expfmod (used to be hard to sample)

Percentage Accurate: 7.0% → 91.6%

Time: 8.4s

Alternatives: 14

Speedup: 3.7×

• could not determine a ground truth

  1. x = -7.742710929056921e+291

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Initial Program: 7.0% 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: 91.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{-2} - 0.25\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(t\_0 \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{e^{x}}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(1 \bmod \left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot \left(x \cdot x\right)\right)\right) \cdot t\_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 \bmod \left(\left(t\_0 \cdot x\right) \cdot x\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (pow x -2.0) 0.25)) (t_1 (exp (- x))))
   (if (<= x -5e-32)
     (* (fmod (exp x) (* t_0 (* x x))) (/ 1.0 (exp x)))
     (if (<= x -7.5e-155)
       (*
        (fmod
         1.0
         (* (/ (- (pow x -4.0) 0.0625) (- (pow x -2.0) -0.25)) (* x x)))
        t_1)
       (if (<= x -2e-310)
         (* (fmod 1.0 (* (* t_0 x) x)) t_1)
         (* (fmod x (fma (* x x) -0.25 1.0)) t_1))))))

C

double code(double x) {
	double t_0 = pow(x, -2.0) - 0.25;
	double t_1 = exp(-x);
	double tmp;
	if (x <= -5e-32) {
		tmp = fmod(exp(x), (t_0 * (x * x))) * (1.0 / exp(x));
	} else if (x <= -7.5e-155) {
		tmp = fmod(1.0, (((pow(x, -4.0) - 0.0625) / (pow(x, -2.0) - -0.25)) * (x * x))) * t_1;
	} else if (x <= -2e-310) {
		tmp = fmod(1.0, ((t_0 * x) * x)) * t_1;
	} else {
		tmp = fmod(x, fma((x * x), -0.25, 1.0)) * t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64((x ^ -2.0) - 0.25)
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -5e-32)
		tmp = Float64(rem(exp(x), Float64(t_0 * Float64(x * x))) * Float64(1.0 / exp(x)));
	elseif (x <= -7.5e-155)
		tmp = Float64(rem(1.0, Float64(Float64(Float64((x ^ -4.0) - 0.0625) / Float64((x ^ -2.0) - -0.25)) * Float64(x * x))) * t_1);
	elseif (x <= -2e-310)
		tmp = Float64(rem(1.0, Float64(Float64(t_0 * x) * x)) * t_1);
	else
		tmp = Float64(rem(x, fma(Float64(x * x), -0.25, 1.0)) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -5e-32], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.5e-155], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(N[(N[Power[x, -4.0], $MachinePrecision] - 0.0625), $MachinePrecision] / N[(N[Power[x, -2.0], $MachinePrecision] - -0.25), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $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$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5e-32

   1. Initial program 46.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. exp-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lift-exp.f64 46.6

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

   4. Applied rewrites 46.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 46.6

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

   7. Applied rewrites 46.6%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 50.0

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

   10. Applied rewrites 50.0%

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

   if -5e-32 < x < -7.5000000000000006e-155

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. pow-flip N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 9.0

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

   11. Applied rewrites 9.0%

       \[\leadsto \left(1 \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. flip-- N/A

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

       6. lower-/.f64 N/A

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

       7. frac-times N/A

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

       8. metadata-eval N/A

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

       9. pow-prod-up N/A

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

       10. metadata-eval N/A

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

       11. lower--.f64 N/A

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

       12. pow-flip N/A

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

       13. lower-pow.f64 N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

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

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

       20. lower--.f64 N/A

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

       21. pow-flip N/A

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

       22. metadata-eval N/A

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

       23. lift-pow.f64 70.0

           \[\leadsto \left(1 \bmod \left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

   13. Applied rewrites 70.0%

       \[\leadsto \left(1 \bmod \left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot \left(x \cdot x\right)\right)\right) \cdot e^{-x} \]

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

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. pow-flip N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 5.4

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

   11. Applied rewrites 5.4%

       \[\leadsto \left(1 \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. lift--.f64 100.0

