expfmod (used to be hard to sample)

Percentage Accurate: 7.1% → 95.6%

Time: 8.7s

Alternatives: 15

Speedup: 3.9×

• could not determine a ground truth

  1. x = -6.593505324823951e+167

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Initial Program: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := {x}^{-2} - -0.25\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-52}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{\left({x}^{-6} - 0.015625\right) \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(0.25, t\_1, {x}^{-4}\right)}\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{t\_1} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (- (pow x -2.0) -0.25)))
   (if (<= x -4.2e-52)
     (* (fmod (exp x) (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x)) t_0)
     (if (<= x -8.6e-78)
       (*
        (fmod
         (exp x)
         (/ (* (- (pow x -6.0) 0.015625) (* x x)) (fma 0.25 t_1 (pow x -4.0))))
        t_0)
       (if (<= x -7.5e-155)
         (* (fmod (exp x) (* (* (/ (- (pow x -4.0) 0.0625) t_1) x) x)) t_0)
         (if (<= x -2e-310)
           (* (fmod (exp x) (* (* (- (pow x -2.0) 0.25) x) x)) (- 1.0 x))
           (* (fmod x 1.0) 1.0)))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = pow(x, -2.0) - -0.25;
	double tmp;
	if (x <= -4.2e-52) {
		tmp = fmod(exp(x), (((exp((log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0;
	} else if (x <= -8.6e-78) {
		tmp = fmod(exp(x), (((pow(x, -6.0) - 0.015625) * (x * x)) / fma(0.25, t_1, pow(x, -4.0)))) * t_0;
	} else if (x <= -7.5e-155) {
		tmp = fmod(exp(x), ((((pow(x, -4.0) - 0.0625) / t_1) * x) * x)) * t_0;
	} else if (x <= -2e-310) {
		tmp = fmod(exp(x), (((pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x);
	} else {
		tmp = fmod(x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64((x ^ -2.0) - -0.25)
	tmp = 0.0
	if (x <= -4.2e-52)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)) * t_0);
	elseif (x <= -8.6e-78)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -6.0) - 0.015625) * Float64(x * x)) / fma(0.25, t_1, (x ^ -4.0)))) * t_0);
	elseif (x <= -7.5e-155)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64((x ^ -4.0) - 0.0625) / t_1) * x) * x)) * t_0);
	elseif (x <= -2e-310)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, 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[Power[x, -2.0], $MachinePrecision] - -0.25), $MachinePrecision]}, If[LessEqual[x, -4.2e-52], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -8.6e-78], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -6.0], $MachinePrecision] - 0.015625), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(0.25 * t$95$1 + N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $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] / t$95$1), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -4.1999999999999997e-52

   1. Initial program 30.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.7

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

   5. Applied rewrites 30.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 36.6

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

   8. Applied rewrites 36.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 61.4

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

   10. Applied rewrites 61.4%

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

   if -4.1999999999999997e-52 < 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. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 4.2

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

   8. Applied rewrites 4.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 55.1

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

   10. Applied rewrites 55.1%

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

   12. Applied rewrites 100.0%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\left({x}^{-6} - 0.015625\right) \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(0.25, \color{blue}{{x}^{-2} - -0.25}, {x}^{-4}\right)}\right)\right) \cdot e^{-x} \]

   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. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 9.1

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

   8. Applied rewrites 9.1%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow-flip N/A

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

      9. metadata-eval N/A

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

      10. pow-flip N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 N/A

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

      21. pow-flip N/A

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

      22. metadata-eval N/A

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

      23. lift-pow.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -7.5000000000000006e-155 < x < -1.999999999999994e-310

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. lower--.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-6} - 0.015625}{\left({x}^{-4} + 0.0625\right) + {x}^{-2} \cdot 0.25} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

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
         (exp x)
         (* (* (/ (- (pow x -4.0) 0.0625) (- (pow x -2.0) -0.25)) x) x))
        t_0)
       (if (<= x -2e-310)
         (* (fmod (exp x) (* (* (- (pow x -2.0) 0.25) x) x)) (- 1.0 x))
         (* (fmod x 1.0) 1.0))))))

