expfmod (used to be hard to sample)

Percentage Accurate: 8.7% → 97.4%

Time: 15.6s

Alternatives: 12

Speedup: 4.2×

• could not determine a ground truth

  1. x = -4.202253322041948e-229

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: 8.7% 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: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\left(0.041666666666666664 + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{0.5}{x \cdot x}\right)\right) \cdot e^{\log \left(x \cdot x\right) \cdot 2}}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1e-21)
     (*
      (fmod
       (exp x)
       (sqrt
        (*
         (+
          0.041666666666666664
          (- (/ 1.0 (* (* x x) (* x x))) (/ 0.5 (* x x))))
         (exp (* (log (* x x)) 2.0)))))
      t_0)
     (* (fmod x (fma (* x x) -0.25 1.0)) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1e-21) {
		tmp = fmod(exp(x), sqrt(((0.041666666666666664 + ((1.0 / ((x * x) * (x * x))) - (0.5 / (x * x)))) * exp((log((x * x)) * 2.0))))) * t_0;
	} else {
		tmp = fmod(x, fma((x * x), -0.25, 1.0)) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1e-21)
		tmp = Float64(rem(exp(x), sqrt(Float64(Float64(0.041666666666666664 + Float64(Float64(1.0 / Float64(Float64(x * x) * Float64(x * x))) - Float64(0.5 / Float64(x * x)))) * exp(Float64(log(Float64(x * x)) * 2.0))))) * t_0);
	else
		tmp = Float64(rem(x, fma(Float64(x * x), -0.25, 1.0)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1e-21], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(0.041666666666666664 + N[(N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999908e-22

   1. Initial program 68.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1}\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right)}\right)\right) \cdot e^{-x} \]

      4. negate-sub N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{x}}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right), {\color{blue}{x}}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot \color{blue}{x}, 1\right)}\right)\right) \cdot e^{-x} \]

      10. lower-*.f64 68.5

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot \color{blue}{x}, 1\right)}\right)\right) \cdot e^{-x} \]

   4. Applied rewrites 68.5%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{{x}^{4} \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\left(\frac{1}{24} + \frac{1}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{\color{blue}{4}}}\right)\right) \cdot e^{-x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\left(\frac{1}{24} + \frac{1}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{\color{blue}{4}}}\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 68.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(0.041666666666666664 + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{0.5}{x \cdot x}\right)\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}\right)\right) \cdot e^{-x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)}\right)\right) \cdot e^{-x} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot e^{-x} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot e^{-x} \]

      4. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot {\left(x \cdot x\right)}^{2}}\right)\right) \cdot e^{-x} \]

      5. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot {\left({x}^{2}\right)}^{2}}\right)\right) \cdot e^{-x} \]

      6. pow-to-exp N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot e^{\log \left({x}^{2}\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

      7. lower-exp.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot e^{\log \left({x}^{2}\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot e^{\log \left({x}^{2}\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

      9. lower-log.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot e^{\log \left({x}^{2}\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

      10. pow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot e^{\log \left(x \cdot x\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

      11. lift-*.f64 76.0

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(0.041666666666666664 + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{0.5}{x \cdot x}\right)\right) \cdot e^{\log \left(x \cdot x\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

   9. Applied rewrites 76.0%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(0.041666666666666664 + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{0.5}{x \cdot x}\right)\right) \cdot e^{\log \left(x \cdot x\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

   if -9.99999999999999908e-22 < x

   1. Initial program 5.5%

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

   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 37.2%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 38.2

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

   10. Applied rewrites 38.2%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 98.6%

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

Alternative 2: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} \cdot e^{\log \left(x \cdot x\right) \cdot 2}}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1e-21)
     (*
      (fmod
       (exp x)
       (sqrt (* (/ 1.0 (* (* (* x x) x) x)) (exp (* (log (* x x)) 2.0)))))
      t_0)
     (* (fmod x (fma (* x x) -0.25 1.0)) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1e-21) {
		tmp = fmod(exp(x), sqrt(((1.0 / (((x * x) * x) * x)) * exp((log((x * x)) * 2.0))))) * t_0;
	} else {
		tmp = fmod(x, fma((x * x), -0.25, 1.0)) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1e-21)
		tmp = Float64(rem(exp(x), sqrt(Float64(Float64(1.0 / Float64(Float64(Float64(x * x) * x) * x)) * exp(Float64(log(Float64(x * x)) * 2.0))))) * t_0);
	else
		tmp = Float64(rem(x, fma(Float64(x * x), -0.25, 1.0)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1e-21], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999908e-22

