expfmod (used to be hard to sample)

Percentage Accurate: 6.8% → 58.5%

Time: 8.4s

Alternatives: 17

Speedup: 3.9×

• could not determine a ground truth

  1. x = -3.843748807958351e+100

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: 6.8% 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: 58.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := {\left(x \cdot x\right)}^{-0.5}\\ \mathbf{if}\;x \leq -4 \cdot 10^{-47}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(t\_1 \cdot t\_1 - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;\left(1 \bmod \left(\left(\frac{{x}^{-6} - 0.015625}{\left({x}^{-4} + 0.0625\right) + {x}^{-2} \cdot 0.25} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (pow (* x x) -0.5)))
   (if (<= x -4e-47)
     (* (fmod (exp x) (* (* (- (* t_1 t_1) 0.25) x) x)) t_0)
     (if (<= x -8.6e-78)
       (*
        (fmod
         1.0
         (*
          (*
           (/
            (- (pow x -6.0) 0.015625)
            (+ (+ (pow x -4.0) 0.0625) (* (pow x -2.0) 0.25)))
           x)
          x))
        t_0)
       (if (<= x -7.5e-155)
         (*
          (fmod
           1.0
           (* (* (/ (- (pow x -4.0) 0.0625) (- (pow x -2.0) -0.25)) x) x))
          t_0)
         (*
          (fmod 1.0 (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x))
          t_0))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = pow((x * x), -0.5);
	double tmp;
	if (x <= -4e-47) {
		tmp = fmod(exp(x), ((((t_1 * t_1) - 0.25) * x) * x)) * t_0;
	} else if (x <= -8.6e-78) {
		tmp = fmod(1.0, ((((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 <= -7.5e-155) {
		tmp = fmod(1.0, ((((pow(x, -4.0) - 0.0625) / (pow(x, -2.0) - -0.25)) * x) * x)) * t_0;
	} else {
		tmp = fmod(1.0, (((exp((log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-x)
    t_1 = (x * x) ** (-0.5d0)
    if (x <= (-4d-47)) then
        tmp = mod(exp(x), ((((t_1 * t_1) - 0.25d0) * x) * x)) * t_0
    else if (x <= (-8.6d-78)) then
        tmp = mod(1.0d0, (((((x ** (-6.0d0)) - 0.015625d0) / (((x ** (-4.0d0)) + 0.0625d0) + ((x ** (-2.0d0)) * 0.25d0))) * x) * x)) * t_0
    else if (x <= (-7.5d-155)) then
        tmp = mod(1.0d0, (((((x ** (-4.0d0)) - 0.0625d0) / ((x ** (-2.0d0)) - (-0.25d0))) * x) * x)) * t_0
    else
        tmp = mod(1.0d0, (((exp((log((x * x)) * (-1.0d0))) - 0.25d0) * x) * x)) * t_0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	t_1 = math.pow((x * x), -0.5)
	tmp = 0
	if x <= -4e-47:
		tmp = math.fmod(math.exp(x), ((((t_1 * t_1) - 0.25) * x) * x)) * t_0
	elif x <= -8.6e-78:
		tmp = math.fmod(1.0, ((((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 <= -7.5e-155:
		tmp = math.fmod(1.0, ((((math.pow(x, -4.0) - 0.0625) / (math.pow(x, -2.0) - -0.25)) * x) * x)) * t_0
	else:
		tmp = math.fmod(1.0, (((math.exp((math.log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(x * x) ^ -0.5
	tmp = 0.0
	if (x <= -4e-47)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64(t_1 * t_1) - 0.25) * x) * x)) * t_0);
	elseif (x <= -8.6e-78)
		tmp = Float64(rem(1.0, 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 <= -7.5e-155)
		tmp = Float64(rem(1.0, Float64(Float64(Float64(Float64((x ^ -4.0) - 0.0625) / Float64((x ^ -2.0) - -0.25)) * x) * x)) * t_0);
	else
		tmp = Float64(rem(1.0, Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x * x), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[x, -4e-47], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -8.6e-78], N[(N[With[{TMP1 = 1.0, 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, -7.5e-155], N[(N[With[{TMP1 = 1.0, 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], N[(N[With[{TMP1 = 1.0, 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]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.9999999999999999e-47

   1. Initial program 47.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 47.9

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

   5. Applied rewrites 47.9%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 48.4

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

   8. Applied rewrites 48.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   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. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 67.7