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(exp(x), ((((pow(x, -4.0) - 0.0625) / (pow(x, -2.0) - -0.25)) * x) * x)) * t_0;
	} else if (x <= -2e-310) {
		tmp = fmod(exp(x), (((pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x);
	} else {
		tmp = fmod(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 (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(exp(x), (((((x ** (-4.0d0)) - 0.0625d0) / ((x ** (-2.0d0)) - (-0.25d0))) * x) * x)) * t_0
    else if (x <= (-2d-310)) then
        tmp = mod(exp(x), ((((x ** (-2.0d0)) - 0.25d0) * x) * x)) * (1.0d0 - x)
    else
        tmp = mod(x, 1.0d0) * 1.0d0
    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(math.exp(x), ((((math.pow(x, -4.0) - 0.0625) / (math.pow(x, -2.0) - -0.25)) * x) * x)) * t_0
	elif x <= -2e-310:
		tmp = math.fmod(math.exp(x), (((math.pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x)
	else:
		tmp = math.fmod(x, 1.0) * 1.0
	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(Float64((x ^ -4.0) + 0.0625) + Float64((x ^ -2.0) * 0.25))) * x) * x)) * t_0);
	elseif (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);
	elseif (x <= -2e-310)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, 1.0) * 1.0);
	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[(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 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(N[Power[x, -4.0], $MachinePrecision] - 0.0625), $MachinePrecision] / N[(N[Power[x, -2.0], $MachinePrecision] - -0.25), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.59999999999999987e-78

   1. Initial program 20.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 20.7

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

   5. Applied rewrites 20.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 24.8

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

   8. Applied rewrites 24.8%

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

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

      7. lower--.f64 N/A

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

      8. pow-flip N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. pow2 N/A

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

      16. frac-times N/A

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

      17. metadata-eval N/A

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

      18. swap-sqr N/A

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

   10. Applied rewrites 68.3%

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

   if -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. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 9.1

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

   8. Applied rewrites 9.1%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow-flip N/A

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

      9. metadata-eval N/A

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

      10. pow-flip N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 N/A

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

      21. pow-flip N/A

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

      22. metadata-eval N/A

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

      23. lift-pow.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -7.5000000000000006e-155 < x < -1.999999999999994e-310

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. lower--.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

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

Alternative 3: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-77}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1e-77)
     (* (fmod (exp x) (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x)) t_0)
     (if (<= x -7.5e-155)
       (*
        (fmod
         (exp x)
         (* (* (/ (- (pow x -4.0) 0.0625) (- (pow x -2.0) -0.25)) x) x))
        t_0)
       (if (<= x -2e-310)
         (* (fmod (exp x) (* (* (- (pow x -2.0) 0.25) x) x)) (- 1.0 x))
         (* (fmod x 1.0) 1.0))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1e-77) {
		tmp = fmod(exp(x), (((exp((log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0;
	} else 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 if (x <= -2e-310) {
		tmp = fmod(exp(x), (((pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x);
	} else {
		tmp = fmod(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 (x <= (-1d-77)) then
        tmp = mod(exp(x), (((exp((log((x * x)) * (-1.0d0))) - 0.25d0) * x) * x)) * t_0
    else 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 if (x <= (-2d-310)) then
        tmp = mod(exp(x), ((((x ** (-2.0d0)) - 0.25d0) * x) * x)) * (1.0d0 - x)
    else
        tmp = mod(x, 1.0d0) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -1e-77:
		tmp = math.fmod(math.exp(x), (((math.exp((math.log((x * x)) * -1.0)) - 0.25) * x) * x)) * t_0
	elif 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
	elif x <= -2e-310:
		tmp = math.fmod(math.exp(x), (((math.pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x)
	else:
		tmp = math.fmod(x, 1.0) * 1.0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1e-77)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)) * t_0);
	elseif (x <= -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);
	elseif (x <= -2e-310)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1e-77], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -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], If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.9999999999999993e-78

   1. Initial program 20.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 20.7

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

   5. Applied rewrites 20.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 24.8

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

   8. Applied rewrites 24.8%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   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. lower-*.f64 59.1

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

   10. Applied rewrites 59.1%

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

   if -9.9999999999999993e-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. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 9.1

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

   8. Applied rewrites 9.1%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow-flip N/A

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

      9. metadata-eval N/A

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

      10. pow-flip N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 N/A

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

      21. pow-flip N/A

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

      22. metadata-eval N/A

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

      23. lift-pow.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -7.5000000000000006e-155 < x < -1.999999999999994e-310

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. lower--.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

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

Alternative 4: 89.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (*
    (fmod (exp x) (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x))
    (exp (- x)))
   (* (fmod x 1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = fmod(exp(x), (((exp((log((x * x)) * -1.0)) - 0.25) * x) * x)) * exp(-x);
	} else {
		tmp = fmod(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 (x <= (-2d-310)) then
        tmp = mod(exp(x), (((exp((log((x * x)) * (-1.0d0))) - 0.25d0) * x) * x)) * exp(-x)
    else
        tmp = mod(x, 1.0d0) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = math.fmod(math.exp(x), (((math.exp((math.log((x * x)) * -1.0)) - 0.25) * x) * x)) * math.exp(-x)
	else:
		tmp = math.fmod(x, 1.0) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)) * exp(Float64(-x)));
	else
		tmp = Float64(rem(x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 8.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 8.2