   1. Initial program 68.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1}\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right)}\right)\right) \cdot e^{-x} \]

      4. negate-sub N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{x}}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right), {\color{blue}{x}}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot \color{blue}{x}, 1\right)}\right)\right) \cdot e^{-x} \]

      10. lower-*.f64 68.5

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot \color{blue}{x}, 1\right)}\right)\right) \cdot e^{-x} \]

   4. Applied rewrites 68.5%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{{x}^{4} \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\left(\frac{1}{24} + \frac{1}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{\color{blue}{4}}}\right)\right) \cdot e^{-x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\left(\frac{1}{24} + \frac{1}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{\color{blue}{4}}}\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 68.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(0.041666666666666664 + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{0.5}{x \cdot x}\right)\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}\right)\right) \cdot e^{-x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)}\right)\right) \cdot e^{-x} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot e^{-x} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot e^{-x} \]

      4. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot {\left(x \cdot x\right)}^{2}}\right)\right) \cdot e^{-x} \]

      5. pow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot {\left({x}^{2}\right)}^{2}}\right)\right) \cdot e^{-x} \]

      6. pow-to-exp N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot e^{\log \left({x}^{2}\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

      7. lower-exp.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot e^{\log \left({x}^{2}\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot e^{\log \left({x}^{2}\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

      9. lower-log.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot e^{\log \left({x}^{2}\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

      10. pow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x \cdot x}\right)\right) \cdot e^{\log \left(x \cdot x\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

      11. lift-*.f64 76.0

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(0.041666666666666664 + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{0.5}{x \cdot x}\right)\right) \cdot e^{\log \left(x \cdot x\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

   9. Applied rewrites 76.0%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(0.041666666666666664 + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{0.5}{x \cdot x}\right)\right) \cdot e^{\log \left(x \cdot x\right) \cdot 2}}\right)\right) \cdot e^{-x} \]

   10. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-prod-up N/A

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

       4. pow2 N/A

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

       5. associate-*r* N/A

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

       6. pow2 N/A

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

       7. pow3 N/A

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

       8. lower-*.f64 N/A

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

       9. pow3 N/A

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

       10. pow2 N/A

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

       11. lower-*.f64 N/A

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

       12. pow2 N/A

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

       13. lift-*.f64 76.0

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

   12. Applied rewrites 76.0%

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

   if -9.99999999999999908e-22 < x

   1. Initial program 5.5%

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

   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 37.2%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 38.2

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

   10. Applied rewrites 38.2%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 98.6%

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

Alternative 3: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{\frac{1}{x \cdot x} - 0.5}{x \cdot x} + 0.041666666666666664\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1e-21)
     (*
      (fmod
       (exp x)
       (sqrt
        (*
         (+ (/ (- (/ 1.0 (* x x)) 0.5) (* x x)) 0.041666666666666664)
         (* (* (* x x) x) x))))
      t_0)
     (* (fmod x (fma (* x x) -0.25 1.0)) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1e-21) {
		tmp = fmod(exp(x), sqrt((((((1.0 / (x * x)) - 0.5) / (x * x)) + 0.041666666666666664) * (((x * x) * x) * x)))) * t_0;
	} else {
		tmp = fmod(x, fma((x * x), -0.25, 1.0)) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1e-21)
		tmp = Float64(rem(exp(x), sqrt(Float64(Float64(Float64(Float64(Float64(1.0 / Float64(x * x)) - 0.5) / Float64(x * x)) + 0.041666666666666664) * Float64(Float64(Float64(x * x) * x) * x)))) * t_0);
	else
		tmp = Float64(rem(x, fma(Float64(x * x), -0.25, 1.0)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1e-21], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999908e-22

   1. Initial program 68.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1}\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right)}\right)\right) \cdot e^{-x} \]

      4. negate-sub N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{x}}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right), {\color{blue}{x}}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot \color{blue}{x}, 1\right)}\right)\right) \cdot e^{-x} \]

      10. lower-*.f64 68.5

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot \color{blue}{x}, 1\right)}\right)\right) \cdot e^{-x} \]