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({\left(x \cdot x\right)}^{-0.5} \cdot {\left(x \cdot x\right)}^{-0.5} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 67.7%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({\left(x \cdot x\right)}^{-0.5} \cdot {\left(x \cdot x\right)}^{-0.5} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -3.9999999999999999e-47 < x < -8.59999999999999987e-78

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 13.4

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

   8. Applied rewrites 13.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 13.4%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-flip N/A

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

       4. lower--.f64 N/A

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

       5. flip3-- N/A

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

       6. lower-/.f64 N/A

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

       7. lower--.f64 N/A

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

       8. pow-flip N/A

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

       9. metadata-eval N/A

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

       10. pow-pow N/A

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

       11. lower-pow.f64 N/A

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

       12. metadata-eval N/A

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

       13. metadata-eval N/A

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

       14. associate-+r+ N/A

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

       15. lower-+.f64 N/A

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

   13. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\left(\frac{{x}^{-6} - 0.015625}{\left({x}^{-4} + 0.0625\right) + {x}^{-2} \cdot 0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -8.59999999999999987e-78 < x < -7.5000000000000006e-155

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 21.1

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

   8. Applied rewrites 21.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 21.1%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-flip N/A

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

       4. lower--.f64 N/A

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

       5. flip-- N/A

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

       6. lower-/.f64 N/A

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

       7. lower--.f64 N/A

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

       8. pow-flip N/A

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

       9. metadata-eval N/A

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

       10. pow-flip N/A

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

       11. metadata-eval N/A

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

       12. pow-prod-up N/A

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

       13. lower-pow.f64 N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

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

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

       20. lower--.f64 N/A

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

       21. pow-flip N/A

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

       22. metadata-eval N/A

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

       23. lift-pow.f64 100.0

           \[\leadsto \left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   13. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -7.5000000000000006e-155 < x

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 6.0

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

   5. Applied rewrites 6.0%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 27.4

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

   8. Applied rewrites 27.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.6%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-pow N/A

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

       4. pow-to-exp N/A

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

       5. lower-exp.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-log.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 50.6

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

   13. Applied rewrites 50.6%

       \[\leadsto \left(1 \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 47.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(1 \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* (fmod (exp x) (sqrt (cos x))) t_0)))
   (if (<= t_1 0.0)
     (* (fmod 1.0 (* (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) x) x)) t_0)
     (if (<= t_1 2.0)
       (*
        (fmod
         (exp x)
         (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
        t_0)
       (* (fmod 1.0 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x))) * t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fmod(1.0, ((fma((-1.0 / x), (-1.0 / x), -0.25) * x) * x)) * t_0;
	} else if (t_1 <= 2.0) {
		tmp = fmod(exp(x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(rem(exp(x), sqrt(cos(x))) * t_0)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(rem(1.0, Float64(Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * x) * x)) * t_0);
	elseif (t_1 <= 2.0)
		tmp = Float64(rem(exp(x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[With[{TMP1 = 1.0, 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] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.1

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

   5. Applied rewrites 4.1%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 30.4

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

   8. Applied rewrites 30.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 30.4%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow2 N/A

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

       9. times-frac N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 32.7

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

   13. Applied rewrites 32.7%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 72.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, \color{blue}{{x}^{2}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{2} \cdot \frac{1}{2}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

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

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} + \frac{-1}{2} \cdot \frac{1}{2}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} + \frac{-1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, {x}^{2}, \frac{-1}{4}\right), {\color{blue}{x}}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      11. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot e^{-x} \]

      12. lower-*.f64 67.9

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 67.9%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 0.0%

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

   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 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

Alternative 3: 47.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(1 \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* (fmod (exp x) (sqrt (cos x))) t_0)))
   (if (<= t_1 0.0)
     (* (fmod 1.0 (* (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) x) x)) t_0)
     (if (<= t_1 2.0)
       (* (fmod (exp x) (fma (* x x) -0.25 1.0)) t_0)
       (* (fmod 1.0 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x))) * t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fmod(1.0, ((fma((-1.0 / x), (-1.0 / x), -0.25) * x) * x)) * t_0;
	} else if (t_1 <= 2.0) {
		tmp = fmod(exp(x), fma((x * x), -0.25, 1.0)) * t_0;
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(rem(exp(x), sqrt(cos(x))) * t_0)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(rem(1.0, Float64(Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * x) * x)) * t_0);
	elseif (t_1 <= 2.0)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) * t_0);
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[With[{TMP1 = 1.0, 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] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.1

