expfmod (used to be hard to sample)

Percentage Accurate: 7.0% → 95.3%

Time: 8.9s

Alternatives: 16

Speedup: 3.6×

• could not determine a ground truth

  1. x = -2.4677848568112167e+98

Specification

Math

\[\begin{array}{l} \\ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = mod(exp(x), sqrt(cos(x))) * exp(-x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)

Julia

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Initial Program: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = mod(exp(x), sqrt(cos(x))) * exp(-x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)

Julia

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-49}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-6} - 0.015625}{\left({x}^{-4} + 0.0625\right) + {x}^{-2} \cdot 0.25} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-148}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -2e-49)
     (* (fmod (exp x) (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x)) t_0)
     (if (<= x -2e-77)
       (*
        (fmod
         (exp x)
         (*
          (*
           (/
            (- (pow x -6.0) 0.015625)
            (+ (+ (pow x -4.0) 0.0625) (* (pow x -2.0) 0.25)))
           x)
          x))
        t_0)
       (if (<= x -4e-148)
         (*
          (fmod
           (exp x)
           (* (* (/ (- (pow x -4.0) 0.0625) (- (pow x -2.0) -0.25)) x) x))
          t_0)
         (if (<= x -1e-309)
           (* (fmod (exp x) (* (* (- (pow x -2.0) 0.25) x) x)) (- 1.0 x))
           (/ (fmod x (sqrt 1.0)) (exp x))))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -2e-49) {
		tmp = fmod(exp(x), (((exp((log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0;
	} else if (x <= -2e-77) {
		tmp = fmod(exp(x), ((((pow(x, -6.0) - 0.015625) / ((pow(x, -4.0) + 0.0625) + (pow(x, -2.0) * 0.25))) * x) * x)) * t_0;
	} else if (x <= -4e-148) {
		tmp = fmod(exp(x), ((((pow(x, -4.0) - 0.0625) / (pow(x, -2.0) - -0.25)) * x) * x)) * t_0;
	} else if (x <= -1e-309) {
		tmp = fmod(exp(x), (((pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x);
	} else {
		tmp = fmod(x, sqrt(1.0)) / exp(x);
	}
	return tmp;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-2d-49)) then
        tmp = mod(exp(x), (((exp((log((x * x)) * (-1.0d0))) - 0.25d0) * x) * x)) * t_0
    else if (x <= (-2d-77)) then
        tmp = mod(exp(x), (((((x ** (-6.0d0)) - 0.015625d0) / (((x ** (-4.0d0)) + 0.0625d0) + ((x ** (-2.0d0)) * 0.25d0))) * x) * x)) * t_0
    else if (x <= (-4d-148)) then
        tmp = mod(exp(x), (((((x ** (-4.0d0)) - 0.0625d0) / ((x ** (-2.0d0)) - (-0.25d0))) * x) * x)) * t_0
    else if (x <= (-1d-309)) then
        tmp = mod(exp(x), ((((x ** (-2.0d0)) - 0.25d0) * x) * x)) * (1.0d0 - x)
    else
        tmp = mod(x, sqrt(1.0d0)) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -2e-49:
		tmp = math.fmod(math.exp(x), (((math.exp((math.log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0
	elif x <= -2e-77:
		tmp = math.fmod(math.exp(x), ((((math.pow(x, -6.0) - 0.015625) / ((math.pow(x, -4.0) + 0.0625) + (math.pow(x, -2.0) * 0.25))) * x) * x)) * t_0
	elif x <= -4e-148:
		tmp = math.fmod(math.exp(x), ((((math.pow(x, -4.0) - 0.0625) / (math.pow(x, -2.0) - -0.25)) * x) * x)) * t_0
	elif x <= -1e-309:
		tmp = math.fmod(math.exp(x), (((math.pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x)
	else:
		tmp = math.fmod(x, math.sqrt(1.0)) / math.exp(x)
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -2e-49)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)) * t_0);
	elseif (x <= -2e-77)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64((x ^ -6.0) - 0.015625) / Float64(Float64((x ^ -4.0) + 0.0625) + Float64((x ^ -2.0) * 0.25))) * x) * x)) * t_0);
	elseif (x <= -4e-148)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64((x ^ -4.0) - 0.0625) / Float64((x ^ -2.0) - -0.25)) * x) * x)) * t_0);
	elseif (x <= -1e-309)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, sqrt(1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -2e-49], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -2e-77], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(N[Power[x, -6.0], $MachinePrecision] - 0.015625), $MachinePrecision] / N[(N[(N[Power[x, -4.0], $MachinePrecision] + 0.0625), $MachinePrecision] + N[(N[Power[x, -2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -4e-148], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(N[Power[x, -4.0], $MachinePrecision] - 0.0625), $MachinePrecision] / N[(N[Power[x, -2.0], $MachinePrecision] - -0.25), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.99999999999999987e-49