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

   13. Applied rewrites 100.0%

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

   if -1.999999999999994e-310 < x

   1. Initial program 6.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.5

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 37.5

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

   11. Applied rewrites 37.5%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 98.1%

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

Alternative 2: 25.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in x around 0

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

      1. rec-exp N/A

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

      2. mul-1-neg N/A

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

      3. flip-+ N/A

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

      4. sqr-neg-rev N/A

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

      5. flip3-- N/A

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

      6. metadata-eval N/A

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

      7. cube-neg N/A

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

      8. sqr-pow N/A

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

      9. distribute-lft-neg-in N/A

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

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

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

      11. metadata-eval N/A

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

      12. sqr-pow N/A

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

      13. sqr-neg-rev N/A

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

      14. unpow2 N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. distribute-lft-neg-out N/A

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

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

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

      18. unpow2 N/A

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

   8. Applied rewrites 7.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 97.9

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

   8. Applied rewrites 97.9%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. lower-fma.f64 2.5

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

   11. Applied rewrites 2.5%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 97.9%

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

Alternative 3: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod t\_0\right) \cdot 1}{e^{x}}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-154}:\\ \;\;\;\;\left(1 \bmod \left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod t\_0\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)) (t_1 (exp (- x))))
   (if (<= x -6e-17)
     (/ (* (fmod (exp x) t_0) 1.0) (exp x))
     (if (<= x -1e-154)
       (* (fmod 1.0 (* (- (exp (* (log (* x x)) -1.0)) 0.25) (* x x))) t_1)
       (if (<= x -2e-310)
         (* (fmod 1.0 (* (* (- (pow x -2.0) 0.25) x) x)) t_1)
         (* (fmod x t_0) t_1))))))

C

double code(double x) {
	double t_0 = fma((x * x), -0.25, 1.0);
	double t_1 = exp(-x);
	double tmp;
	if (x <= -6e-17) {
		tmp = (fmod(exp(x), t_0) * 1.0) / exp(x);
	} else if (x <= -1e-154) {
		tmp = fmod(1.0, ((exp((log((x * x)) * -1.0)) - 0.25) * (x * x))) * t_1;
	} else if (x <= -2e-310) {
		tmp = fmod(1.0, (((pow(x, -2.0) - 0.25) * x) * x)) * t_1;
	} else {
		tmp = fmod(x, t_0) * t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), -0.25, 1.0)
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -6e-17)
		tmp = Float64(Float64(rem(exp(x), t_0) * 1.0) / exp(x));
	elseif (x <= -1e-154)
		tmp = Float64(rem(1.0, Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * Float64(x * x))) * t_1);
	elseif (x <= -2e-310)
		tmp = Float64(rem(1.0, Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * t_1);
	else
		tmp = Float64(rem(x, t_0) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -6e-17], N[(N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-154], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.00000000000000012e-17

   1. Initial program 83.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. exp-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lift-exp.f64 83.8

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

   4. Applied rewrites 83.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-exp.f64 84.0

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

   9. Applied rewrites 84.0%

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

   if -6.00000000000000012e-17 < x < -9.9999999999999997e-155

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. pow-flip N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 9.3

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

   11. Applied rewrites 9.3%

       \[\leadsto \left(1 \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-pow N/A

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

       4. pow-to-exp N/A

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

       5. lower-exp.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-log.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 55.7

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

   13. Applied rewrites 55.7%

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

   if -9.9999999999999997e-155 < x < -1.999999999999994e-310

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. pow-flip N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 5.4

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

   11. Applied rewrites 5.4%

       \[\leadsto \left(1 \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. lift--.f64 99.8