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

   5. Applied rewrites 4.1%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 30.4

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

   8. Applied rewrites 30.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 30.4%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow2 N/A

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

       9. times-frac N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 32.7

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

   13. Applied rewrites 32.7%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 72.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 0.0%

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

   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 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

Alternative 4: 47.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{-1}{x} \cdot \frac{-1}{x} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (* (fmod (exp x) (* (* (- (* (/ -1.0 x) (/ -1.0 x)) 0.25) x) x)) t_0)
     (* (fmod 1.0 1.0) 1.0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = fmod(exp(x), (((((-1.0 / x) * (-1.0 / x)) - 0.25) * x) * x)) * t_0;
	} else {
		tmp = fmod(1.0, 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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if ((mod(exp(x), sqrt(cos(x))) * t_0) <= 2.0d0) then
        tmp = mod(exp(x), ((((((-1.0d0) / x) * ((-1.0d0) / x)) - 0.25d0) * x) * x)) * t_0
    else
        tmp = mod(1.0d0, 1.0d0) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if (math.fmod(math.exp(x), math.sqrt(math.cos(x))) * t_0) <= 2.0:
		tmp = math.fmod(math.exp(x), (((((-1.0 / x) * (-1.0 / x)) - 0.25) * x) * x)) * t_0
	else:
		tmp = math.fmod(1.0, 1.0) * 1.0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64(Float64(-1.0 / x) * Float64(-1.0 / x)) - 0.25) * x) * x)) * t_0);
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 10.4%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 9.9

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

   5. Applied rewrites 9.9%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 33.7

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

   8. Applied rewrites 33.7%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 35.7

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

   10. Applied rewrites 35.7%

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

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 0.0%

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

   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 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

Alternative 5: 46.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{x \cdot x} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (* (fmod (exp x) (* (* (- (/ 1.0 (* x x)) 0.25) x) x)) t_0)
     (* (fmod 1.0 1.0) 1.0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = fmod(exp(x), ((((1.0 / (x * x)) - 0.25) * x) * x)) * t_0;
	} else {
		tmp = fmod(1.0, 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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if ((mod(exp(x), sqrt(cos(x))) * t_0) <= 2.0d0) then
        tmp = mod(exp(x), ((((1.0d0 / (x * x)) - 0.25d0) * x) * x)) * t_0
    else
        tmp = mod(1.0d0, 1.0d0) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if (math.fmod(math.exp(x), math.sqrt(math.cos(x))) * t_0) <= 2.0:
		tmp = math.fmod(math.exp(x), ((((1.0 / (x * x)) - 0.25) * x) * x)) * t_0
	else:
		tmp = math.fmod(1.0, 1.0) * 1.0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64(1.0 / Float64(x * x)) - 0.25) * x) * x)) * t_0);
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], 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] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 10.4%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 9.9

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

   5. Applied rewrites 9.9%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 33.7

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

   8. Applied rewrites 33.7%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   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 34.8

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

   10. Applied rewrites 34.8%

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

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 0.0%

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

   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 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

Alternative 6: 45.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 0:\\ \;\;\;\;\left(1 \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 0.0)
     (* (fmod 1.0 (* (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) x) x)) t_0)
     (* (fmod (+ 1.0 x) 1.0) 1.0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 0.0) {
		tmp = fmod(1.0, ((fma((-1.0 / x), (-1.0 / x), -0.25) * x) * x)) * t_0;
	} else {
		tmp = fmod((1.0 + x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 0.0)
		tmp = Float64(rem(1.0, Float64(Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * x) * x)) * t_0);
	else
		tmp = Float64(rem(Float64(1.0 + x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 0.0], N[(N[With[{TMP1 = 1.0, 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] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = N[(1.0 + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.1

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

   5. Applied rewrites 4.1%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 30.4

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

   8. Applied rewrites 30.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 30.4%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow2 N/A

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

       9. times-frac N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 32.7

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

   13. Applied rewrites 32.7%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 20.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 17.7%

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

   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 13.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 83.9

