expfmod (used to be hard to sample)

Percentage Accurate: 6.8% → 63.0%

Time: 8.6s

Alternatives: 9

Speedup: 3.6×

• could not determine a ground truth

  1. x = -3.910319236051393e+192

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1.6e-162)
     (* (fmod (exp x) (* (- (pow x -2.0) 0.25) (* x x))) t_0)
     (* (fmod x (sqrt (cos x))) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1.6e-162) {
		tmp = fmod(exp(x), ((pow(x, -2.0) - 0.25) * (x * x))) * t_0;
	} else {
		tmp = fmod(x, sqrt(cos(x))) * t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-1.6d-162)) then
        tmp = mod(exp(x), (((x ** (-2.0d0)) - 0.25d0) * (x * x))) * t_0
    else
        tmp = mod(x, sqrt(cos(x))) * t_0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -1.6e-162:
		tmp = math.fmod(math.exp(x), ((math.pow(x, -2.0) - 0.25) * (x * x))) * t_0
	else:
		tmp = math.fmod(x, math.sqrt(math.cos(x))) * t_0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1.6e-162)
		tmp = Float64(rem(exp(x), Float64(Float64((x ^ -2.0) - 0.25) * Float64(x * x))) * t_0);
	else
		tmp = Float64(rem(x, sqrt(cos(x))) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1.6e-162], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999988e-162

   1. Initial program 14.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 14.7

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

   5. Applied rewrites 14.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. 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 23.3

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

   8. Applied rewrites 23.3%

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

   if -1.59999999999999988e-162 < x

   1. Initial program 5.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 11.5

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

   5. Applied rewrites 11.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left({x}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 24.1

          \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot {x}^{3}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 24.1%

      \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot \color{blue}{{x}^{3}}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 77.8%

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

Alternative 2: 24.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (* (fmod (exp x) 1.0) 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), 1.0) * 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 ((mod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0d0) then
        tmp = mod(exp(x), 1.0d0) * 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 (math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)) <= 2.0:
		tmp = math.fmod(math.exp(x), 1.0) * 1.0
	else:
		tmp = math.fmod(1.0, math.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), 1.0) * 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 = 1.0}, 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 2 regimes

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

   1. Initial program 9.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 8.7%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg 6.6

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

      2. exp-prod 6.6

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

      3. lift-exp.f64 N/A

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

      4. sqr-pow N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 6.6

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

   8. Applied rewrites 6.6%

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

   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 97.7%

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

   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 97.6%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. lower-fma.f64 97.7

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

   11. Applied rewrites 97.7%

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

   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 97.7%

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

4. Final simplification 21.6%

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

Alternative 3: 60.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -5e-310)
     (* (fmod (exp x) 1.0) t_0)
     (* (fmod x (sqrt (cos x))) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(exp(x), 1.0) * t_0;
	} else {
		tmp = fmod(x, sqrt(cos(x))) * t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-5d-310)) then
        tmp = mod(exp(x), 1.0d0) * t_0
    else
        tmp = mod(x, sqrt(cos(x))) * t_0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -5e-310:
		tmp = math.fmod(math.exp(x), 1.0) * t_0
	else:
		tmp = math.fmod(x, math.sqrt(math.cos(x))) * t_0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(exp(x), 1.0) * t_0);
	else
		tmp = Float64(rem(x, sqrt(cos(x))) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 9.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 9.8%

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

   if -4.999999999999985e-310 < x

   1. Initial program 6.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 13.8

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

   5. Applied rewrites 13.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left({x}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 30.8

          \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot {x}^{3}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 30.8%

      \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot \color{blue}{{x}^{3}}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.6%

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

Alternative 4: 60.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -5e-310) (* (fmod (exp x) 1.0) t_0) (* (fmod x 1.0) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(exp(x), 1.0) * t_0;
	} else {
		tmp = fmod(x, 1.0) * t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-5d-310)) then
        tmp = mod(exp(x), 1.0d0) * t_0
    else
        tmp = mod(x, 1.0d0) * t_0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -5e-310:
		tmp = math.fmod(math.exp(x), 1.0) * t_0
	else:
		tmp = math.fmod(x, 1.0) * t_0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(exp(x), 1.0) * t_0);
	else
		tmp = Float64(rem(x, 1.0) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 9.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 9.8%

