expfmod (used to be hard to sample)

Percentage Accurate: 6.7% → 62.2%

Time: 8.5s

Alternatives: 7

Speedup: 3.6×

• could not determine a ground truth

  1. x = -4.8119349653704684e+160

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-78}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \left(-0.5 \cdot x\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\sqrt{1}\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 x -1.0) x 1.0)))
   (if (<= x -8.5e-78)
     (* (fmod (exp x) (* (- (exp (* (log (* x x)) -1.0)) 0.25) (* x x))) t_0)
     (if (<= x -7.5e-155)
       (*
        (fmod
         (exp x)
         (* (/ (- (pow x -4.0) 0.0625) (- (pow x -2.0) -0.25)) (* x x)))
        t_0)
       (if (<= x -5e-310)
         (* (fmod (exp x) (* (* (- (pow x -2.0) 0.25) x) x)) t_0)
         (if (<= x 1.1e-16)
           (* (fmod (exp x) (* (fma 0.5 x 1.0) (* -0.5 x))) (exp (- x)))
           (* (fmod (- x -1.0) (sqrt 1.0)) (fma -1.0 x 1.0))))))))

C

double code(double x) {
	double t_0 = fma(fma(0.5, x, -1.0), x, 1.0);
	double tmp;
	if (x <= -8.5e-78) {
		tmp = fmod(exp(x), ((exp((log((x * x)) * -1.0)) - 0.25) * (x * x))) * t_0;
	} else if (x <= -7.5e-155) {
		tmp = fmod(exp(x), (((pow(x, -4.0) - 0.0625) / (pow(x, -2.0) - -0.25)) * (x * x))) * t_0;
	} else if (x <= -5e-310) {
		tmp = fmod(exp(x), (((pow(x, -2.0) - 0.25) * x) * x)) * t_0;
	} else if (x <= 1.1e-16) {
		tmp = fmod(exp(x), (fma(0.5, x, 1.0) * (-0.5 * x))) * exp(-x);
	} else {
		tmp = fmod((x - -1.0), sqrt(1.0)) * fma(-1.0, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(fma(0.5, x, -1.0), x, 1.0)
	tmp = 0.0
	if (x <= -8.5e-78)
		tmp = Float64(rem(exp(x), Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * Float64(x * x))) * t_0);
	elseif (x <= -7.5e-155)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -4.0) - 0.0625) / Float64((x ^ -2.0) - -0.25)) * Float64(x * x))) * t_0);
	elseif (x <= -5e-310)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * t_0);
	elseif (x <= 1.1e-16)
		tmp = Float64(rem(exp(x), Float64(fma(0.5, x, 1.0) * Float64(-0.5 * x))) * exp(Float64(-x)));
	else
		tmp = Float64(rem(Float64(x - -1.0), sqrt(1.0)) * fma(-1.0, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]}, If[LessEqual[x, -8.5e-78], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -7.5e-155], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -4.0], $MachinePrecision] - 0.0625), $MachinePrecision] / N[(N[Power[x, -2.0], $MachinePrecision] - -0.25), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 1.1e-16], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -8.49999999999999957e-78

   1. Initial program 21.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 21.5

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

   4. Applied rewrites 21.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 17.7

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

   7. Applied rewrites 17.7%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 22.7

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

   10. Applied rewrites 22.7%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]
   11. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-pow N/A

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

       4. pow-to-exp N/A

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

       5. lower-exp.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-log.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 58.0

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

   12. Applied rewrites 58.0%

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

   if -8.49999999999999957e-78 < 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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   4. Applied rewrites 3.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 3.1

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 9.6

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

   10. Applied rewrites 9.6%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]
   11. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. flip-- N/A

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

       6. lower-/.f64 N/A

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

       7. lower--.f64 N/A

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

       8. pow-flip N/A

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

       9. metadata-eval N/A

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

       10. pow-flip N/A

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

       11. metadata-eval N/A

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

       12. pow-prod-up N/A

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

       13. lower-pow.f64 N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

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

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

       20. lower--.f64 N/A

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

       21. pow-flip N/A

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

       22. metadata-eval N/A

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

       23. lift-pow.f64 100.0

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]

