expfmod (used to be hard to sample)

Percentage Accurate: 6.6% → 58.5%

Time: 9.5s

Alternatives: 13

Speedup: 3.6×

• could not determine a ground truth

  1. x = -1.1672880777134159e+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: 6.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 58.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-77}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left({x}^{-2} - 0.25\right) \cdot e^{\log \left(x \cdot x\right)}\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} + 0.25} \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -4.8e-77)
     (* (fmod (exp x) (* (- (pow x -2.0) 0.25) (exp (log (* 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 50.0)
         (* (fmod (exp x) (* (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) x) x)) t_0)
         (* (fmod 1.0 (sqrt 1.0)) 1.0))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -4.8e-77) {
		tmp = fmod(exp(x), ((pow(x, -2.0) - 0.25) * exp(log((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 <= 50.0) {
		tmp = fmod(exp(x), ((fma((-1.0 / x), (-1.0 / x), -0.25) * x) * x)) * t_0;
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -4.8e-77)
		tmp = Float64(rem(exp(x), Float64(Float64((x ^ -2.0) - 0.25) * exp(log(Float64(x * x))))) * t_0);
	elseif (x <= -7.5e-155)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -4.0) - 0.0625) / Float64((x ^ -2.0) + 0.25)) * Float64(x * x))) * t_0);
	elseif (x <= 50.0)
		tmp = Float64(rem(exp(x), Float64(Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * x) * x)) * t_0);
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -4.8e-77], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * N[Exp[N[Log[N[(x * x), $MachinePrecision]], $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[Power[x, -4.0], $MachinePrecision] - 0.0625), $MachinePrecision] / N[(N[Power[x, -2.0], $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 50.0], 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] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.7999999999999998e-77

   1. Initial program 17.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 17.4

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

   5. Applied rewrites 17.4%

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

   6. Taylor expanded in x around inf

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

      1. *-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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 19.2

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

   8. Applied rewrites 19.2%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 0.0

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

   10. Applied rewrites 0.0%

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

       1. lift-*.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. *-commutative N/A

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

       4. log-pow-rev N/A

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

       5. lower-log.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 63.3

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

   12. Applied rewrites 63.3%

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

   if -4.7999999999999998e-77 < 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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 4.2

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

   8. Applied rewrites 4.2%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 N/A

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

      17. pow-flip N/A

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

      18. metadata-eval N/A

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

      19. lift-pow.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -7.5000000000000006e-155 < x < 50

   1. Initial program 7.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 7.3

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 5.9

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

   8. Applied rewrites 5.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift--.f64 39.4

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

   10. Applied rewrites 39.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   11. 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. frac-times 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. lift-/.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. lift-/.f64 39.5

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

   12. Applied rewrites 39.5%

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

   if 50 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. sinh-cosh N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-77}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left({x}^{-2} - 0.25\right) \cdot e^{\log \left(x \cdot x\right)}\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} + 0.25} \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 26.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (*
    (fmod (exp x) (fma -0.25 (* x x) 1.0))
    (fma (fma (fma -0.16666666666666666 x 0.5) x -1.0) x 1.0))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = fmod(exp(x), fma(-0.25, (x * x), 1.0)) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(rem(exp(x), fma(-0.25, Float64(x * x), 1.0)) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 7.9

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2} + 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, \color{blue}{{x}^{2}}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {\color{blue}{x}}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      6. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      8. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      9. lift-*.f64 7.9

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]

   8. Applied rewrites 7.9%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{4}, \color{blue}{x} \cdot x, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 7.8%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, \color{blue}{x} \cdot x, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. sinh-cosh N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 25.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (*
    (fmod (fma (fma 0.5 x 1.0) x 1.0) (fma (* x x) -0.25 1.0))
    (fma (- (* 0.5 x) 1.0) x 1.0))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = fmod(fma(fma(0.5, x, 1.0), x, 1.0), fma((x * x), -0.25, 1.0)) * fma(((0.5 * x) - 1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(rem(fma(fma(0.5, x, 1.0), x, 1.0), fma(Float64(x * x), -0.25, 1.0)) * fma(Float64(Float64(0.5 * x) - 1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(0.5 * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 8.1