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

   11. Applied rewrites 83.9%

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

Alternative 7: 44.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 0:\\ \;\;\;\;\left(1 \bmod \left(\left(\left(\frac{1}{x \cdot x} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 0.0)
     (* (fmod 1.0 (* (* (- (/ 1.0 (* x x)) 0.25) x) x)) t_0)
     (* (fmod (+ 1.0 x) 1.0) 1.0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 0.0) {
		tmp = fmod(1.0, ((((1.0 / (x * x)) - 0.25) * x) * x)) * t_0;
	} else {
		tmp = fmod((1.0 + x), 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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if ((mod(exp(x), sqrt(cos(x))) * t_0) <= 0.0d0) then
        tmp = mod(1.0d0, ((((1.0d0 / (x * x)) - 0.25d0) * x) * x)) * t_0
    else
        tmp = mod((1.0d0 + x), 1.0d0) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if (math.fmod(math.exp(x), math.sqrt(math.cos(x))) * t_0) <= 0.0:
		tmp = math.fmod(1.0, ((((1.0 / (x * x)) - 0.25) * x) * x)) * t_0
	else:
		tmp = math.fmod((1.0 + x), 1.0) * 1.0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 0.0)
		tmp = Float64(rem(1.0, Float64(Float64(Float64(Float64(1.0 / Float64(x * x)) - 0.25) * x) * x)) * t_0);
	else
		tmp = Float64(rem(Float64(1.0 + x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 0.0], N[(N[With[{TMP1 = 1.0, 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] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = N[(1.0 + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.1

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

   5. Applied rewrites 4.1%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 30.4

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

   8. Applied rewrites 30.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 30.4%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-flip N/A

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

       4. lower-/.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 31.6

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

   13. Applied rewrites 31.6%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 20.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 17.7%

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

   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 13.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 83.9

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

   11. Applied rewrites 83.9%

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

Alternative 8: 58.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-6} - 0.015625}{{x}^{-4} + \left(0.0625 + {x}^{-2} \cdot 0.25\right)} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -8.6e-78)
     (*
      (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 -7.5e-155)
       (*
        (fmod
         1.0
         (* (* (/ (- (pow x -4.0) 0.0625) (- (pow x -2.0) -0.25)) x) x))
        t_0)
       (* (fmod 1.0 (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x)) t_0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -8.6e-78) {
		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 <= -7.5e-155) {
		tmp = fmod(1.0, ((((pow(x, -4.0) - 0.0625) / (pow(x, -2.0) - -0.25)) * x) * x)) * t_0;
	} else {
		tmp = fmod(1.0, (((exp((log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-8.6d-78)) 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 <= (-7.5d-155)) then
        tmp = mod(1.0d0, (((((x ** (-4.0d0)) - 0.0625d0) / ((x ** (-2.0d0)) - (-0.25d0))) * x) * x)) * t_0
    else
        tmp = mod(1.0d0, (((exp((log((x * x)) * (-1.0d0))) - 0.25d0) * x) * x)) * t_0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -8.6e-78:
		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 <= -7.5e-155:
		tmp = math.fmod(1.0, ((((math.pow(x, -4.0) - 0.0625) / (math.pow(x, -2.0) - -0.25)) * x) * x)) * t_0
	else:
		tmp = math.fmod(1.0, (((math.exp((math.log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -8.6e-78)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64((x ^ -6.0) - 0.015625) / Float64((x ^ -4.0) + Float64(0.0625 + Float64((x ^ -2.0) * 0.25)))) * x) * x)) * t_0);
	elseif (x <= -7.5e-155)
		tmp = Float64(rem(1.0, Float64(Float64(Float64(Float64((x ^ -4.0) - 0.0625) / Float64((x ^ -2.0) - -0.25)) * x) * x)) * t_0);
	else
		tmp = Float64(rem(1.0, Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -8.6e-78], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(N[Power[x, -6.0], $MachinePrecision] - 0.015625), $MachinePrecision] / N[(N[Power[x, -4.0], $MachinePrecision] + N[(0.0625 + N[(N[Power[x, -2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -7.5e-155], N[(N[With[{TMP1 = 1.0, 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], N[(N[With[{TMP1 = 1.0, 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]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.59999999999999987e-78

   1. Initial program 28.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 28.0

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

   5. Applied rewrites 28.0%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 32.8

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

   8. Applied rewrites 32.8%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   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. pow-flip N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-flip N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. pow-flip N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 N/A

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

   10. Applied rewrites 75.9%

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

   if -8.59999999999999987e-78 < x < -7.5000000000000006e-155

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 21.1

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

   8. Applied rewrites 21.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 21.1%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-flip N/A