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

   13. Applied rewrites 99.8%

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

   if -1.999999999999994e-310 < x

   1. Initial program 6.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.5

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 37.5

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

   11. Applied rewrites 37.5%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 98.1%

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

Alternative 4: 84.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod t\_1\right) \cdot 1}{e^{x}}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod t\_1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (fma (* x x) -0.25 1.0)))
   (if (<= x -5e-17)
     (/ (* (fmod (exp x) t_1) 1.0) (exp x))
     (if (<= x -2e-151)
       (* (fmod 1.0 (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) (* x x))) t_0)
       (if (<= x -2e-310)
         (* (fmod 1.0 (* (* (- (pow x -2.0) 0.25) x) x)) t_0)
         (* (fmod x t_1) t_0))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fma((x * x), -0.25, 1.0);
	double tmp;
	if (x <= -5e-17) {
		tmp = (fmod(exp(x), t_1) * 1.0) / exp(x);
	} else if (x <= -2e-151) {
		tmp = fmod(1.0, (fma((-1.0 / x), (-1.0 / x), -0.25) * (x * x))) * t_0;
	} else if (x <= -2e-310) {
		tmp = fmod(1.0, (((pow(x, -2.0) - 0.25) * x) * x)) * t_0;
	} else {
		tmp = fmod(x, t_1) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = fma(Float64(x * x), -0.25, 1.0)
	tmp = 0.0
	if (x <= -5e-17)
		tmp = Float64(Float64(rem(exp(x), t_1) * 1.0) / exp(x));
	elseif (x <= -2e-151)
		tmp = Float64(rem(1.0, Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * Float64(x * x))) * t_0);
	elseif (x <= -2e-310)
		tmp = Float64(rem(1.0, Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * t_0);
	else
		tmp = Float64(rem(x, t_1) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, If[LessEqual[x, -5e-17], N[(N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-151], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.9999999999999999e-17

   1. Initial program 83.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. exp-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lift-exp.f64 83.8

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

   4. Applied rewrites 83.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-exp.f64 84.0

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

   9. Applied rewrites 84.0%

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

   if -4.9999999999999999e-17 < x < -1.9999999999999999e-151

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. pow-flip N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 9.2

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

   11. Applied rewrites 9.2%

       \[\leadsto \left(1 \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow2 N/A

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

       9. times-frac N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 22.8

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

   13. Applied rewrites 22.8%

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

   if -1.9999999999999999e-151 < x < -1.999999999999994e-310

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. pow-flip N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 5.6

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

   11. Applied rewrites 5.6%

       \[\leadsto \left(1 \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. lift--.f64 97.8

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

   13. Applied rewrites 97.8%

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

   if -1.999999999999994e-310 < x

   1. Initial program 6.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.5

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 37.5

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

   11. Applied rewrites 37.5%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 98.1%

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

Alternative 5: 84.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \frac{1}{e^{x}}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -5e-17)
     (* (fmod (exp x) 1.0) (/ 1.0 (exp x)))
     (if (<= x -2e-151)
       (* (fmod 1.0 (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) (* x x))) t_0)
       (if (<= x -2e-310)
         (* (fmod 1.0 (* (* (- (pow x -2.0) 0.25) 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 <= -5e-17) {
		tmp = fmod(exp(x), 1.0) * (1.0 / exp(x));
	} else if (x <= -2e-151) {
		tmp = fmod(1.0, (fma((-1.0 / x), (-1.0 / x), -0.25) * (x * x))) * t_0;
	} else if (x <= -2e-310) {
		tmp = fmod(1.0, (((pow(x, -2.0) - 0.25) * 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 <= -5e-17)
		tmp = Float64(rem(exp(x), 1.0) * Float64(1.0 / exp(x)));
	elseif (x <= -2e-151)
		tmp = Float64(rem(1.0, Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * Float64(x * x))) * t_0);
	elseif (x <= -2e-310)
		tmp = Float64(rem(1.0, Float64(Float64(Float64((x ^ -2.0) - 0.25) * 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, -5e-17], 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], If[LessEqual[x, -2e-151], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 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 4 regimes

2. if x < -4.9999999999999999e-17

   1. Initial program 83.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. rec-exp N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-/.f64 83.8

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

   7. Applied rewrites 83.8%

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

   if -4.9999999999999999e-17 < x < -1.9999999999999999e-151

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. pow-flip N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 9.2

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

   11. Applied rewrites 9.2%

       \[\leadsto \left(1 \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow2 N/A

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

       9. times-frac N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 22.8

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

   13. Applied rewrites 22.8%

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

   if -1.9999999999999999e-151 < x < -1.999999999999994e-310

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. pow-flip N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 5.6

