lift-fma.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-*.f64 53.9%

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

11. Applied rewrites 53.9%

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

Alternative 11: 53.9% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sinh-undef N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in K around 0

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

6. Applied rewrites 80.5%

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

7. Taylor expanded in l around 0

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

9. Applied rewrites 53.9%

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

Alternative 12: 36.8% accurate, 68.7× speedup

Math

\[U \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

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

Python

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

Julia

function code(J, l, K, U)
	return U
end

MATLAB

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

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 86.7%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 36.8%

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

Reproduce

herbie shell --seed 2025182 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))