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

   5. Applied rewrites 8.1%

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

   6. Taylor expanded in x around 0

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

      1. rec-exp N/A

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

      2. sinh-cosh N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 7.7

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right) \]

   8. Applied rewrites 7.7%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)} \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 7.7

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right) \]

   11. Applied rewrites 7.7%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right) \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. sinh-cosh N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 25.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (* (fmod (- x -1.0) (fma (* x x) -0.25 1.0)) (fma (- (* 0.5 x) 1.0) x 1.0))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = fmod((x - -1.0), fma((x * x), -0.25, 1.0)) * fma(((0.5 * x) - 1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(rem(Float64(x - -1.0), fma(Float64(x * x), -0.25, 1.0)) * fma(Float64(Float64(0.5 * x) - 1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(0.5 * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 8.1

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

   5. Applied rewrites 8.1%

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

   6. Taylor expanded in x around 0

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

      1. rec-exp N/A

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

      2. sinh-cosh N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 7.7

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right) \]

   8. Applied rewrites 7.7%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)} \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

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

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. lower--.f64 7.1

          \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right) \]

   11. Applied rewrites 7.1%

       \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right) \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. sinh-cosh N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -4 \cdot 10^{-75}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} + 0.25} \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -4e-75)
     (* (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 50.0)
         (* (fmod (exp x) (* (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) x) x)) t_0)
         (* (fmod 1.0 (sqrt 1.0)) 1.0))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -4e-75) {
		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 <= 50.0) {
		tmp = fmod(exp(x), ((fma((-1.0 / x), (-1.0 / x), -0.25) * x) * x)) * t_0;
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -4e-75)
		tmp = Float64(rem(exp(x), Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * Float64(x * x))) * t_0);
	elseif (x <= -7.5e-155)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -4.0) - 0.0625) / Float64((x ^ -2.0) + 0.25)) * Float64(x * x))) * t_0);
	elseif (x <= 50.0)
		tmp = Float64(rem(exp(x), Float64(Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * x) * x)) * t_0);
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -4e-75], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $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[Power[x, -4.0], $MachinePrecision] - 0.0625), $MachinePrecision] / N[(N[Power[x, -2.0], $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 50.0], 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] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.9999999999999998e-75

   1. Initial program 17.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 17.4

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

   5. Applied rewrites 17.4%

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

   6. Taylor expanded in x around inf

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

      1. *-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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 19.2

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

   8. Applied rewrites 19.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 60.0

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

   10. Applied rewrites 60.0%

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

   if -3.9999999999999998e-75 < 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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 4.2

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

   8. Applied rewrites 4.2%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 N/A

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

      17. pow-flip N/A

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

      18. metadata-eval N/A

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

      19. lift-pow.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -7.5000000000000006e-155 < x < 50

   1. Initial program 7.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 7.3

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 5.9

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

   8. Applied rewrites 5.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift--.f64 39.4

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

   10. Applied rewrites 39.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   11. 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. frac-times 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. lift-/.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. lift-/.f64 39.5

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

   12. Applied rewrites 39.5%

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

   if 50 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. sinh-cosh N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-75}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} + 0.25} \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right) \cdot x\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(\frac{-1}{x}\right) \cdot 2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot \frac{{t\_0}^{2} - 1}{t\_0 - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fma (fma -0.16666666666666666 x 0.5) x -1.0) x)))
   (if (<= x -5e-310)
     (*
      (fmod (exp x) (* (* (- (exp (* (log (/ -1.0 x)) 2.0)) 0.25) x) x))
      (exp (- x)))
     (if (<= x 1.0)
       (*
        (fmod
         (exp x)
         (fma (- (* -0.010416666666666666 (* x x)) 0.25) (* x x) 1.0))
        (/ (- (pow t_0 2.0) 1.0) (- t_0 1.0)))
       (* (fmod 1.0 (sqrt 1.0)) 1.0)))))