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

       4. lower--.f64 N/A

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

       5. flip-- N/A

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

       6. lower-/.f64 N/A

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

       7. lower--.f64 N/A

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

       8. pow-flip N/A

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

       9. metadata-eval N/A

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

       10. pow-flip N/A

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

       11. metadata-eval N/A

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

       12. pow-prod-up N/A

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

       13. lower-pow.f64 N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

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

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

       20. lower--.f64 N/A

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

       21. pow-flip N/A

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

       22. metadata-eval N/A

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

       23. lift-pow.f64 100.0

           \[\leadsto \left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   13. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -7.5000000000000006e-155 < x

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 6.0

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

   5. Applied rewrites 6.0%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 27.4

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

   8. Applied rewrites 27.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.6%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-pow N/A

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

       4. pow-to-exp N/A

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

       5. lower-exp.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-log.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 50.6

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

   13. Applied rewrites 50.6%

       \[\leadsto \left(1 \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 44.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(1 \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 0.0)
   (* (fmod 1.0 (* (* (- (pow x -2.0) 0.25) x) x)) (- 1.0 x))
   (* (fmod (+ 1.0 x) 1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 0.0) {
		tmp = fmod(1.0, (((pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x);
	} else {
		tmp = fmod((1.0 + x), 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 ((mod(exp(x), sqrt(cos(x))) * exp(-x)) <= 0.0d0) then
        tmp = mod(1.0d0, ((((x ** (-2.0d0)) - 0.25d0) * x) * x)) * (1.0d0 - x)
    else
        tmp = mod((1.0d0 + x), 1.0d0) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if (math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)) <= 0.0:
		tmp = math.fmod(1.0, (((math.pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x)
	else:
		tmp = math.fmod((1.0 + x), 1.0) * 1.0
	return tmp

Julia

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

Wolfram

code[x_] := If[LessEqual[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], 0.0], N[(N[With[{TMP1 = 1.0, 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 = N[(1.0 + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.1

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

   5. Applied rewrites 4.1%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 30.4

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

   8. Applied rewrites 30.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 30.4%

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

   12. Taylor expanded in x around 0

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. lower--.f64 30.4

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

   14. Applied rewrites 30.4%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 20.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 17.7%

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

   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 13.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 83.9

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

   11. Applied rewrites 83.9%

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

Alternative 10: 55.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} + 0.25} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -7.5e-155)
     (*
      (fmod
       (exp x)
       (* (* (/ (- (pow x -4.0) 0.0625) (+ (pow x -2.0) 0.25)) x) x))
      t_0)
     (* (fmod 1.0 (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x)) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -7.5e-155) {
		tmp = fmod(exp(x), ((((pow(x, -4.0) - 0.0625) / (pow(x, -2.0) + 0.25)) * x) * x)) * t_0;
	} else {
		tmp = fmod(1.0, (((exp((log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-7.5d-155)) then
        tmp = mod(exp(x), (((((x ** (-4.0d0)) - 0.0625d0) / ((x ** (-2.0d0)) + 0.25d0)) * x) * x)) * t_0
    else
        tmp = mod(1.0d0, (((exp((log((x * x)) * (-1.0d0))) - 0.25d0) * x) * x)) * t_0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -7.5e-155:
		tmp = math.fmod(math.exp(x), ((((math.pow(x, -4.0) - 0.0625) / (math.pow(x, -2.0) + 0.25)) * x) * x)) * t_0
	else:
		tmp = math.fmod(1.0, (((math.exp((math.log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -7.5e-155)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64((x ^ -4.0) - 0.0625) / Float64((x ^ -2.0) + 0.25)) * x) * x)) * t_0);
	else
		tmp = Float64(rem(1.0, Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -7.5e-155], 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], N[(N[With[{TMP1 = 1.0, 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]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000006e-155

   1. Initial program 14.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 14.9

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

   5. Applied rewrites 14.9%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 26.6

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

   8. Applied rewrites 26.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   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. lower-+.f64 N/A

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

      17. pow-flip N/A

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

      18. metadata-eval N/A

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

      19. lift-pow.f64 72.2

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

   10. Applied rewrites 72.2%

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

   if -7.5000000000000006e-155 < x

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 6.0

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

   5. Applied rewrites 6.0%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 27.4

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

   8. Applied rewrites 27.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.6%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-pow N/A

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

       4. pow-to-exp N/A

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

       5. lower-exp.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-log.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 50.6