   1. Initial program 33.7%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 33.7

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 33.7%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right)}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval 34.6

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 34.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(2 \cdot -1\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      3. pow-pow N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({\left({x}^{2}\right)}^{-1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. pow-to-exp N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left({x}^{2}\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-exp.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left({x}^{2}\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left({x}^{2}\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. lower-log.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left({x}^{2}\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. lift-*.f64 66.4

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 66.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -1.99999999999999987e-49 < x < -1.9999999999999999e-77

   1. Initial program 3.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 3.1

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 3.1%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right)}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval 6.2

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 6.2%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. flip3-- N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{\left(\frac{1}{{x}^{2}}\right)}^{3} - {\frac{1}{4}}^{3}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \frac{1}{{x}^{2}} \cdot \frac{1}{4}\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{\left(\frac{1}{{x}^{2}}\right)}^{3} - {\frac{1}{4}}^{3}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \frac{1}{{x}^{2}} \cdot \frac{1}{4}\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{\left(\frac{1}{{x}^{2}}\right)}^{3} - {\frac{1}{4}}^{3}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \frac{1}{{x}^{2}} \cdot \frac{1}{4}\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{\left({x}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}^{3} - {\frac{1}{4}}^{3}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \frac{1}{{x}^{2}} \cdot \frac{1}{4}\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{\left({x}^{-2}\right)}^{3} - {\frac{1}{4}}^{3}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \frac{1}{{x}^{2}} \cdot \frac{1}{4}\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      10. pow-pow N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{\left(-2 \cdot 3\right)} - {\frac{1}{4}}^{3}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \frac{1}{{x}^{2}} \cdot \frac{1}{4}\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      11. lower-pow.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{\left(-2 \cdot 3\right)} - {\frac{1}{4}}^{3}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \frac{1}{{x}^{2}} \cdot \frac{1}{4}\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-6} - {\frac{1}{4}}^{3}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \frac{1}{{x}^{2}} \cdot \frac{1}{4}\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-6} - \frac{1}{64}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \frac{1}{{x}^{2}} \cdot \frac{1}{4}\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      14. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-6} - \frac{1}{64}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \frac{1 \cdot 1}{{x}^{2}} \cdot \frac{1}{4}\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      15. pow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-6} - \frac{1}{64}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \frac{1 \cdot 1}{x \cdot x} \cdot \frac{1}{4}\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      16. frac-times N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-6} - \frac{1}{64}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \frac{1}{4}\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-6} - \frac{1}{64}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      18. swap-sqr N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-6} - \frac{1}{64}}{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{4} \cdot \frac{1}{4} + \left(\frac{1}{x} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{x} \cdot \frac{1}{2}\right)\right)} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 100.0%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-6} - 0.015625}{\left({x}^{-4} + 0.0625\right) + {x}^{-2} \cdot 0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -1.9999999999999999e-77 < x < -3.99999999999999974e-148

   1. Initial program 3.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 3.1

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 3.1%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right)}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval 8.6

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 8.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. flip-- N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot \frac{1}{{x}^{2}} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-2} \cdot \frac{1}{{x}^{2}} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      10. pow-flip N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-2} \cdot {x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-2} \cdot {x}^{-2} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      12. pow-prod-up N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{\left(-2 + -2\right)} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      13. lower-pow.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{\left(-2 + -2\right)} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      14. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      16. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{\frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{2}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      17. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{\frac{1}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{2}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{\frac{1}{{x}^{2}} - \frac{-1}{2} \cdot \frac{1}{2}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      19. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{\frac{1}{{x}^{2}} - \frac{-1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      20. lower--.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{\frac{1}{{x}^{2}} - \frac{-1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      21. pow-flip N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{{x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{-1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      22. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{{x}^{-2} - \frac{-1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      23. lift-pow.f64 100.0