C

double code(double x) {
	double t_0 = fma(fma(-0.16666666666666666, x, 0.5), x, -1.0) * x;
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(exp(x), (((exp((log((-1.0 / x)) * 2.0)) - 0.25) * x) * x)) * exp(-x);
	} else if (x <= 1.0) {
		tmp = fmod(exp(x), fma(((-0.010416666666666666 * (x * x)) - 0.25), (x * x), 1.0)) * ((pow(t_0, 2.0) - 1.0) / (t_0 - 1.0));
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0) * x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(exp(Float64(log(Float64(-1.0 / x)) * 2.0)) - 0.25) * x) * x)) * exp(Float64(-x)));
	elseif (x <= 1.0)
		tmp = Float64(rem(exp(x), fma(Float64(Float64(-0.010416666666666666 * Float64(x * x)) - 0.25), Float64(x * x), 1.0)) * Float64(Float64((t_0 ^ 2.0) - 1.0) / Float64(t_0 - 1.0)));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Exp[N[(N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - 1.0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 6.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 6.9

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

   5. Applied rewrites 6.9%

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

   6. Taylor expanded in x around inf

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

      1. *-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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 8.0

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

   8. Applied rewrites 8.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift--.f64 56.6

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

   10. Applied rewrites 56.6%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-flip N/A

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

       4. metadata-eval N/A

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

       5. pow2 N/A

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

       6. frac-times N/A

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

       7. pow2 N/A

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

       8. pow-to-exp N/A

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

       9. lower-exp.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-log.f64 N/A

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

       12. lift-/.f64 78.2

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

   12. Applied rewrites 78.2%

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

   if -4.999999999999985e-310 < x < 1

   1. Initial program 9.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 9.4

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

   5. Applied rewrites 9.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2} + 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, \color{blue}{{x}^{2}}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {\color{blue}{x}}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      6. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      8. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      9. lift-*.f64 9.4

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]

   8. Applied rewrites 9.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right) \cdot x + \color{blue}{1}\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{6} \cdot x + \frac{1}{2}, x, -1\right) \cdot x + 1\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot \left(\left(\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right) \cdot x + -1\right) \cdot x + 1\right) \]

      4. flip-+ N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot \frac{\left(\left(\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right) \cdot x + -1\right) \cdot x\right) \cdot \left(\left(\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right) \cdot x + -1\right) \cdot x\right) - 1 \cdot 1}{\color{blue}{\left(\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right) \cdot x + -1\right) \cdot x - 1}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot \frac{\left(\left(\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right) \cdot x + -1\right) \cdot x\right) \cdot \left(\left(\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right) \cdot x + -1\right) \cdot x\right) - 1 \cdot 1}{\color{blue}{\left(\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right) \cdot x + -1\right) \cdot x - 1}} \]

   10. Applied rewrites 9.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right) \cdot x\right)}^{2} - 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right) \cdot x - 1}} \]

   if 1 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. sinh-cosh N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(e^{\log \left(\frac{-1}{x}\right) \cdot 2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right) \cdot x\right)}^{2} - 1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right) \cdot x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1e-153)
     (* (fmod (exp x) (* (- (exp (* (log (* x x)) -1.0)) 0.25) (* x x))) t_0)
     (if (<= x 50.0)
       (* (fmod (exp x) (* (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) x) x)) t_0)
       (* (fmod 1.0 (sqrt 1.0)) 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1e-153) {
		tmp = fmod(exp(x), ((exp((log((x * x)) * -1.0)) - 0.25) * (x * x))) * t_0;
	} else if (x <= 50.0) {
		tmp = fmod(exp(x), ((fma((-1.0 / x), (-1.0 / x), -0.25) * x) * x)) * t_0;
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1e-153)
		tmp = Float64(rem(exp(x), Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * Float64(x * x))) * t_0);
	elseif (x <= 50.0)
		tmp = Float64(rem(exp(x), Float64(Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * x) * x)) * t_0);
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1e-153], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 50.0], 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] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000004e-153

   1. Initial program 10.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 10.5

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

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 12.0

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

   8. Applied rewrites 12.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 56.8

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

   10. Applied rewrites 56.8%

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

   if -1.00000000000000004e-153 < x < 50

   1. Initial program 7.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 7.3

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 5.9

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

   8. Applied rewrites 5.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift--.f64 39.4

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

   10. Applied rewrites 39.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   11. 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. frac-times 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. lift-/.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. lift-/.f64 39.5