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

   13. Applied rewrites 50.6%

       \[\leadsto \left(1 \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 58.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -5 \cdot 10^{-78}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod t\_1\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod t\_1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))
        (t_1 (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x)))
   (if (<= x -5e-78)
     (* (fmod (exp x) t_1) t_0)
     (if (<= x -7.5e-155)
       (*
        (fmod
         1.0
         (* (* (/ (- (pow x -4.0) 0.0625) (- (pow x -2.0) -0.25)) x) x))
        t_0)
       (* (fmod 1.0 t_1) t_0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = ((exp((log((x * x)) * -1.0)) - 0.25) * x) * x;
	double tmp;
	if (x <= -5e-78) {
		tmp = fmod(exp(x), t_1) * t_0;
	} else if (x <= -7.5e-155) {
		tmp = fmod(1.0, ((((pow(x, -4.0) - 0.0625) / (pow(x, -2.0) - -0.25)) * x) * x)) * t_0;
	} else {
		tmp = fmod(1.0, t_1) * t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-x)
    t_1 = ((exp((log((x * x)) * (-1.0d0))) - 0.25d0) * x) * x
    if (x <= (-5d-78)) then
        tmp = mod(exp(x), t_1) * t_0
    else if (x <= (-7.5d-155)) then
        tmp = mod(1.0d0, (((((x ** (-4.0d0)) - 0.0625d0) / ((x ** (-2.0d0)) - (-0.25d0))) * x) * x)) * t_0
    else
        tmp = mod(1.0d0, t_1) * t_0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	t_1 = ((math.exp((math.log((x * x)) * -1.0)) - 0.25) * x) * x
	tmp = 0
	if x <= -5e-78:
		tmp = math.fmod(math.exp(x), t_1) * t_0
	elif x <= -7.5e-155:
		tmp = math.fmod(1.0, ((((math.pow(x, -4.0) - 0.0625) / (math.pow(x, -2.0) - -0.25)) * x) * x)) * t_0
	else:
		tmp = math.fmod(1.0, t_1) * t_0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)
	tmp = 0.0
	if (x <= -5e-78)
		tmp = Float64(rem(exp(x), t_1) * t_0);
	elseif (x <= -7.5e-155)
		tmp = Float64(rem(1.0, Float64(Float64(Float64(Float64((x ^ -4.0) - 0.0625) / Float64((x ^ -2.0) - -0.25)) * x) * x)) * t_0);
	else
		tmp = Float64(rem(1.0, t_1) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -5e-78], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -7.5e-155], N[(N[With[{TMP1 = 1.0, 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], N[(N[With[{TMP1 = 1.0, TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999996e-78

   1. Initial program 28.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 28.0

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

   5. Applied rewrites 28.0%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 32.8

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

   8. Applied rewrites 32.8%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   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 53.1

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

   10. Applied rewrites 53.1%

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

   if -4.9999999999999996e-78 < x < -7.5000000000000006e-155

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 21.1

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

   8. Applied rewrites 21.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 21.1%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-flip N/A

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

       4. lower--.f64 N/A

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

       5. flip-- N/A

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

       6. lower-/.f64 N/A

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

       7. lower--.f64 N/A

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

       8. pow-flip N/A

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

       9. metadata-eval N/A

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

       10. pow-flip N/A

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

       11. metadata-eval N/A

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

       12. pow-prod-up N/A

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

       13. lower-pow.f64 N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

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

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

       20. lower--.f64 N/A

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

       21. pow-flip N/A

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

       22. metadata-eval N/A

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

       23. lift-pow.f64 100.0

           \[\leadsto \left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   13. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   if -7.5000000000000006e-155 < x

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 6.0

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

   5. Applied rewrites 6.0%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 27.4

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

   8. Applied rewrites 27.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.6%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-pow N/A

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

       4. pow-to-exp N/A

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

       5. lower-exp.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-log.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 50.6