   12. Applied rewrites 100.0%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{{x}^{-4} - 0.0625}{{x}^{-2} - -0.25} \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]

   if -7.5000000000000006e-155 < x < -4.999999999999985e-310

   1. Initial program 3.1%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   4. Applied rewrites 3.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 3.1

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 5.8

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

   10. Applied rewrites 5.8%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. lift--.f64 100.0

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

   12. Applied rewrites 100.0%

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

   if -4.999999999999985e-310 < x < 1.1e-16

   1. Initial program 5.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 5.5

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

   4. Applied rewrites 5.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. swap-sqr N/A

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

      9. metadata-eval N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 5.5

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

   6. Applied rewrites 5.5%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 17.1

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

   9. Applied rewrites 17.1%

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

   if 1.1e-16 < x

   1. Initial program 6.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 89.2%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 89.2

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

   10. Applied rewrites 89.2%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

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

       6. lower--.f64 89.3

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

   13. Applied rewrites 89.3%

       \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{1}\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 29.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \left(-0.5 \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\sqrt{1}\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 0.0)
   (*
    (fmod (exp x) (* (fma 0.5 x 1.0) (* -0.5 x)))
    (fma (fma 0.5 x -1.0) x 1.0))
   (* (fmod (- x -1.0) (sqrt 1.0)) (fma -1.0 x 1.0))))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 0.0) {
		tmp = fmod(exp(x), (fma(0.5, x, 1.0) * (-0.5 * x))) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod((x - -1.0), sqrt(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))) <= 0.0)
		tmp = Float64(rem(exp(x), Float64(fma(0.5, x, 1.0) * Float64(-0.5 * x))) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(Float64(x - -1.0), sqrt(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], 0.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[Sqrt[1.0], $MachinePrecision]}, 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))) < 0.0

   1. Initial program 4.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 4.3

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

   4. Applied rewrites 4.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 4.3

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

   7. Applied rewrites 4.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. difference-of-squares-rev N/A

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

      13. +-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. lower-fma.f64 4.3

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

   9. Applied rewrites 4.3%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 11.0

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

   12. Applied rewrites 11.0%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 14.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 82.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 81.1%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 81.2

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

   10. Applied rewrites 81.2%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

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

       6. lower--.f64 86.6

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

   13. Applied rewrites 86.6%

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

Alternative 3: 57.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-152}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(e^{\log \left(x \cdot x\right) \cdot -1} - 0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \left(-0.5 \cdot x\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\sqrt{1}\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 x -1.0) x 1.0)))
   (if (<= x -1e-152)
     (* (fmod (exp x) (* (- (exp (* (log (* x x)) -1.0)) 0.25) (* x x))) t_0)
     (if (<= x -5e-310)
       (* (fmod (exp x) (* (* (- (pow x -2.0) 0.25) x) x)) t_0)
       (if (<= x 1.1e-16)
         (* (fmod (exp x) (* (fma 0.5 x 1.0) (* -0.5 x))) (exp (- x)))
         (* (fmod (- x -1.0) (sqrt 1.0)) (fma -1.0 x 1.0)))))))

C

double code(double x) {
	double t_0 = fma(fma(0.5, x, -1.0), x, 1.0);
	double tmp;
	if (x <= -1e-152) {
		tmp = fmod(exp(x), ((exp((log((x * x)) * -1.0)) - 0.25) * (x * x))) * t_0;
	} else if (x <= -5e-310) {
		tmp = fmod(exp(x), (((pow(x, -2.0) - 0.25) * x) * x)) * t_0;
	} else if (x <= 1.1e-16) {
		tmp = fmod(exp(x), (fma(0.5, x, 1.0) * (-0.5 * x))) * exp(-x);
	} else {
		tmp = fmod((x - -1.0), sqrt(1.0)) * fma(-1.0, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(fma(0.5, x, -1.0), x, 1.0)
	tmp = 0.0
	if (x <= -1e-152)
		tmp = Float64(rem(exp(x), Float64(Float64(exp(Float64(log(Float64(x * x)) * -1.0)) - 0.25) * Float64(x * x))) * t_0);
	elseif (x <= -5e-310)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * t_0);
	elseif (x <= 1.1e-16)
		tmp = Float64(rem(exp(x), Float64(fma(0.5, x, 1.0) * Float64(-0.5 * x))) * exp(Float64(-x)));
	else
		tmp = Float64(rem(Float64(x - -1.0), sqrt(1.0)) * fma(-1.0, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]}, If[LessEqual[x, -1e-152], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 1.1e-16], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.00000000000000007e-152