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

   5. Applied rewrites 8.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 59.5

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

   8. Applied rewrites 59.5%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \cdot e^{-x} \]
   9. 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. lower-*.f64 76.5

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

   10. Applied rewrites 76.5%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

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

Alternative 5: 88.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-40}:\\ \;\;\;\;\left(\left(\left(\left({x}^{-2} + 0.5\right) + {x}^{-1}\right) \cdot \left(x \cdot x\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 \bmod \left(\left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.6e-40)
   (*
    (fmod (* (+ (+ (pow x -2.0) 0.5) (pow x -1.0)) (* x x)) 1.0)
    (fma (fma 0.5 x -1.0) x 1.0))
   (if (<= x -2e-310)
     (*
      (fmod 1.0 (* (* (- (exp (* (log (* x x)) -1.0)) 0.25) x) x))
      (exp (- x)))
     (* (fmod x 1.0) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -1.6e-40) {
		tmp = fmod((((pow(x, -2.0) + 0.5) + pow(x, -1.0)) * (x * x)), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else if (x <= -2e-310) {
		tmp = fmod(1.0, (((exp((log((x * x)) * -1.0)) - 0.25) * x) * x)) * exp(-x);
	} else {
		tmp = fmod(x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.6e-40)
		tmp = Float64(rem(Float64(Float64(Float64((x ^ -2.0) + 0.5) + (x ^ -1.0)) * Float64(x * x)), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0));
	elseif (x <= -2e-310)
		tmp = Float64(rem(1.0, Float64(Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * x) * x)) * exp(Float64(-x)));
	else
		tmp = Float64(rem(x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.6e-40], N[(N[With[{TMP1 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] + 0.5), $MachinePrecision] + N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-310], 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] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.60000000000000001e-40

   1. Initial program 33.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 33.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 31.7

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

   8. Applied rewrites 31.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 32.4

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

   11. Applied rewrites 32.4%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-+r+ N/A

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

       5. lower-+.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. pow-flip N/A

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

       9. metadata-eval N/A

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

       10. lift-pow.f64 N/A

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

       11. inv-pow N/A

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

       12. lower-pow.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 62.4

           \[\leadsto \left(\left(\left(\left({x}^{-2} + 0.5\right) + {x}^{-1}\right) \cdot \left(x \cdot x\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]

   14. Applied rewrites 62.4%

       \[\leadsto \left(\left(\left(\left({x}^{-2} + 0.5\right) + {x}^{-1}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]

   if -1.60000000000000001e-40 < x < -1.999999999999994e-310

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 63.3

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

   8. Applied rewrites 63.3%

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

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

      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. lower-*.f64 80.3

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

   10. Applied rewrites 80.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 80.3%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

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

Alternative 6: 83.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\left(\left(\left(\left({x}^{-2} + 0.5\right) + {x}^{-1}\right) \cdot \left(x \cdot x\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-77)
   (*
    (fmod (* (+ (+ (pow x -2.0) 0.5) (pow x -1.0)) (* x x)) 1.0)
    (fma (fma 0.5 x -1.0) x 1.0))
   (if (<= x -2e-310)
     (*
      (fmod (exp x) (* (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) x) x))
      (exp (- x)))
     (* (fmod x 1.0) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-77) {
		tmp = fmod((((pow(x, -2.0) + 0.5) + pow(x, -1.0)) * (x * x)), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else if (x <= -2e-310) {
		tmp = fmod(exp(x), ((fma((-1.0 / x), (-1.0 / x), -0.25) * x) * x)) * exp(-x);
	} else {
		tmp = fmod(x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-77)
		tmp = Float64(rem(Float64(Float64(Float64((x ^ -2.0) + 0.5) + (x ^ -1.0)) * Float64(x * x)), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0));
	elseif (x <= -2e-310)
		tmp = Float64(rem(exp(x), Float64(Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * x) * x)) * exp(Float64(-x)));
	else
		tmp = Float64(rem(x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-77], N[(N[With[{TMP1 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] + 0.5), $MachinePrecision] + N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision] + -0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9999999999999999e-77

   1. Initial program 21.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 21.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 20.1