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

   13. Applied rewrites 50.6%

       \[\leadsto \left(1 \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 58.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{-1}{x} \cdot \frac{-1}{x} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-78} \lor \neg \left(x \leq -7.5 \cdot 10^{-155}\right):\\ \;\;\;\;\left(1 \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -3.1e-24)
     (* (fmod (exp x) (* (* (- (* (/ -1.0 x) (/ -1.0 x)) 0.25) x) x)) t_0)
     (if (or (<= x -5e-78) (not (<= x -7.5e-155)))
       (* (fmod 1.0 (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x)) t_0)
       (*
        (fmod
         1.0
         (* (* (/ (- (pow x -4.0) 0.0625) (- (pow x -2.0) -0.25)) x) x))
        t_0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -3.1e-24) {
		tmp = fmod(exp(x), (((((-1.0 / x) * (-1.0 / x)) - 0.25) * x) * x)) * t_0;
	} else if ((x <= -5e-78) || !(x <= -7.5e-155)) {
		tmp = fmod(1.0, (((exp((log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0;
	} else {
		tmp = fmod(1.0, ((((pow(x, -4.0) - 0.0625) / (pow(x, -2.0) - -0.25)) * x) * x)) * t_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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-3.1d-24)) then
        tmp = mod(exp(x), ((((((-1.0d0) / x) * ((-1.0d0) / x)) - 0.25d0) * x) * x)) * t_0
    else if ((x <= (-5d-78)) .or. (.not. (x <= (-7.5d-155)))) then
        tmp = mod(1.0d0, (((exp((log((x * x)) * (-1.0d0))) - 0.25d0) * x) * x)) * t_0
    else
        tmp = mod(1.0d0, (((((x ** (-4.0d0)) - 0.0625d0) / ((x ** (-2.0d0)) - (-0.25d0))) * x) * x)) * t_0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -3.1e-24:
		tmp = math.fmod(math.exp(x), (((((-1.0 / x) * (-1.0 / x)) - 0.25) * x) * x)) * t_0
	elif (x <= -5e-78) or not (x <= -7.5e-155):
		tmp = math.fmod(1.0, (((math.exp((math.log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0
	else:
		tmp = math.fmod(1.0, ((((math.pow(x, -4.0) - 0.0625) / (math.pow(x, -2.0) - -0.25)) * x) * x)) * t_0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -3.1e-24)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64(Float64(-1.0 / x) * Float64(-1.0 / x)) - 0.25) * x) * x)) * t_0);
	elseif ((x <= -5e-78) || !(x <= -7.5e-155))
		tmp = Float64(rem(1.0, Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)) * t_0);
	else
		tmp = Float64(rem(1.0, Float64(Float64(Float64(Float64((x ^ -4.0) - 0.0625) / Float64((x ^ -2.0) - -0.25)) * x) * x)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -3.1e-24], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[Or[LessEqual[x, -5e-78], N[Not[LessEqual[x, -7.5e-155]], $MachinePrecision]], N[(N[With[{TMP1 = 1.0, 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], N[(N[With[{TMP1 = 1.0, 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]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1e-24

   1. Initial program 70.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 70.3

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

   8. Applied rewrites 70.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 89.7

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

   10. Applied rewrites 89.7%

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

   if -3.1e-24 < x < -4.9999999999999996e-78 or -7.5000000000000006e-155 < x

   1. Initial program 6.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 5.8

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

   5. Applied rewrites 5.8%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 26.1

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

   8. Applied rewrites 26.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 46.6%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-pow N/A

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

       4. pow-to-exp N/A

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

       5. lower-exp.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-log.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 50.3

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

   13. Applied rewrites 50.3%

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

   if -4.9999999999999996e-78 < x < -7.5000000000000006e-155

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 21.1

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

   8. Applied rewrites 21.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 21.1%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-flip N/A

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

       4. lower--.f64 N/A

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

       5. flip-- N/A

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

       6. lower-/.f64 N/A

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

       7. lower--.f64 N/A

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

       8. pow-flip N/A

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

       9. metadata-eval N/A

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

       10. pow-flip N/A

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

       11. metadata-eval N/A

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

       12. pow-prod-up N/A

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

       13. lower-pow.f64 N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

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

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

       20. lower--.f64 N/A

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

       21. pow-flip N/A

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

       22. metadata-eval N/A

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

       23. lift-pow.f64 100.0

           \[\leadsto \left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]

   13. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{-1}{x} \cdot \frac{-1}{x} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-78} \lor \neg \left(x \leq -7.5 \cdot 10^{-155}\right):\\ \;\;\;\;\left(1 \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{-1}{x} \cdot \frac{-1}{x} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -3.1e-24)
     (* (fmod (exp x) (* (* (- (* (/ -1.0 x) (/ -1.0 x)) 0.25) x) x)) t_0)
     (* (fmod 1.0 (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x)) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -3.1e-24) {
		tmp = fmod(exp(x), (((((-1.0 / x) * (-1.0 / x)) - 0.25) * x) * x)) * t_0;
	} else {
		tmp = fmod(1.0, (((exp((log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-3.1d-24)) then
        tmp = mod(exp(x), ((((((-1.0d0) / x) * ((-1.0d0) / x)) - 0.25d0) * x) * x)) * t_0
    else
        tmp = mod(1.0d0, (((exp((log((x * x)) * (-1.0d0))) - 0.25d0) * x) * x)) * t_0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -3.1e-24:
		tmp = math.fmod(math.exp(x), (((((-1.0 / x) * (-1.0 / x)) - 0.25) * x) * x)) * t_0
	else:
		tmp = math.fmod(1.0, (((math.exp((math.log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -3.1e-24)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64(Float64(-1.0 / x) * Float64(-1.0 / x)) - 0.25) * x) * x)) * t_0);
	else
		tmp = Float64(rem(1.0, Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -3.1e-24], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, 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]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e-24