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 100.0%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -3.99999999999999974e-148 < x < -1.000000000000002e-309

   1. Initial program 3.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 3.1

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 3.1%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right)}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval 100.0

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 100.0%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   10. Step-by-step derivation

       1. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right) \]

       2. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - 1 \cdot x\right) \]

       3. *-lft-identity N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right) \]

       4. lower--.f64 100.0

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - \color{blue}{x}\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}} \]

       2. lift-neg.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

       3. lift-exp.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}} \]

       4. exp-neg N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

       5. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

   13. Applied rewrites 96.1%

       \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-49}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-6} - 0.015625}{\left({x}^{-4} + 0.0625\right) + {x}^{-2} \cdot 0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-148}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-148}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -2e-77)
     (* (fmod (exp x) (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x)) t_0)
     (if (<= x -4e-148)
       (*
        (fmod
         (exp x)
         (* (* (/ (- (pow x -4.0) 0.0625) (- (pow x -2.0) -0.25)) x) x))
        t_0)
       (if (<= x -1e-309)
         (* (fmod (exp x) (* (* (- (pow x -2.0) 0.25) x) x)) (- 1.0 x))
         (/ (fmod x (sqrt 1.0)) (exp x)))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -2e-77) {
		tmp = fmod(exp(x), (((exp((log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0;
	} else if (x <= -4e-148) {
		tmp = fmod(exp(x), ((((pow(x, -4.0) - 0.0625) / (pow(x, -2.0) - -0.25)) * x) * x)) * t_0;
	} else if (x <= -1e-309) {
		tmp = fmod(exp(x), (((pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x);
	} else {
		tmp = fmod(x, sqrt(1.0)) / exp(x);
	}
	return tmp;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-2d-77)) then
        tmp = mod(exp(x), (((exp((log((x * x)) * (-1.0d0))) - 0.25d0) * x) * x)) * t_0
    else if (x <= (-4d-148)) then
        tmp = mod(exp(x), (((((x ** (-4.0d0)) - 0.0625d0) / ((x ** (-2.0d0)) - (-0.25d0))) * x) * x)) * t_0
    else if (x <= (-1d-309)) then
        tmp = mod(exp(x), ((((x ** (-2.0d0)) - 0.25d0) * x) * x)) * (1.0d0 - x)
    else
        tmp = mod(x, sqrt(1.0d0)) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -2e-77:
		tmp = math.fmod(math.exp(x), (((math.exp((math.log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0
	elif x <= -4e-148:
		tmp = math.fmod(math.exp(x), ((((math.pow(x, -4.0) - 0.0625) / (math.pow(x, -2.0) - -0.25)) * x) * x)) * t_0
	elif x <= -1e-309:
		tmp = math.fmod(math.exp(x), (((math.pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x)
	else:
		tmp = math.fmod(x, math.sqrt(1.0)) / math.exp(x)
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -2e-77)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)) * t_0);
	elseif (x <= -4e-148)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64((x ^ -4.0) - 0.0625) / Float64((x ^ -2.0) - -0.25)) * x) * x)) * t_0);
	elseif (x <= -1e-309)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, sqrt(1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -2e-77], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -4e-148], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(N[Power[x, -4.0], $MachinePrecision] - 0.0625), $MachinePrecision] / N[(N[Power[x, -2.0], $MachinePrecision] - -0.25), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.9999999999999999e-77

   1. Initial program 23.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 23.5

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 23.5%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right)}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval 25.1

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 25.1%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(2 \cdot -1\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      3. pow-pow N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({\left({x}^{2}\right)}^{-1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. pow-to-exp N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left({x}^{2}\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-exp.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left({x}^{2}\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left({x}^{2}\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. lower-log.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left({x}^{2}\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. lift-*.f64 61.2

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 61.2%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -1.9999999999999999e-77 < x < -3.99999999999999974e-148

   1. Initial program 3.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 3.1

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 3.1%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right)}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval 8.6