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

   12. Applied rewrites 39.5%

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

   if 50 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. sinh-cosh N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 50.0)
   (* (fmod (exp x) (* (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) x) x)) (exp (- x)))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 50.0) {
		tmp = fmod(exp(x), ((fma((-1.0 / x), (-1.0 / x), -0.25) * x) * x)) * exp(-x);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 50.0)
		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(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 50.0], 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 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 50

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 8.1

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

   5. Applied rewrites 8.1%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 7.5

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

   8. Applied rewrites 7.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift--.f64 32.9

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

   10. Applied rewrites 32.9%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x} \]
   11. 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. frac-times 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. lift-/.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. lift-/.f64 36.7

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

   12. Applied rewrites 36.7%

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

   if 50 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. sinh-cosh N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{x \cdot x} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 50.0)
   (* (fmod (exp x) (* (* (- (/ 1.0 (* x x)) 0.25) x) x)) (exp (- x)))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 50.0) {
		tmp = fmod(exp(x), ((((1.0 / (x * x)) - 0.25) * x) * x)) * exp(-x);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 50.0d0) then
        tmp = mod(exp(x), ((((1.0d0 / (x * x)) - 0.25d0) * x) * x)) * exp(-x)
    else
        tmp = mod(1.0d0, sqrt(1.0d0)) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 50.0:
		tmp = math.fmod(math.exp(x), ((((1.0 / (x * x)) - 0.25) * x) * x)) * math.exp(-x)
	else:
		tmp = math.fmod(1.0, math.sqrt(1.0)) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 50.0)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64(1.0 / Float64(x * x)) - 0.25) * x) * x)) * exp(Float64(-x)));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 50.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 50

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 8.1

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

   5. Applied rewrites 8.1%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 7.5

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

   8. Applied rewrites 7.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift--.f64 32.9

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

   10. Applied rewrites 32.9%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-flip N/A

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

       4. lower-/.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 33.8

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

   12. Applied rewrites 33.8%

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

   if 50 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. sinh-cosh N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{x \cdot x} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{x} \cdot \frac{-1}{x} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-140)
   (* (fmod (exp x) (* (- (* (/ -1.0 x) (/ -1.0 x)) 0.25) (* x x))) 1.0)
   (if (<= x 50.0)
     (* (fmod (exp x) (* (* (- (pow x -2.0) 0.25) x) x)) 1.0)
     (* (fmod 1.0 (sqrt 1.0)) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-140) {
		tmp = fmod(exp(x), ((((-1.0 / x) * (-1.0 / x)) - 0.25) * (x * x))) * 1.0;
	} else if (x <= 50.0) {
		tmp = fmod(exp(x), (((pow(x, -2.0) - 0.25) * x) * x)) * 1.0;
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-140)) then
        tmp = mod(exp(x), (((((-1.0d0) / x) * ((-1.0d0) / x)) - 0.25d0) * (x * x))) * 1.0d0
    else if (x <= 50.0d0) then
        tmp = mod(exp(x), ((((x ** (-2.0d0)) - 0.25d0) * x) * x)) * 1.0d0
    else
        tmp = mod(1.0d0, sqrt(1.0d0)) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-140:
		tmp = math.fmod(math.exp(x), ((((-1.0 / x) * (-1.0 / x)) - 0.25) * (x * x))) * 1.0
	elif x <= 50.0:
		tmp = math.fmod(math.exp(x), (((math.pow(x, -2.0) - 0.25) * x) * x)) * 1.0
	else:
		tmp = math.fmod(1.0, math.sqrt(1.0)) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-140)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64(-1.0 / x) * Float64(-1.0 / x)) - 0.25) * Float64(x * x))) * 1.0);
	elseif (x <= 50.0)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * 1.0);
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-140], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, 50.0], 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] * 1.0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999998e-141

   1. Initial program 11.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 11.0

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

   5. Applied rewrites 11.0%

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

   6. Taylor expanded in x around inf

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

      1. *-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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 12.5