   4. Applied rewrites 68.5%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{{x}^{4} \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\left(\frac{1}{24} + \frac{1}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{\color{blue}{4}}}\right)\right) \cdot e^{-x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\left(\frac{1}{24} + \frac{1}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{\color{blue}{4}}}\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 68.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(0.041666666666666664 + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{0.5}{x \cdot x}\right)\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}\right)\right) \cdot e^{-x} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{\frac{1}{x \cdot x} - 0.5}{x \cdot x} + 0.041666666666666664\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{x}\right)}\right)\right) \cdot e^{-x} \]

   if -9.99999999999999908e-22 < x

   1. Initial program 5.5%

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

   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 37.2%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 38.2

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

   10. Applied rewrites 38.2%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 98.6%

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

Alternative 4: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1e-21)
     (*
      (fmod (exp x) (sqrt (* (/ 1.0 (* (* (* x x) x) x)) (* (* x x) (* x x)))))
      t_0)
     (* (fmod x (fma (* x x) -0.25 1.0)) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1e-21) {
		tmp = fmod(exp(x), sqrt(((1.0 / (((x * x) * x) * x)) * ((x * x) * (x * x))))) * t_0;
	} else {
		tmp = fmod(x, fma((x * x), -0.25, 1.0)) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1e-21)
		tmp = Float64(rem(exp(x), sqrt(Float64(Float64(1.0 / Float64(Float64(Float64(x * x) * x) * x)) * Float64(Float64(x * x) * Float64(x * x))))) * t_0);
	else
		tmp = Float64(rem(x, fma(Float64(x * x), -0.25, 1.0)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1e-21], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999908e-22

   1. Initial program 68.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1}\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right)}\right)\right) \cdot e^{-x} \]

      4. negate-sub N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{x}}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right), {\color{blue}{x}}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot \color{blue}{x}, 1\right)}\right)\right) \cdot e^{-x} \]

      10. lower-*.f64 68.5

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot \color{blue}{x}, 1\right)}\right)\right) \cdot e^{-x} \]

   4. Applied rewrites 68.5%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{{x}^{4} \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\left(\frac{1}{24} + \frac{1}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{\color{blue}{4}}}\right)\right) \cdot e^{-x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\left(\frac{1}{24} + \frac{1}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{\color{blue}{4}}}\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 68.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(0.041666666666666664 + \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \frac{0.5}{x \cdot x}\right)\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}\right)\right) \cdot e^{-x} \]

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow3 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 68.7

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

   10. Applied rewrites 68.7%

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

   if -9.99999999999999908e-22 < x

   1. Initial program 5.5%

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

   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 37.2%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 38.2

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

   10. Applied rewrites 38.2%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 98.6%

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

Alternative 5: 97.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \frac{1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-21)
   (* (fmod (exp x) 1.0) (/ 1.0 (exp x)))
   (* (fmod x (fma (* x x) -0.25 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-21) {
		tmp = fmod(exp(x), 1.0) * (1.0 / exp(x));
	} else {
		tmp = fmod(x, fma((x * x), -0.25, 1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-21)
		tmp = Float64(rem(exp(x), 1.0) * Float64(1.0 / exp(x)));
	else
		tmp = Float64(rem(x, fma(Float64(x * x), -0.25, 1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-21], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 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 < -9.99999999999999908e-22

   1. Initial program 68.5%

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

   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 68.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. exp-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lift-exp.f64 68.6

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

   6. Applied rewrites 68.6%

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

   if -9.99999999999999908e-22 < x

   1. Initial program 5.5%

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

   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 37.2%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 38.2

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

   10. Applied rewrites 38.2%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 98.6%

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

Alternative 6: 97.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1e-21)
     (* (fmod (exp x) 1.0) t_0)
     (* (fmod x (fma (* x x) -0.25 1.0)) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1e-21) {
		tmp = fmod(exp(x), 1.0) * t_0;
	} else {
		tmp = fmod(x, fma((x * x), -0.25, 1.0)) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1e-21)
		tmp = Float64(rem(exp(x), 1.0) * t_0);
	else
		tmp = Float64(rem(x, fma(Float64(x * x), -0.25, 1.0)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1e-21], 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[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999908e-22

   1. Initial program 68.5%

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

   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 68.5%

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

   if -9.99999999999999908e-22 < x

   1. Initial program 5.5%

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

   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 37.2%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 38.2

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

   10. Applied rewrites 38.2%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 98.6%

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

Alternative 7: 96.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-21)
   (* (fmod (exp x) 1.0) (fma (fma 0.5 x -1.0) x 1.0))
   (* (fmod x (fma (* x x) -0.25 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-21) {
		tmp = fmod(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod(x, fma((x * x), -0.25, 1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-21)
		tmp = Float64(rem(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(x, fma(Float64(x * x), -0.25, 1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-21], 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[(N[(x * x), $MachinePrecision] * -0.25 + 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 < -9.99999999999999908e-22