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 8.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. flip-- N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{\frac{1}{{x}^{2}} \cdot \frac{1}{{x}^{2}} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot \frac{1}{{x}^{2}} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-2} \cdot \frac{1}{{x}^{2}} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      10. pow-flip N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-2} \cdot {x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-2} \cdot {x}^{-2} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      12. pow-prod-up N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{\left(-2 + -2\right)} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      13. lower-pow.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{\left(-2 + -2\right)} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      14. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{\frac{1}{{x}^{2}} + \frac{1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      16. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{\frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{2}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      17. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{\frac{1}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{2}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{\frac{1}{{x}^{2}} - \frac{-1}{2} \cdot \frac{1}{2}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      19. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{\frac{1}{{x}^{2}} - \frac{-1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      20. lower--.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{\frac{1}{{x}^{2}} - \frac{-1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      21. pow-flip N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{{x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{-1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      22. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - \frac{1}{16}}{{x}^{-2} - \frac{-1}{4}} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      23. lift-pow.f64 100.0

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 100.0%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -3.99999999999999974e-148 < x < -1.000000000000002e-309

   1. Initial program 3.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 3.1

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 3.1%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right)}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval 100.0

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 100.0%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   10. Step-by-step derivation

       1. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right) \]

       2. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - 1 \cdot x\right) \]

       3. *-lft-identity N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right) \]

       4. lower--.f64 100.0

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - \color{blue}{x}\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}} \]

       2. lift-neg.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

       3. lift-exp.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}} \]

       4. exp-neg N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

       5. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

   13. Applied rewrites 96.1%

       \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-148}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (*
    (fmod (exp x) (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x))
    (exp (- x)))
   (/ (fmod x (sqrt 1.0)) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), (((exp((log((x * x)) * -1.0)) - 0.25) * x) * x)) * exp(-x);
	} else {
		tmp = fmod(x, sqrt(1.0)) / exp(x);
	}
	return tmp;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-309)) then
        tmp = mod(exp(x), (((exp((log((x * x)) * (-1.0d0))) - 0.25d0) * x) * x)) * exp(-x)
    else
        tmp = mod(x, sqrt(1.0d0)) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-309:
		tmp = math.fmod(math.exp(x), (((math.exp((math.log((x * x)) * -1.0)) - 0.25) * x) * x)) * math.exp(-x)
	else:
		tmp = math.fmod(x, math.sqrt(1.0)) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)) * exp(Float64(-x)));
	else
		tmp = Float64(rem(x, sqrt(1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 9.3

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 9.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right)}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval 60.2

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 60.2%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(2 \cdot -1\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      3. pow-pow N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({\left({x}^{2}\right)}^{-1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. pow-to-exp N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left({x}^{2}\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-exp.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left({x}^{2}\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left({x}^{2}\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. lower-log.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left({x}^{2}\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. lift-*.f64 78.2

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 78.2%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}} \]

       2. lift-neg.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

       3. lift-exp.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}} \]

       4. exp-neg N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

       5. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

   13. Applied rewrites 96.1%

       \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (* (fmod (exp x) (* (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) x) x)) (exp (- x)))
   (/ (fmod x (sqrt 1.0)) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), ((fma((-1.0 / x), (-1.0 / x), -0.25) * x) * x)) * exp(-x);
	} else {
		tmp = fmod(x, sqrt(1.0)) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), Float64(Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * x) * x)) * exp(Float64(-x)));
	else
		tmp = Float64(rem(x, sqrt(1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision] + -0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 9.3

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 9.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right)}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval 60.2

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 60.2%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{2} \cdot \frac{1}{2}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{-1 \cdot -1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{-1 \cdot -1}{x \cdot x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. times-frac N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{-1}{x} \cdot \frac{-1}{x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{-1}{x} \cdot \frac{-1}{x} + \frac{-1}{2} \cdot \frac{1}{2}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{-1}{x} \cdot \frac{-1}{x} + \frac{-1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, \frac{-1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, \frac{-1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      14. lower-/.f64 64.7

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 64.7%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}} \]

       2. lift-neg.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

       3. lift-exp.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}} \]