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

   8. Applied rewrites 20.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 20.5

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

   11. Applied rewrites 20.5%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-+r+ N/A

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

       5. lower-+.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. pow-flip N/A

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

       9. metadata-eval N/A

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

       10. lift-pow.f64 N/A

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

       11. inv-pow N/A

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

       12. lower-pow.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 53.2

           \[\leadsto \left(\left(\left(\left({x}^{-2} + 0.5\right) + {x}^{-1}\right) \cdot \left(x \cdot x\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]

   14. Applied rewrites 53.2%

       \[\leadsto \left(\left(\left(\left({x}^{-2} + 0.5\right) + {x}^{-1}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]

   if -1.9999999999999999e-77 < x < -1.999999999999994e-310

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 72.6

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

   8. Applied rewrites 72.6%

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

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

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

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 79.4

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

   10. Applied rewrites 79.4%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

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

Alternative 7: 84.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\left(\left(\left(\left({x}^{-2} + 0.5\right) + {x}^{-1}\right) \cdot \left(x \cdot x\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-147)
   (*
    (fmod (* (+ (+ (pow x -2.0) 0.5) (pow x -1.0)) (* x x)) 1.0)
    (fma (fma 0.5 x -1.0) x 1.0))
   (if (<= x -2e-310)
     (* (fmod (exp x) (* (* (- (pow x -2.0) 0.25) x) x)) (- 1.0 x))
     (* (fmod x 1.0) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-147) {
		tmp = fmod((((pow(x, -2.0) + 0.5) + pow(x, -1.0)) * (x * x)), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else if (x <= -2e-310) {
		tmp = fmod(exp(x), (((pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x);
	} else {
		tmp = fmod(x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-147)
		tmp = Float64(rem(Float64(Float64(Float64((x ^ -2.0) + 0.5) + (x ^ -1.0)) * Float64(x * x)), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0));
	elseif (x <= -2e-310)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-147], N[(N[With[{TMP1 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] + 0.5), $MachinePrecision] + N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999997e-148

   1. Initial program 13.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 13.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 13.0

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

   8. Applied rewrites 13.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 13.2

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

   11. Applied rewrites 13.2%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-+r+ N/A

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

       5. lower-+.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. pow-flip N/A

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

       9. metadata-eval N/A

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

       10. lift-pow.f64 N/A

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

       11. inv-pow N/A

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

       12. lower-pow.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 42.8

           \[\leadsto \left(\left(\left(\left({x}^{-2} + 0.5\right) + {x}^{-1}\right) \cdot \left(x \cdot x\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]

   14. Applied rewrites 42.8%

       \[\leadsto \left(\left(\left(\left({x}^{-2} + 0.5\right) + {x}^{-1}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]

   if -9.9999999999999997e-148 < x < -1.999999999999994e-310

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 96.9

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

   8. Applied rewrites 96.9%

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

   9. Taylor expanded in x around 0

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. lower--.f64 96.9

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

   11. Applied rewrites 96.9%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

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

Alternative 8: 80.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (* (fmod (exp x) (* (* (- (pow x -2.0) 0.25) x) x)) (- 1.0 x))
   (* (fmod x 1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = fmod(exp(x), (((pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x);
	} else {
		tmp = fmod(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 (x <= (-2d-310)) then
        tmp = mod(exp(x), ((((x ** (-2.0d0)) - 0.25d0) * x) * x)) * (1.0d0 - x)
    else
        tmp = mod(x, 1.0d0) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = math.fmod(math.exp(x), (((math.pow(x, -2.0) - 0.25) * x) * x)) * (1.0 - x)
	else:
		tmp = math.fmod(x, 1.0) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 8.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 8.2

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

   5. Applied rewrites 8.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 59.5

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

   8. Applied rewrites 59.5%

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

   9. Taylor expanded in x around 0

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. lower--.f64 58.9

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

   11. Applied rewrites 58.9%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

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

Alternative 9: 65.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(x - \frac{-1}{x} \cdot x\right) \bmod 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6e-309)
   (* (fmod (- x (* (/ -1.0 x) x)) 1.0) 1.0)
   (* (fmod x 1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -6e-309) {
		tmp = fmod((x - ((-1.0 / x) * x)), 1.0) * 1.0;
	} else {
		tmp = fmod(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 (x <= (-6d-309)) then
        tmp = mod((x - (((-1.0d0) / x) * x)), 1.0d0) * 1.0d0
    else
        tmp = mod(x, 1.0d0) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -6e-309:
		tmp = math.fmod((x - ((-1.0 / x) * x)), 1.0) * 1.0
	else:
		tmp = math.fmod(x, 1.0) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -6e-309)
		tmp = Float64(rem(Float64(x - Float64(Float64(-1.0 / x) * x)), 1.0) * 1.0);
	else
		tmp = Float64(rem(x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -6e-309], N[(N[With[{TMP1 = N[(x - N[(N[(-1.0 / x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.000000000000001e-309