   1. Initial program 12.4%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 12.4

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

   4. Applied rewrites 12.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 10.5

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 16.3

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

   10. Applied rewrites 16.3%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]
   11. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-pow N/A

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

       4. pow-to-exp N/A

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

       5. lower-exp.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-log.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 56.4

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

   12. Applied rewrites 56.4%

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

   if -1.00000000000000007e-152 < x < -4.999999999999985e-310

   1. Initial program 3.1%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   4. Applied rewrites 3.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 3.1

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 5.8

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

   10. Applied rewrites 5.8%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. lift--.f64 98.8

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

   12. Applied rewrites 98.8%

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

   if -4.999999999999985e-310 < x < 1.1e-16

   1. Initial program 5.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 5.5

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

   4. Applied rewrites 5.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. swap-sqr N/A

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

      9. metadata-eval N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 5.5

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

   6. Applied rewrites 5.5%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 17.1

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

   9. Applied rewrites 17.1%

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

   if 1.1e-16 < x

   1. Initial program 6.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 89.2%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 89.2

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

   10. Applied rewrites 89.2%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

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

       6. lower--.f64 89.3

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

   13. Applied rewrites 89.3%

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

Alternative 4: 51.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-152}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{-2} - 0.25\right) \cdot x\right) \cdot x\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \left(-0.5 \cdot x\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\sqrt{1}\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 x -1.0) x 1.0)))
   (if (<= x -1e-152)
     (* (fmod (exp x) (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) (* x x))) t_0)
     (if (<= x -5e-310)
       (* (fmod (exp x) (* (* (- (pow x -2.0) 0.25) x) x)) t_0)
       (if (<= x 1.1e-16)
         (* (fmod (exp x) (* (fma 0.5 x 1.0) (* -0.5 x))) (exp (- x)))
         (* (fmod (- x -1.0) (sqrt 1.0)) (fma -1.0 x 1.0)))))))

C

double code(double x) {
	double t_0 = fma(fma(0.5, x, -1.0), x, 1.0);
	double tmp;
	if (x <= -1e-152) {
		tmp = fmod(exp(x), (fma((-1.0 / x), (-1.0 / x), -0.25) * (x * x))) * t_0;
	} else if (x <= -5e-310) {
		tmp = fmod(exp(x), (((pow(x, -2.0) - 0.25) * x) * x)) * t_0;
	} else if (x <= 1.1e-16) {
		tmp = fmod(exp(x), (fma(0.5, x, 1.0) * (-0.5 * x))) * exp(-x);
	} else {
		tmp = fmod((x - -1.0), sqrt(1.0)) * fma(-1.0, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(fma(0.5, x, -1.0), x, 1.0)
	tmp = 0.0
	if (x <= -1e-152)
		tmp = Float64(rem(exp(x), Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * Float64(x * x))) * t_0);
	elseif (x <= -5e-310)
		tmp = Float64(rem(exp(x), Float64(Float64(Float64((x ^ -2.0) - 0.25) * x) * x)) * t_0);
	elseif (x <= 1.1e-16)
		tmp = Float64(rem(exp(x), Float64(fma(0.5, x, 1.0) * Float64(-0.5 * x))) * exp(Float64(-x)));
	else
		tmp = Float64(rem(Float64(x - -1.0), sqrt(1.0)) * fma(-1.0, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]}, If[LessEqual[x, -1e-152], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], 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, -5e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[Power[x, -2.0], $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 1.1e-16], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.00000000000000007e-152

   1. Initial program 12.4%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 12.4

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

   4. Applied rewrites 12.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 10.5

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 16.3

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

   10. Applied rewrites 16.3%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]
   11. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow2 N/A

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

       9. times-frac N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 28.0

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

   12. Applied rewrites 28.0%

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

   if -1.00000000000000007e-152 < x < -4.999999999999985e-310

   1. Initial program 3.1%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   4. Applied rewrites 3.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 3.1