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

   11. Applied rewrites 5.6%

       \[\leadsto \left(1 \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. lift--.f64 97.8

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

   13. Applied rewrites 97.8%

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

   if -1.999999999999994e-310 < x

   1. Initial program 6.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.5

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 37.5

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

   11. Applied rewrites 37.5%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 98.1%

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

Alternative 6: 66.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \frac{1}{e^{x}}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -5e-17)
     (* (fmod (exp x) 1.0) (/ 1.0 (exp x)))
     (if (<= x -1.55e-162)
       (* (fmod 1.0 (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) (* 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 <= -5e-17) {
		tmp = fmod(exp(x), 1.0) * (1.0 / exp(x));
	} else if (x <= -1.55e-162) {
		tmp = fmod(1.0, (fma((-1.0 / x), (-1.0 / x), -0.25) * (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 <= -5e-17)
		tmp = Float64(rem(exp(x), 1.0) * Float64(1.0 / exp(x)));
	elseif (x <= -1.55e-162)
		tmp = Float64(rem(1.0, Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * 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, -5e-17], 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], If[LessEqual[x, -1.55e-162], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = N[(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 3 regimes

2. if x < -4.9999999999999999e-17

   1. Initial program 83.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. rec-exp N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-/.f64 83.8

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

   7. Applied rewrites 83.8%

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

   if -4.9999999999999999e-17 < x < -1.5499999999999999e-162

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. pow-flip N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 14.4

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

   11. Applied rewrites 14.4%

       \[\leadsto \left(1 \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow2 N/A

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

       9. times-frac N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 27.2

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

   13. Applied rewrites 27.2%

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

   if -1.5499999999999999e-162 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 28.5

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

   8. Applied rewrites 28.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 29.3

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

   11. Applied rewrites 29.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 75.2%

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

Alternative 7: 65.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod t\_0\right) \cdot t\_1\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod t\_0\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)) (t_1 (exp (- x))))
   (if (<= x -5e-17)
     (* (fmod (fma (fma 0.5 x 1.0) x 1.0) t_0) t_1)
     (if (<= x -1.55e-162)
       (* (fmod 1.0 (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) (* x x))) t_1)
       (* (fmod x t_0) t_1)))))

C

double code(double x) {
	double t_0 = fma((x * x), -0.25, 1.0);
	double t_1 = exp(-x);
	double tmp;
	if (x <= -5e-17) {
		tmp = fmod(fma(fma(0.5, x, 1.0), x, 1.0), t_0) * t_1;
	} else if (x <= -1.55e-162) {
		tmp = fmod(1.0, (fma((-1.0 / x), (-1.0 / x), -0.25) * (x * x))) * t_1;
	} else {
		tmp = fmod(x, t_0) * t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), -0.25, 1.0)
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -5e-17)
		tmp = Float64(rem(fma(fma(0.5, x, 1.0), x, 1.0), t_0) * t_1);
	elseif (x <= -1.55e-162)
		tmp = Float64(rem(1.0, Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * Float64(x * x))) * t_1);
	else
		tmp = Float64(rem(x, t_0) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -5e-17], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x, -1.55e-162], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999999e-17

   1. Initial program 83.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 6.5

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

   8. Applied rewrites 6.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 69.1

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

   11. Applied rewrites 69.1%

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

   if -4.9999999999999999e-17 < x < -1.5499999999999999e-162

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. pow-flip N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 14.4

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

   11. Applied rewrites 14.4%

       \[\leadsto \left(1 \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow2 N/A

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

       9. times-frac N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 27.2

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

   13. Applied rewrites 27.2%

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

   if -1.5499999999999999e-162 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 28.5

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

   8. Applied rewrites 28.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 29.3

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

   11. Applied rewrites 29.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 75.2%

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

Alternative 8: 62.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod t\_0\right) \cdot t\_1\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(1 \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod t\_0\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)) (t_1 (exp (- x))))
   (if (<= x -1e-10)
     (* (fmod (fma (fma 0.5 x 1.0) x 1.0) t_0) t_1)
     (if (<= x -1.55e-162)
       (* (fmod 1.0 (* (- (pow x -2.0) 0.25) (* x x))) 1.0)
       (* (fmod x t_0) t_1)))))