       4. exp-neg N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

       5. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

   13. Applied rewrites 96.1%

       \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{x \cdot x} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (* (fmod (exp x) (* (* (- (/ 1.0 (* x x)) 0.25) x) x)) (exp (- x)))
   (/ (fmod x (sqrt 1.0)) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), ((((1.0 / (x * x)) - 0.25) * x) * x)) * exp(-x);
	} else {
		tmp = fmod(x, sqrt(1.0)) / exp(x);
	}
	return tmp;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-309)) then
        tmp = mod(exp(x), ((((1.0d0 / (x * x)) - 0.25d0) * x) * x)) * exp(-x)
    else
        tmp = mod(x, sqrt(1.0d0)) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-309:
		tmp = math.fmod(math.exp(x), ((((1.0 / (x * x)) - 0.25) * x) * x)) * math.exp(-x)
	else:
		tmp = math.fmod(x, math.sqrt(1.0)) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64(1.0 / Float64(x * x)) - 0.25) * x) * x)) * exp(Float64(-x)));
	else
		tmp = Float64(rem(x, sqrt(1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 9.3

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 9.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right)}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval 60.2

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 60.2%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      3. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{x \cdot x} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lift-*.f64 62.1

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{x \cdot x} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 62.1%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{x \cdot x} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}} \]

       2. lift-neg.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

       3. lift-exp.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}} \]

       4. exp-neg N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

       5. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

   13. Applied rewrites 96.1%

       \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{x \cdot x} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (* (fmod (exp x) (* (* (- (pow x -2.0) 0.25) x) x)) (- 1.0 x))
   (/ (fmod x (sqrt 1.0)) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), (((pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x);
	} else {
		tmp = fmod(x, sqrt(1.0)) / exp(x);
	}
	return tmp;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-309)) then
        tmp = mod(exp(x), ((((x ** (-2.0d0)) - 0.25d0) * x) * x)) * (1.0d0 - x)
    else
        tmp = mod(x, sqrt(1.0d0)) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-309:
		tmp = math.fmod(math.exp(x), (((math.pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x)
	else:
		tmp = math.fmod(x, math.sqrt(1.0)) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, sqrt(1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 9.3

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 9.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right)}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{{x}^{2}} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      7. pow-flip N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{\left(\mathsf{neg}\left(2\right)\right)} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

      9. metadata-eval 60.2

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 60.2%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   10. Step-by-step derivation

       1. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right) \]

       2. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - 1 \cdot x\right) \]

       3. *-lft-identity N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - \frac{1}{4}\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right) \]

       4. lower--.f64 59.4

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - \color{blue}{x}\right) \]

   11. Applied rewrites 59.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}} \]

       2. lift-neg.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

       3. lift-exp.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}} \]

       4. exp-neg N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

       5. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

   13. Applied rewrites 96.1%

       \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (* (fmod (exp x) 1.0) (exp (- x)))
   (/ (fmod x (sqrt 1.0)) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), 1.0) * exp(-x);
	} else {
		tmp = fmod(x, sqrt(1.0)) / exp(x);
	}
	return tmp;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-309)) then
        tmp = mod(exp(x), 1.0d0) * exp(-x)
    else
        tmp = mod(x, sqrt(1.0d0)) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-309:
		tmp = math.fmod(math.exp(x), 1.0) * math.exp(-x)
	else:
		tmp = math.fmod(x, math.sqrt(1.0)) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), 1.0) * exp(Float64(-x)));
	else
		tmp = Float64(rem(x, sqrt(1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

   5. Applied rewrites 9.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}} \]

       2. lift-neg.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

       3. lift-exp.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}} \]

       4. exp-neg N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

       5. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

   13. Applied rewrites 96.1%

       \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (* (fmod (fma (fma 0.5 x 1.0) x 1.0) (sqrt (cos x))) (- 1.0 x))
   (/ (fmod x (sqrt 1.0)) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(fma(fma(0.5, x, 1.0), x, 1.0), sqrt(cos(x))) * (1.0 - x);
	} else {
		tmp = fmod(x, sqrt(1.0)) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(fma(fma(0.5, x, 1.0), x, 1.0), sqrt(cos(x))) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, sqrt(1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + \frac{1}{2} \cdot x\right) \cdot x + 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(1 + \frac{1}{2} \cdot x, \color{blue}{x}, 1\right)\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2} \cdot x + 1, x, 1\right)\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. lower-fma.f64 8.6