   1. Initial program 70.3%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 70.3

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

   8. Applied rewrites 70.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 89.7

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

   10. Applied rewrites 89.7%

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

   if -3.1e-24 < x

   1. Initial program 5.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 5.5

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

   5. Applied rewrites 5.5%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 25.5

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

   8. Applied rewrites 25.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 43.4%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-pow N/A

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

       4. pow-to-exp N/A

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

       5. lower-exp.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-log.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 50.9

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

   13. Applied rewrites 50.9%

       \[\leadsto \left(1 \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 46.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-17)
   (/ (fmod (exp x) 1.0) (exp x))
   (* (fmod 1.0 (* (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) x) x)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-17) {
		tmp = fmod(exp(x), 1.0) / exp(x);
	} else {
		tmp = fmod(1.0, ((fma((-1.0 / x), (-1.0 / x), -0.25) * x) * x)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-17)
		tmp = Float64(rem(exp(x), 1.0) / exp(x));
	else
		tmp = Float64(rem(1.0, Float64(Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * x) * x)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-17], 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 = 1.0, 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]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999999e-17

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. exp-neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   9. Applied rewrites 100.0%

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

   if -4.9999999999999999e-17 < x

   1. Initial program 5.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 5.4

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

   5. Applied rewrites 5.4%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 25.2

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

   8. Applied rewrites 25.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 43.0%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow2 N/A

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

       9. times-frac N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 44.9

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

   13. Applied rewrites 44.9%

       \[\leadsto \left(1 \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 46.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -5e-17)
     (* (fmod (exp x) 1.0) t_0)
     (* (fmod 1.0 (* (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) x) x)) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -5e-17) {
		tmp = fmod(exp(x), 1.0) * t_0;
	} else {
		tmp = fmod(1.0, ((fma((-1.0 / x), (-1.0 / x), -0.25) * x) * x)) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -5e-17)
		tmp = Float64(rem(exp(x), 1.0) * t_0);
	else
		tmp = Float64(rem(1.0, Float64(Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * x) * x)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -5e-17], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, 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] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999999e-17

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.1%

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

   if -4.9999999999999999e-17 < x

   1. Initial program 5.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 5.4

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

   5. Applied rewrites 5.4%

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

   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. pow2 N/A

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

      2. *-commutative N/A

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

      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 25.2

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

   8. Applied rewrites 25.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 43.0%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow2 N/A

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

       9. times-frac N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 44.9

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

   13. Applied rewrites 44.9%

       \[\leadsto \left(1 \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 23.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(1 + x\right) \bmod 1\right) \cdot 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod (+ 1.0 x) 1.0) 1.0))

C

double code(double x) {
	return fmod((1.0 + x), 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((1.0d0 + x), 1.0d0) * 1.0d0
end function

Python

def code(x):
	return math.fmod((1.0 + x), 1.0) * 1.0

Julia

function code(x)
	return Float64(rem(Float64(1.0 + x), 1.0) * 1.0)
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[(1.0 + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 8.4%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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 7.7%

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

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 6.6%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 25.3

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

11. Applied rewrites 25.3%

    \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod 1\right) \cdot 1 \]
12. Add Preprocessing

Alternative 17: 22.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \left(1 \bmod 1\right) \cdot 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod 1.0 1.0) 1.0))

C

double code(double x) {
	return fmod(1.0, 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(1.0d0, 1.0d0) * 1.0d0
end function

Python

def code(x):
	return math.fmod(1.0, 1.0) * 1.0

Julia

function code(x)
	return Float64(rem(1.0, 1.0) * 1.0)
end

Wolfram

code[x_] := N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 8.4%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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 7.7%

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

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 6.6%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 22.4%

    \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025082 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))