C

double code(double x) {
	double t_0 = fma((x * x), -0.25, 1.0);
	double t_1 = exp(-x);
	double tmp;
	if (x <= -1e-10) {
		tmp = fmod(fma(fma(0.5, x, 1.0), x, 1.0), t_0) * t_1;
	} else if (x <= -1.55e-162) {
		tmp = fmod(1.0, ((pow(x, -2.0) - 0.25) * (x * x))) * 1.0;
	} else {
		tmp = fmod(x, t_0) * t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), -0.25, 1.0)
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1e-10)
		tmp = Float64(rem(fma(fma(0.5, x, 1.0), x, 1.0), t_0) * t_1);
	elseif (x <= -1.55e-162)
		tmp = Float64(rem(1.0, Float64(Float64((x ^ -2.0) - 0.25) * Float64(x * x))) * 1.0);
	else
		tmp = Float64(rem(x, t_0) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1e-10], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x, -1.55e-162], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000004e-10

   1. Initial program 74.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 8.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 8.4

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

   8. Applied rewrites 8.4%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 52.1

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

   11. Applied rewrites 52.1%

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

   if -1.00000000000000004e-10 < x < -1.5499999999999999e-162

   1. Initial program 7.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. pow-flip N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 14.0

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

   11. Applied rewrites 14.0%

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

   12. Taylor expanded in x around 0

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

       1. rec-exp 14.0

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

   14. Applied rewrites 14.0%

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

   if -1.5499999999999999e-162 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 28.5

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

   8. Applied rewrites 28.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 29.3

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

   11. Applied rewrites 29.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 75.2%

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

Alternative 9: 62.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(1 \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-10)
   (* (fmod (exp x) 1.0) (fma (fma 0.5 x -1.0) x 1.0))
   (if (<= x -1.55e-162)
     (* (fmod 1.0 (* (- (pow x -2.0) 0.25) (* x x))) 1.0)
     (* (fmod x (fma (* x x) -0.25 1.0)) (exp (- x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-10) {
		tmp = fmod(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else if (x <= -1.55e-162) {
		tmp = fmod(1.0, ((pow(x, -2.0) - 0.25) * (x * 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-10)
		tmp = Float64(rem(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0));
	elseif (x <= -1.55e-162)
		tmp = Float64(rem(1.0, Float64(Float64((x ^ -2.0) - 0.25) * Float64(x * 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-10], 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], If[LessEqual[x, -1.55e-162], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $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] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000004e-10

   1. Initial program 74.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 51.7

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.7

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

   8. Applied rewrites 51.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 51.7%

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

   if -1.00000000000000004e-10 < x < -1.5499999999999999e-162

   1. Initial program 7.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. pow-flip N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 14.0

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

   11. Applied rewrites 14.0%

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

   12. Taylor expanded in x around 0

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

       1. rec-exp 14.0

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

   14. Applied rewrites 14.0%

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

   if -1.5499999999999999e-162 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 28.5

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

   8. Applied rewrites 28.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 29.3

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

   11. Applied rewrites 29.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 75.2%

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

Alternative 10: 61.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (* (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 <= -2e-310) {
		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 <= -2e-310)
		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, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 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 < -1.999999999999994e-310

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 7.4

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

   5. Applied rewrites 7.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 7.4

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

   8. Applied rewrites 7.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 7.4%

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

   if -1.999999999999994e-310 < x

   1. Initial program 6.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.5

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 37.5

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

   11. Applied rewrites 37.5%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 98.1%

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

Alternative 11: 25.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 23.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 23.0