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 8.6%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   7. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(1 - 1 \cdot x\right) \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(1 - x\right) \]

      4. lower--.f64 8.8

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(1 - \color{blue}{x}\right) \]

   8. Applied rewrites 8.8%

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}} \]

       2. lift-neg.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

       3. lift-exp.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}} \]

       4. exp-neg N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

       5. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

   13. Applied rewrites 96.1%

       \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (*
    (fmod (exp x) 1.0)
    (fma (fma (fma -0.16666666666666666 x 0.5) x -1.0) x 1.0))
   (/ (fmod x (sqrt 1.0)) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), 1.0) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0);
	} else {
		tmp = fmod(x, sqrt(1.0)) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), 1.0) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(x, sqrt(1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

   5. Applied rewrites 9.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) \cdot x + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, \color{blue}{x}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1 \cdot 1, x, 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - \left(\mathsf{neg}\left(-1\right)\right) \cdot 1, x, 1\right) \]

      6. fp-cancel-sign-sub N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + -1 \cdot 1, x, 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + -1, x, 1\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x + -1, x, 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + \frac{-1}{6} \cdot x, x, -1\right), x, 1\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6} \cdot x + \frac{1}{2}, x, -1\right), x, 1\right) \]

      11. lower-fma.f64 8.6

          \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]

   8. Applied rewrites 8.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}} \]

       2. lift-neg.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

       3. lift-exp.f64 N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}} \]

       4. exp-neg N/A

          \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

       5. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]

   13. Applied rewrites 96.1%

       \[\leadsto \color{blue}{\frac{\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1}{e^{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{1}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (*
    (fmod (exp x) 1.0)
    (fma (fma (fma -0.16666666666666666 x 0.5) x -1.0) x 1.0))
   (* (fmod x (sqrt 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), 1.0) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0);
	} else {
		tmp = fmod(x, sqrt(1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), 1.0) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(x, sqrt(1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

   5. Applied rewrites 9.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) \cdot x + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, \color{blue}{x}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1 \cdot 1, x, 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - \left(\mathsf{neg}\left(-1\right)\right) \cdot 1, x, 1\right) \]

      6. fp-cancel-sign-sub N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + -1 \cdot 1, x, 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + -1, x, 1\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x + -1, x, 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + \frac{-1}{6} \cdot x, x, -1\right), x, 1\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6} \cdot x + \frac{1}{2}, x, -1\right), x, 1\right) \]

      11. lower-fma.f64 8.6

          \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]

   8. Applied rewrites 8.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 61.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (* (fmod (exp x) 1.0) (fma (fma (* -0.16666666666666666 x) x -1.0) x 1.0))
   (* (fmod x (sqrt 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), 1.0) * fma(fma((-0.16666666666666666 * x), x, -1.0), x, 1.0);
	} else {
		tmp = fmod(x, sqrt(1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), 1.0) * fma(fma(Float64(-0.16666666666666666 * x), x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(x, sqrt(1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

   5. Applied rewrites 9.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) \cdot x + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, \color{blue}{x}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1 \cdot 1, x, 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - \left(\mathsf{neg}\left(-1\right)\right) \cdot 1, x, 1\right) \]

      6. fp-cancel-sign-sub N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + -1 \cdot 1, x, 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + -1, x, 1\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x + -1, x, 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + \frac{-1}{6} \cdot x, x, -1\right), x, 1\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6} \cdot x + \frac{1}{2}, x, -1\right), x, 1\right) \]

      11. lower-fma.f64 8.6

          \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]

   8. Applied rewrites 8.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6} \cdot x, x, -1\right), x, 1\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 8.6

          \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, -1\right), x, 1\right) \]

   11. Applied rewrites 8.6%

       \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, -1\right), x, 1\right) \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 61.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (* (fmod (exp x) 1.0) (fma (fma 0.5 x -1.0) x 1.0))
   (* (fmod x (sqrt 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod(x, sqrt(1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(x, sqrt(1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

   5. Applied rewrites 9.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} \cdot x - 1\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(\left(\frac{1}{2} \cdot x - 1\right) \cdot x + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot x - 1, \color{blue}{x}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot x - -1 \cdot -1, x, 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(1\right)\right) \cdot -1, x, 1\right) \]

