Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 100.0%

Time: 6.8s

Alternatives: 7

Speedup: 1.2×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Applied rewrites 100.0%

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

Alternative 2: 87.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.986:\\ \;\;\;\;U\\ \mathbf{elif}\;t\_0 \leq -0.761:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \left(1 + -0.125 \cdot {K}^{2}\right) + U\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\left(\cos \left(K \cdot 0.5\right) \cdot J\right) \cdot \ell, 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J + J, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.986)
     U
     (if (<= t_0 -0.761)
       (+ (* (* J (- (exp l) (exp (- l)))) (+ 1.0 (* -0.125 (pow K 2.0)))) U)
       (if (<= t_0 -0.1)
         (fma (* (* (cos (* K 0.5)) J) l) 2.0 U)
         (fma (+ J J) (sinh l) U))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.986) {
		tmp = U;
	} else if (t_0 <= -0.761) {
		tmp = ((J * (exp(l) - exp(-l))) * (1.0 + (-0.125 * pow(K, 2.0)))) + U;
	} else if (t_0 <= -0.1) {
		tmp = fma(((cos((K * 0.5)) * J) * l), 2.0, U);
	} else {
		tmp = fma((J + J), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.986)
		tmp = U;
	elseif (t_0 <= -0.761)
		tmp = Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * Float64(1.0 + Float64(-0.125 * (K ^ 2.0)))) + U);
	elseif (t_0 <= -0.1)
		tmp = fma(Float64(Float64(cos(Float64(K * 0.5)) * J) * l), 2.0, U);
	else
		tmp = fma(Float64(J + J), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.986], U, If[LessEqual[t$95$0, -0.761], 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], If[LessEqual[t$95$0, -0.1], N[(N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * l), $MachinePrecision] * 2.0 + U), $MachinePrecision], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.98599999999999999

   1. Initial program 86.3%

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

   2. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U} \]

   4. Applied rewrites 37.1%

      \[\leadsto \color{blue}{U} \]

   if -0.98599999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.76100000000000001

   1. Initial program 86.3%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 64.5%

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

   if -0.76100000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.10000000000000001

   1. Initial program 86.3%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in l around 0

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

   6. Applied rewrites 64.2%

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

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

   1. Initial program 86.3%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 73.7%

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

   6. Applied rewrites 81.0%

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

Alternative 3: 86.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (+ (* 2.0 (* J (fma (* (* l -0.125) K) K 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)) <= -0.1) {
		tmp = (2.0 * (J * fma(((l * -0.125) * K), K, 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)) <= -0.1)
		tmp = Float64(Float64(2.0 * Float64(J * fma(Float64(Float64(l * -0.125) * K), K, 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], -0.1], N[(N[(2.0 * N[(J * N[(N[(N[(l * -0.125), $MachinePrecision] * K), $MachinePrecision] * K + 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))) < -0.10000000000000001

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 49.6%

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

   9. Applied rewrites 51.2%

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

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

   1. Initial program 86.3%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 73.7%

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

   6. Applied rewrites 81.0%

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

Alternative 4: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J + J\right) \cdot \sinh \ell\\ t_1 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1000000:\\ \;\;\;\;\mathsf{fma}\left(J + J, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (+ J J) (sinh l)))
        (t_1 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0)))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 1000000.0) (fma (+ J J) l U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (J + J) * sinh(l);
	double t_1 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= 1000000.0) {
		tmp = fma((J + J), l, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J + J) * sinh(l))
	t_1 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= 1000000.0)
		tmp = fma(Float64(J + J), l, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, 1000000.0], N[(N[(J + J), $MachinePrecision] * l + U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -inf.0 or 1e6 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   1. Initial program 86.3%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 73.7%

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

   6. Applied rewrites 81.0%

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

   7. Taylor expanded in J around inf

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

   9. Applied rewrites 39.3%

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

   11. Applied rewrites 46.4%

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

   if -inf.0 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < 1e6

   1. Initial program 86.3%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 73.7%

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

   6. Applied rewrites 81.0%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 54.4%

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

Alternative 5: 80.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -10000:\\ \;\;\;\;\left(J + J\right) \cdot \sinh \ell\\ \mathbf{else}:\\ \;\;\;\;U + \frac{\mathsf{expm1}\left(\ell + \ell\right) \cdot J}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -10000.0) (* (+ J J) (sinh l)) (+ U (/ (* (expm1 (+ l l)) J) 1.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -10000.0) {
		tmp = (J + J) * sinh(l);
	} else {
		tmp = U + ((expm1((l + l)) * J) / 1.0);
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -10000.0) {
		tmp = (J + J) * Math.sinh(l);
	} else {
		tmp = U + ((Math.expm1((l + l)) * J) / 1.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -10000.0:
		tmp = (J + J) * math.sinh(l)
	else:
		tmp = U + ((math.expm1((l + l)) * J) / 1.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -10000.0)
		tmp = Float64(Float64(J + J) * sinh(l));
	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, -10000.0], N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $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 < -1e4

   1. Initial program 86.3%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 73.7%

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

   6. Applied rewrites 81.0%

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

   7. Taylor expanded in J around inf

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

   9. Applied rewrites 39.3%

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

   11. Applied rewrites 46.4%

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

   if -1e4 < l

   1. Initial program 86.3%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 73.7%

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

   6. Applied rewrites 62.5%

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

   7. Taylor expanded in l around 0

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

   9. Applied rewrites 62.8%

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

Alternative 6: 54.4% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

2. Taylor expanded in K around 0

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

4. Applied rewrites 73.7%

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

6. Applied rewrites 81.0%

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

7. Taylor expanded in l around 0

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

9. Applied rewrites 54.4%

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

Alternative 7: 37.1% accurate, 68.7× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

2. Taylor expanded in J around 0

   \[\leadsto \color{blue}{U} \]

4. Applied rewrites 37.1%

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

Reproduce

herbie shell --seed 2025161 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))