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

   8. Applied rewrites 12.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 26.8

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

   10. Applied rewrites 26.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 22.7%

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

   if -9.9999999999999998e-141 < x < 50

   1. Initial program 7.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 7.2

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

   5. Applied rewrites 7.2%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 5.8

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

   8. Applied rewrites 5.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift--.f64 38.7

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

   10. Applied rewrites 38.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 38.1%

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

   if 50 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. sinh-cosh N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{x} \cdot \frac{-1}{x} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{x} \cdot \frac{-1}{x} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.6e-162)
   (* (fmod (exp x) (* (- (* (/ -1.0 x) (/ -1.0 x)) 0.25) (* x x))) 1.0)
   (if (<= x 1.0)
     (*
      (fmod
       (exp x)
       (fma (- (* -0.010416666666666666 (* x x)) 0.25) (* x x) 1.0))
      (fma (fma (fma -0.16666666666666666 x 0.5) x -1.0) x 1.0))
     (* (fmod 1.0 (sqrt 1.0)) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -1.6e-162) {
		tmp = fmod(exp(x), ((((-1.0 / x) * (-1.0 / x)) - 0.25) * (x * x))) * 1.0;
	} else if (x <= 1.0) {
		tmp = fmod(exp(x), fma(((-0.010416666666666666 * (x * x)) - 0.25), (x * x), 1.0)) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.6e-162)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64(-1.0 / x) * Float64(-1.0 / x)) - 0.25) * Float64(x * x))) * 1.0);
	elseif (x <= 1.0)
		tmp = Float64(rem(exp(x), fma(Float64(Float64(-0.010416666666666666 * Float64(x * x)) - 0.25), Float64(x * x), 1.0)) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.6e-162], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.59999999999999988e-162

   1. Initial program 10.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 10.3

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 15.2

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

   8. Applied rewrites 15.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 28.2

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

   10. Applied rewrites 28.2%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 24.4%

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

   if -1.59999999999999988e-162 < x < 1

   1. Initial program 7.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 7.4

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

   5. Applied rewrites 7.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2} + 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, \color{blue}{{x}^{2}}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {\color{blue}{x}}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      6. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      8. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      9. lift-*.f64 7.4

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]

   8. Applied rewrites 7.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]

   if 1 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. sinh-cosh N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{x} \cdot \frac{-1}{x} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{x} \cdot \frac{-1}{x} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.6e-162)
   (* (fmod (exp x) (* (- (* (/ -1.0 x) (/ -1.0 x)) 0.25) (* x x))) 1.0)
   (if (<= x 1.0)
     (*
      (fmod (exp x) (fma -0.25 (* x x) 1.0))
      (fma (fma (fma -0.16666666666666666 x 0.5) x -1.0) x 1.0))
     (* (fmod 1.0 (sqrt 1.0)) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -1.6e-162) {
		tmp = fmod(exp(x), ((((-1.0 / x) * (-1.0 / x)) - 0.25) * (x * x))) * 1.0;
	} else if (x <= 1.0) {
		tmp = fmod(exp(x), fma(-0.25, (x * x), 1.0)) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.6e-162)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64(Float64(-1.0 / x) * Float64(-1.0 / x)) - 0.25) * Float64(x * x))) * 1.0);
	elseif (x <= 1.0)
		tmp = Float64(rem(exp(x), fma(-0.25, Float64(x * x), 1.0)) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.6e-162], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.59999999999999988e-162

   1. Initial program 10.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 10.3

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 15.2

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

   8. Applied rewrites 15.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 28.2

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

   10. Applied rewrites 28.2%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 24.4%

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

   if -1.59999999999999988e-162 < x < 1

   1. Initial program 7.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 7.4

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

   5. Applied rewrites 7.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2} + 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, \color{blue}{{x}^{2}}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {\color{blue}{x}}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      6. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      8. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]

      9. lift-*.f64 7.4

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]

   8. Applied rewrites 7.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{4}, \color{blue}{x} \cdot x, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 7.3%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, \color{blue}{x} \cdot x, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \]

   if 1 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. sinh-cosh N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{x} \cdot \frac{-1}{x} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 23.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fmod(1.0, sqrt(1.0)) * 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = mod(1.0d0, sqrt(1.0d0)) * 1.0d0
end function

Python

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, 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

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

5. Applied rewrites 24.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 23.9%

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

9. Taylor expanded in x around 0

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

    1. rec-exp N/A

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

    2. sinh-cosh N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. lower--.f64 N/A

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

    7. lower-*.f64 12.2

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

11. Applied rewrites 12.2%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 23.9%

    \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]

15. Final simplification 23.9%

    \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))