      6. fp-cancel-sign-sub N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot x + 1 \cdot -1, x, 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot x + -1, x, 1\right) \]

      8. lower-fma.f64 8.6

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]

   8. Applied rewrites 8.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 61.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (* (fmod (exp x) 1.0) (- 1.0 x))
   (* (fmod x (sqrt 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), 1.0) * (1.0 - x);
	} else {
		tmp = fmod(x, sqrt(1.0)) * exp(-x);
	}
	return tmp;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-309)) then
        tmp = mod(exp(x), 1.0d0) * (1.0d0 - x)
    else
        tmp = mod(x, sqrt(1.0d0)) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-309:
		tmp = math.fmod(math.exp(x), 1.0) * (1.0 - x)
	else:
		tmp = math.fmod(x, math.sqrt(1.0)) * math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), 1.0) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, sqrt(1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

   5. Applied rewrites 9.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   7. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - 1 \cdot x\right) \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right) \]

      4. lower--.f64 8.6

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - \color{blue}{x}\right) \]

   8. Applied rewrites 8.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{\left(1 - x\right)} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 60.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (* (fmod (exp x) 1.0) (- 1.0 x))
   (* (fmod x (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), 1.0) * (1.0 - x);
	} else {
		tmp = fmod(x, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-309)) then
        tmp = mod(exp(x), 1.0d0) * (1.0d0 - x)
    else
        tmp = mod(x, sqrt(1.0d0)) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-309:
		tmp = math.fmod(math.exp(x), 1.0) * (1.0 - x)
	else:
		tmp = math.fmod(x, math.sqrt(1.0)) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), 1.0) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

   5. Applied rewrites 9.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   7. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - 1 \cdot x\right) \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right) \]

      4. lower--.f64 8.6

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - \color{blue}{x}\right) \]

   8. Applied rewrites 8.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{\left(1 - x\right)} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{1} \]
   13. Step-by-step derivation

   14. Applied rewrites 93.4%

       \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 60.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309) (* (fmod (exp x) 1.0) 1.0) (* (fmod x (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), 1.0) * 1.0;
	} else {
		tmp = fmod(x, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-309)) then
        tmp = mod(exp(x), 1.0d0) * 1.0d0
    else
        tmp = mod(x, sqrt(1.0d0)) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-309:
		tmp = math.fmod(math.exp(x), 1.0) * 1.0
	else:
		tmp = math.fmod(x, math.sqrt(1.0)) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), 1.0) * 1.0);
	else
		tmp = Float64(rem(x, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 9.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

   5. Applied rewrites 9.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 7.5%

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \color{blue}{1} \]

   if -1.000000000000002e-309 < x

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. lower--.f64 37.9

         \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 37.9%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.1%

       \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{1} \]
   13. Step-by-step derivation

   14. Applied rewrites 93.4%

       \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 58.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(x \bmod \left(\sqrt{1}\right)\right) \cdot 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod x (sqrt 1.0)) 1.0))

C

double code(double x) {
	return fmod(x, sqrt(1.0)) * 1.0;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = mod(x, sqrt(1.0d0)) * 1.0d0
end function

Python

def code(x):
	return math.fmod(x, math.sqrt(1.0)) * 1.0

Julia

function code(x)
	return Float64(rem(x, sqrt(1.0)) * 1.0)
end

Wolfram

code[x_] := N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 8.6%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(x + \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. metadata-eval N/A

      \[\leadsto \left(\left(x + 1 \cdot \color{blue}{1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   4. metadata-eval N/A

      \[\leadsto \left(\left(x - -1 \cdot 1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. lower--.f64 25.5

      \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

5. Applied rewrites 25.5%

   \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

6. Taylor expanded in x around inf

   \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
7. Step-by-step derivation

8. Applied rewrites 56.5%

   \[\leadsto \left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

9. Taylor expanded in x around 0

   \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
10. Step-by-step derivation

11. Applied rewrites 56.5%

    \[\leadsto \left(x \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]

12. Taylor expanded in x around 0

    \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{1} \]
13. Step-by-step derivation

14. Applied rewrites 55.0%

    \[\leadsto \left(x \bmod \left(\sqrt{1}\right)\right) \cdot \color{blue}{1} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025051 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))