   1. Initial program 8.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 8.2%

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

   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 6.5%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 6.5

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

   11. Applied rewrites 6.5%

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

       1. lift--.f64 N/A

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

       2. metadata-eval N/A

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

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

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(1 \cdot x + -1 \cdot -1\right) \bmod 1\right) \cdot 1 \]

       6. metadata-eval N/A

          \[\leadsto \left(\left(1 \cdot x + 1\right) \bmod 1\right) \cdot 1 \]

       7. lft-mult-inverse N/A

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

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

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

       9. *-lft-identity N/A

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

       10. lower--.f64 N/A

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

       11. distribute-frac-neg N/A

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

       12. metadata-eval N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 16.5

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

   13. Applied rewrites 16.5%

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

   if -6.000000000000001e-309 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

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

Alternative 10: 60.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (* (fmod (fma (fma 0.5 x 1.0) x 1.0) 1.0) (- 1.0 x))
   (* (fmod x 1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = fmod(fma(fma(0.5, x, 1.0), x, 1.0), 1.0) * (1.0 - x);
	} else {
		tmp = fmod(x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(rem(fma(fma(0.5, x, 1.0), x, 1.0), 1.0) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 8.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 8.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 7.8

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

   8. Applied rewrites 7.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 8.0

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

   11. Applied rewrites 8.0%

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

   12. Taylor expanded in x around 0

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. lower--.f64 7.6

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

   14. Applied rewrites 7.6%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

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

Alternative 11: 60.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (* (fmod (- x -1.0) 1.0) (fma (fma 0.5 x -1.0) x 1.0))
   (* (fmod x 1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = fmod((x - -1.0), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod(x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(rem(Float64(x - -1.0), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 8.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 8.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 7.8

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

   8. Applied rewrites 7.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

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

       6. lower--.f64 7.6

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

   11. Applied rewrites 7.6%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

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

Alternative 12: 60.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (* (fmod (fma (fma 0.5 x 1.0) x 1.0) 1.0) 1.0)
   (* (fmod x 1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = fmod(fma(fma(0.5, x, 1.0), x, 1.0), 1.0) * 1.0;
	} else {
		tmp = fmod(x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(rem(fma(fma(0.5, x, 1.0), x, 1.0), 1.0) * 1.0);
	else
		tmp = Float64(rem(x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 8.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 8.2%

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

   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 6.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 6.5

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

   11. Applied rewrites 6.5%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

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

Alternative 13: 60.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310) (* (fmod (- x -1.0) 1.0) 1.0) (* (fmod x 1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = fmod((x - -1.0), 1.0) * 1.0;
	} else {
		tmp = fmod(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 (x <= (-2d-310)) then
        tmp = mod((x - (-1.0d0)), 1.0d0) * 1.0d0
    else
        tmp = mod(x, 1.0d0) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = math.fmod((x - -1.0), 1.0) * 1.0
	else:
		tmp = math.fmod(x, 1.0) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(rem(Float64(x - -1.0), 1.0) * 1.0);
	else
		tmp = Float64(rem(x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 8.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 8.2%

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

   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 6.5%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 6.5

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

   11. Applied rewrites 6.5%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 4.8%

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

   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 4.8%

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

   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. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

       6. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 98.5%

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

Alternative 14: 58.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.4%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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 6.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 5.5%

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

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. +-commutative N/A

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

    2. metadata-eval N/A

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

    3. metadata-eval N/A

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

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

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

    5. metadata-eval N/A

       \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

    6. lower--.f64 22.4

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

11. Applied rewrites 22.4%

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

12. Taylor expanded in x around inf

    \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
13. Step-by-step derivation

14. Applied rewrites 55.3%

    \[\leadsto \left(x \bmod 1\right) \cdot 1 \]
15. Add Preprocessing

Alternative 15: 23.2% 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 6.4%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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 6.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 5.5%

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

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. +-commutative N/A

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

    2. metadata-eval N/A

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

    3. metadata-eval N/A

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

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

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

    5. metadata-eval N/A

       \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \cdot 1 \]

    6. lower--.f64 22.4

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

11. Applied rewrites 22.4%

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

12. Taylor expanded in x around 0

    \[\leadsto \left(1 \bmod 1\right) \cdot 1 \]
13. Step-by-step derivation

14. Applied rewrites 20.7%

    \[\leadsto \left(1 \bmod 1\right) \cdot 1 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025083 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))