expfmod (used to be hard to sample)

Percentage Accurate: 9.2% → 96.6%

Time: 17.1s

Alternatives: 7

Speedup: 3.3×

• could not determine a ground truth

  1. x = -2.1678998247228307e-218

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: 9.2% 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: 96.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-14)
   (/ (fmod (exp x) 1.0) (exp x))
   (* (fmod x (sqrt 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-14) {
		tmp = fmod(exp(x), 1.0) / exp(x);
	} else {
		tmp = fmod(x, sqrt(1.0)) * exp(-x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-14)) then
        tmp = mod(exp(x), 1.0d0) / exp(x)
    else
        tmp = mod(x, sqrt(1.0d0)) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-14:
		tmp = math.fmod(math.exp(x), 1.0) / math.exp(x)
	else:
		tmp = math.fmod(x, math.sqrt(1.0)) * math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-14)
		tmp = Float64(rem(exp(x), 1.0) / exp(x));
	else
		tmp = Float64(rem(x, sqrt(1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-14], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-14

   1. Initial program 84.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-exp.f64 84.9

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

   6. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 84.9

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

   8. Applied rewrites 84.9%

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

   if -5.0000000000000002e-14 < x

   1. Initial program 6.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 36.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 37.9

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

   10. Applied rewrites 37.9%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 97.0%

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

Alternative 2: 95.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-14)
   (* (fmod (exp x) 1.0) (fma (fma 0.5 x -1.0) x 1.0))
   (* (fmod x (sqrt 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-14) {
		tmp = fmod(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod(x, sqrt(1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-14)
		tmp = Float64(rem(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(x, sqrt(1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-14], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-14

   1. Initial program 84.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 66.1

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

   7. Applied rewrites 66.1%

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

   if -5.0000000000000002e-14 < x

   1. Initial program 6.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 36.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 37.9

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

   10. Applied rewrites 37.9%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 97.0%

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

Alternative 3: 95.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-14)
   (* (fmod (exp x) (sqrt 1.0)) (- 1.0 x))
   (* (fmod x (sqrt 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-14) {
		tmp = fmod(exp(x), sqrt(1.0)) * (1.0 - x);
	} else {
		tmp = fmod(x, sqrt(1.0)) * exp(-x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-14)) then
        tmp = mod(exp(x), sqrt(1.0d0)) * (1.0d0 - x)
    else
        tmp = mod(x, sqrt(1.0d0)) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-14:
		tmp = math.fmod(math.exp(x), math.sqrt(1.0)) * (1.0 - x)
	else:
		tmp = math.fmod(x, math.sqrt(1.0)) * math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-14)
		tmp = Float64(rem(exp(x), sqrt(1.0)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, sqrt(1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-14], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-14

   1. Initial program 84.6%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 60.1%

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

   if -5.0000000000000002e-14 < x

   1. Initial program 6.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 36.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 37.9

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

   10. Applied rewrites 37.9%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 97.0%

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

Alternative 4: 95.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\sqrt{1}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-14)
   (* (fmod (- x -1.0) (sqrt 1.0)) (fma (fma 0.5 x -1.0) x 1.0))
   (* (fmod x (sqrt 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-14) {
		tmp = fmod((x - -1.0), sqrt(1.0)) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod(x, sqrt(1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-14)
		tmp = Float64(rem(Float64(x - -1.0), sqrt(1.0)) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(x, sqrt(1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-14], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-14

   1. Initial program 84.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 7.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 58.7

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

   10. Applied rewrites 58.7%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 58.8

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

   13. Applied rewrites 58.8%

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

   if -5.0000000000000002e-14 < x

   1. Initial program 6.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 36.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 37.9

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

   10. Applied rewrites 37.9%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 97.0%

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

Alternative 5: 95.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-14)
   (* (fmod (- x -1.0) (sqrt 1.0)) (- (- x) -1.0))
   (* (fmod x (sqrt 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-14) {
		tmp = fmod((x - -1.0), sqrt(1.0)) * (-x - -1.0);
	} else {
		tmp = fmod(x, sqrt(1.0)) * exp(-x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-14)) then
        tmp = mod((x - (-1.0d0)), sqrt(1.0d0)) * (-x - (-1.0d0))
    else
        tmp = mod(x, sqrt(1.0d0)) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-14:
		tmp = math.fmod((x - -1.0), math.sqrt(1.0)) * (-x - -1.0)
	else:
		tmp = math.fmod(x, math.sqrt(1.0)) * math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-14)
		tmp = Float64(rem(Float64(x - -1.0), sqrt(1.0)) * Float64(Float64(-x) - -1.0));
	else
		tmp = Float64(rem(x, sqrt(1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-14], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[((-x) - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-14

   1. Initial program 84.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 7.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 58.7

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

   10. Applied rewrites 58.7%

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

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lift-neg.f64 N/A

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

       3. +-commutative N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. lower--.f64 58.2

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

   13. Applied rewrites 58.2%

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

   if -5.0000000000000002e-14 < x

   1. Initial program 6.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 36.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 37.9

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

   10. Applied rewrites 37.9%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 97.0%

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

Alternative 6: 37.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fmod((x - -1.0), sqrt(1.0)) * (-x - -1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = mod((x - (-1.0d0)), sqrt(1.0d0)) * (-x - (-1.0d0))
end function

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 35.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 34.7%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 38.7

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

10. Applied rewrites 38.7%

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

11. Taylor expanded in x around 0

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

    1. mul-1-neg N/A

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

    2. lift-neg.f64 N/A

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

    3. +-commutative N/A

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

    4. metadata-eval N/A

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

    5. metadata-eval N/A

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

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

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

    7. metadata-eval N/A

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

    8. lower--.f64 37.6

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

13. Applied rewrites 37.6%

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

Alternative 7: 36.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 35.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 34.7%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 38.7

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

10. Applied rewrites 38.7%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 36.9%

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

Reproduce

herbie shell --seed 2025120 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))