   1. Initial program 68.5%

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

   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 68.5%

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

   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. negate-sub N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 57.6

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

   7. Applied rewrites 57.6%

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

   if -9.99999999999999908e-22 < x

   1. Initial program 5.5%

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

   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 37.2%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 38.2

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

   10. Applied rewrites 38.2%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 98.6%

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

Alternative 8: 96.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-21)
   (* (fmod (exp x) 1.0) (- 1.0 x))
   (* (fmod x (fma (* x x) -0.25 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-21) {
		tmp = fmod(exp(x), 1.0) * (1.0 - x);
	} else {
		tmp = fmod(x, fma((x * x), -0.25, 1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-21)
		tmp = Float64(rem(exp(x), 1.0) * Float64(1.0 - x));
	else
		tmp = Float64(rem(x, fma(Float64(x * x), -0.25, 1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-21], 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 = N[(N[(x * x), $MachinePrecision] * -0.25 + 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 < -9.99999999999999908e-22

   1. Initial program 68.5%

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

   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. mul-1-neg N/A

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

      2. negate-sub N/A

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

      3. lower--.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 53.0

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

   7. Applied rewrites 53.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 53.0%

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

   if -9.99999999999999908e-22 < x

   1. Initial program 5.5%

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

   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 37.2%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 38.2

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

   10. Applied rewrites 38.2%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 98.6%

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

Alternative 9: 39.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\left(x \cdot x\right) \cdot -0.25\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.6)
   (* (fmod (exp x) 1.0) (- 1.0 x))
   (* (fmod 1.0 (* (* x x) -0.25)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = fmod(exp(x), 1.0) * (1.0 - x);
	} else {
		tmp = fmod(1.0, ((x * x) * -0.25)) * 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 <= 0.6d0) then
        tmp = mod(exp(x), 1.0d0) * (1.0d0 - x)
    else
        tmp = mod(1.0d0, ((x * x) * (-0.25d0))) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 0.6:
		tmp = math.fmod(math.exp(x), 1.0) * (1.0 - x)
	else:
		tmp = math.fmod(1.0, ((x * x) * -0.25)) * math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64(rem(exp(x), 1.0) * Float64(1.0 - x));
	else
		tmp = Float64(rem(1.0, Float64(Float64(x * x) * -0.25)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.6], 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 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.599999999999999978

   1. Initial program 12.6%

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

   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. mul-1-neg N/A

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

      2. negate-sub N/A

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

      3. lower--.f64 11.0

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

   4. Applied rewrites 11.0%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 10.9

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

   7. Applied rewrites 10.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 10.7%

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

   if 0.599999999999999978 < x

   1. Initial program 0.5%

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

   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 99.5%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

Alternative 10: 39.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 12.7%

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

   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. mul-1-neg N/A

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

      2. negate-sub N/A

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

      3. lower--.f64 11.0

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

   4. Applied rewrites 11.0%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 11.0

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

   7. Applied rewrites 11.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 10.7%

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

   if 1 < x

   1. Initial program 0.2%

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

   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 99.7%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.6%

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

Alternative 11: 38.3% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 10.0) (* (fmod (exp x) 1.0) 1.0) (* (fmod 1.0 1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 10.0) {
		tmp = fmod(exp(x), 1.0) * 1.0;
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 10.0d0) then
        tmp = mod(exp(x), 1.0d0) * 1.0d0
    else
        tmp = mod(1.0d0, 1.0d0) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 10.0:
		tmp = math.fmod(math.exp(x), 1.0) * 1.0
	else:
		tmp = math.fmod(1.0, 1.0) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 10.0)
		tmp = Float64(rem(exp(x), 1.0) * 1.0);
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 10.0], 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 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 10

   1. Initial program 12.8%

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

   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 11.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 9.6%

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

   if 10 < x

   1. Initial program 0.1%

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

   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 99.8%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

      2. lift-neg.f64 N/A

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

      3. sin-+PI/2 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.8%

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

Alternative 12: 35.3% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

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 35.6%

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

5. Taylor expanded in x around 0

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

   1. cos-neg-rev N/A

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

   2. lift-neg.f64 N/A

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

   3. sin-+PI/2 N/A

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

   4. lift-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 35.6

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

7. Applied rewrites 35.6%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 35.3%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 35.3%

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

Reproduce

herbie shell --seed 2025110 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))