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 5.8

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

   10. Applied rewrites 5.8%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. lift--.f64 98.8

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

   12. Applied rewrites 98.8%

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

   if -4.999999999999985e-310 < x < 1.1e-16

   1. Initial program 5.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 5.5

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

   4. Applied rewrites 5.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. swap-sqr N/A

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

      9. metadata-eval N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 5.5

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

   6. Applied rewrites 5.5%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 17.1

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

   9. Applied rewrites 17.1%

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

   if 1.1e-16 < x

   1. Initial program 6.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 89.2%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 89.2

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

   10. Applied rewrites 89.2%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

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

       6. lower--.f64 89.3

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

   13. Applied rewrites 89.3%

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

Alternative 5: 33.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, -0.25\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \left(-0.5 \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\sqrt{1}\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 x -1.0) x 1.0)))
   (if (<= x -1.55e-162)
     (* (fmod (exp x) (* (fma (/ -1.0 x) (/ -1.0 x) -0.25) (* x x))) t_0)
     (if (<= x 1.1e-16)
       (* (fmod (exp x) (* (fma 0.5 x 1.0) (* -0.5 x))) t_0)
       (* (fmod (- x -1.0) (sqrt 1.0)) (fma -1.0 x 1.0))))))

C

double code(double x) {
	double t_0 = fma(fma(0.5, x, -1.0), x, 1.0);
	double tmp;
	if (x <= -1.55e-162) {
		tmp = fmod(exp(x), (fma((-1.0 / x), (-1.0 / x), -0.25) * (x * x))) * t_0;
	} else if (x <= 1.1e-16) {
		tmp = fmod(exp(x), (fma(0.5, x, 1.0) * (-0.5 * x))) * t_0;
	} else {
		tmp = fmod((x - -1.0), sqrt(1.0)) * fma(-1.0, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(fma(0.5, x, -1.0), x, 1.0)
	tmp = 0.0
	if (x <= -1.55e-162)
		tmp = Float64(rem(exp(x), Float64(fma(Float64(-1.0 / x), Float64(-1.0 / x), -0.25) * Float64(x * x))) * t_0);
	elseif (x <= 1.1e-16)
		tmp = Float64(rem(exp(x), Float64(fma(0.5, x, 1.0) * Float64(-0.5 * x))) * t_0);
	else
		tmp = Float64(rem(Float64(x - -1.0), sqrt(1.0)) * fma(-1.0, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.55e-162], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], 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, 1.1e-16], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5499999999999999e-162

   1. Initial program 11.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 11.8

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

   4. Applied rewrites 11.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 10.0

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

   7. Applied rewrites 10.0%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 20.6

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

   10. Applied rewrites 20.6%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left({x}^{-2} - 0.25\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]
   11. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-flip N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow2 N/A

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

       9. times-frac N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 31.7

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

   12. Applied rewrites 31.7%

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

   if -1.5499999999999999e-162 < x < 1.1e-16

   1. Initial program 4.7%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 4.7

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

   4. Applied rewrites 4.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 4.7

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

   7. Applied rewrites 4.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. difference-of-squares-rev N/A

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

      13. +-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. lower-fma.f64 4.7

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

   9. Applied rewrites 4.7%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 12.7

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

   12. Applied rewrites 12.7%

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

   if 1.1e-16 < x

   1. Initial program 6.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 89.2%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 89.2

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

   10. Applied rewrites 89.2%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

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

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

       5. metadata-eval N/A

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

       6. lower--.f64 89.3

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

   13. Applied rewrites 89.3%

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

Alternative 6: 24.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 23.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 23.2%

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

8. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 23.3

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

10. Applied rewrites 23.3%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

    3. metadata-eval N/A

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

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

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

    5. metadata-eval N/A

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

    6. lower--.f64 24.6

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

13. Applied rewrites 24.6%

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

Alternative 7: 23.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 23.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 23.2%

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

8. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 23.3

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

10. Applied rewrites 23.3%

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

11. Taylor expanded in x around 0

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

    1. rec-exp 23.3

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

13. Applied rewrites 23.3%

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

Reproduce

herbie shell --seed 2025093 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))