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

8. Applied rewrites 23.0%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 25.1

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

11. Applied rewrites 25.1%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 25.1%

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

Alternative 12: 25.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.01:\\ \;\;\;\;\left(\left(1 + x\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.01)
   (*
    (fmod (+ 1.0 x) (fma (* x x) -0.25 1.0))
    (fma (fma (fma -0.16666666666666666 x 0.5) x -1.0) x 1.0))
   (* (fmod 1.0 1.0) (fma -1.0 x 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.01) {
		tmp = fmod((1.0 + x), fma((x * x), -0.25, 1.0)) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, 1.0) * fma(-1.0, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.01)
		tmp = Float64(rem(Float64(1.0 + x), fma(Float64(x * x), -0.25, 1.0)) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, 1.0) * fma(-1.0, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.01], N[(N[With[{TMP1 = N[(1.0 + x), $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0100000000000000002

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 4.5

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

   8. Applied rewrites 4.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 7.1

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

   11. Applied rewrites 7.1%

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

   12. Taylor expanded in x around 0

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

       1. rec-exp N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

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

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 7.1

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

   14. Applied rewrites 7.1%

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

   if 0.0100000000000000002 < x

   1. Initial program 2.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 97.7

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. lower-fma.f64 2.5

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

   11. Applied rewrites 2.5%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 97.3%

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

Alternative 13: 24.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.01:\\ \;\;\;\;\left(\left(1 + x\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.01)
   (* (fmod (+ 1.0 x) (fma (* x x) -0.25 1.0)) 1.0)
   (* (fmod 1.0 1.0) (fma -1.0 x 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.01) {
		tmp = fmod((1.0 + x), fma((x * x), -0.25, 1.0)) * 1.0;
	} else {
		tmp = fmod(1.0, 1.0) * fma(-1.0, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.01)
		tmp = Float64(rem(Float64(1.0 + x), fma(Float64(x * x), -0.25, 1.0)) * 1.0);
	else
		tmp = Float64(rem(1.0, 1.0) * fma(-1.0, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.01], N[(N[With[{TMP1 = N[(1.0 + x), $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, 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] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0100000000000000002

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

   5. Applied rewrites 4.5%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 4.5

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 4.5%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

       1. lower-+.f64 7.1

          \[\leadsto \left(\left(1 + \color{blue}{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   11. Applied rewrites 7.1%

       \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\left(1 + x\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{1} \]
   13. Step-by-step derivation

       1. rec-exp 6.6

          \[\leadsto \left(\left(1 + x\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot 1 \]

   14. Applied rewrites 6.6%

       \[\leadsto \left(\left(1 + x\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{1} \]

   if 0.0100000000000000002 < x

   1. Initial program 2.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.7%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 97.7

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 97.7%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \color{blue}{1}\right) \]

       3. mul-1-neg N/A

          \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(-1 \cdot x + 1\right) \]

       4. lower-fma.f64 2.5

          \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(-1, \color{blue}{x}, 1\right) \]

   11. Applied rewrites 2.5%

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1, x, 1\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.3%

       \[\leadsto \left(1 \bmod 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 22.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \left(1 \bmod 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod 1.0 1.0) (fma -1.0 x 1.0)))

C

double code(double x) {
	return fmod(1.0, 1.0) * fma(-1.0, x, 1.0);
}

Julia

function code(x)
	return Float64(rem(1.0, 1.0) * fma(-1.0, x, 1.0))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
4. Step-by-step derivation

5. Applied rewrites 23.0%

   \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{1}\right)\right) \cdot e^{-x} \]

   2. *-commutative N/A

      \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \frac{-1}{4} + 1\right)\right) \cdot e^{-x} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

   4. unpow2 N/A

      \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

   5. lower-*.f64 23.0

      \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

8. Applied rewrites 23.0%

   \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

9. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

    2. +-commutative N/A

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \color{blue}{1}\right) \]

    3. mul-1-neg N/A

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(-1 \cdot x + 1\right) \]

    4. lower-fma.f64 4.1

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(-1, \color{blue}{x}, 1\right) \]

11. Applied rewrites 4.1%

    \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1, x, 1\right)} \]

12. Taylor expanded in x around 0

    \[\leadsto \left(1 \bmod 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right) \]
13. Step-by-step derivation

14. Applied rewrites 22.7%

    \[\leadsto \left(1 \bmod 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025089 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))