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

   if -4.999999999999985e-310 < x

   1. Initial program 6.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 13.8

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

   5. Applied rewrites 13.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left({x}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 30.8

          \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot {x}^{3}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 30.8%

      \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot \color{blue}{{x}^{3}}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.6%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 98.6%

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

Alternative 5: 60.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (*
    (fmod (exp x) 1.0)
    (fma (fma (fma -0.16666666666666666 x 0.5) x -1.0) x 1.0))
   (* (fmod x 1.0) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(exp(x), 1.0) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0);
	} else {
		tmp = fmod(x, 1.0) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(exp(x), 1.0) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(x, 1.0) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 9.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. exp-prod N/A

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

      3. lift-exp.f64 N/A

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

      4. sqr-pow N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 7.4

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

   8. Applied rewrites 7.4%

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

   if -4.999999999999985e-310 < x

   1. Initial program 6.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 13.8

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

   5. Applied rewrites 13.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left({x}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 30.8

          \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot {x}^{3}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 30.8%

      \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot \color{blue}{{x}^{3}}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.6%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 98.6%

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

Alternative 6: 60.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (* (fmod (exp x) 1.0) (fma (fma 0.5 x -1.0) x 1.0))
   (* (fmod x 1.0) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod(x, 1.0) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(x, 1.0) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 9.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. exp-prod N/A

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

      3. lift-exp.f64 N/A

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

      4. sqr-pow N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 7.1

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

   8. Applied rewrites 7.1%

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

   if -4.999999999999985e-310 < x

   1. Initial program 6.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 13.8

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

   5. Applied rewrites 13.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left({x}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 30.8

          \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot {x}^{3}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 30.8%

      \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot \color{blue}{{x}^{3}}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.6%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 98.6%

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

Alternative 7: 60.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (* (fmod (exp x) 1.0) (- 1.0 x))
   (* (fmod x 1.0) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(exp(x), 1.0) * (1.0 - x);
	} else {
		tmp = fmod(x, 1.0) * exp(-x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = mod(exp(x), 1.0d0) * (1.0d0 - x)
    else
        tmp = mod(x, 1.0d0) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = math.fmod(math.exp(x), 1.0) * (1.0 - x)
	else:
		tmp = math.fmod(x, 1.0) * math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(exp(x), 1.0) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, 1.0) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 9.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. exp-prod N/A

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

      3. lift-exp.f64 N/A

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

      4. sqr-pow N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. cancel-sign-sub N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 6.3

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

   8. Applied rewrites 6.3%

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

   if -4.999999999999985e-310 < x

   1. Initial program 6.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 13.8

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

   5. Applied rewrites 13.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left({x}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 30.8

          \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot {x}^{3}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 30.8%

      \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot \color{blue}{{x}^{3}}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.6%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 98.6%

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

Alternative 8: 58.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 11.2

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

5. Applied rewrites 11.2%

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

6. Taylor expanded in x around inf

   \[\leadsto \left(\left({x}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. pow-flip N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-pow.f64 19.1

       \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot {x}^{3}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

8. Applied rewrites 19.1%

   \[\leadsto \left(\left(\left(\left(0.16666666666666666 + {x}^{-2}\right) + \frac{0.5}{x}\right) \cdot \color{blue}{{x}^{3}}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 60.6%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 60.6%

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

Alternative 9: 22.7% 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 7.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 19.9%

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

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 19.6%

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

9. Taylor expanded in x around 0

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

    1. mul-1-neg N/A

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

    2. +-commutative N/A

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

    3. mul-1-neg N/A

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

    4. lower-fma.f64 19.6

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

11. Applied rewrites 19.6%

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

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 19.6%

    \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]

15. Final simplification 19.6%

    \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2025028 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))