expfmod (used to be hard to sample)

Percentage Accurate: 6.7% → 67.8%

Time: 11.7s

Alternatives: 6

Speedup: N/A×

• could not determine a ground truth

  1. x = -7.891928806195459e+179

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: 67.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ t_1 := {\left({\cos x}^{-0.25}\right)}^{-1}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(e^{x}\right) \bmod \left(t\_1 \cdot t\_1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -1\right), x, \left(\left(\left(\left({x}^{-1} + 0.5\right) + {x}^{-2}\right) \cdot \left(x \cdot x\right)\right) \bmod t\_0\right)\right)\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(\left({x}^{-1} + 1\right) \cdot x\right) \bmod t\_0\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod t\_0\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))) (t_1 (pow (pow (cos x) -0.25) -1.0)))
   (if (<= x -1e-90)
     (fma
      (* (fmod (exp x) (* t_1 t_1)) (fma 0.5 x -1.0))
      x
      (fmod (* (+ (+ (pow x -1.0) 0.5) (pow x -2.0)) (* x x)) t_0))
     (if (<= x -6e-309)
       (* (fmod (* (+ (pow x -1.0) 1.0) x) t_0) 1.0)
       (* (fmod x t_0) (exp (- x)))))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double t_1 = pow(pow(cos(x), -0.25), -1.0);
	double tmp;
	if (x <= -1e-90) {
		tmp = fma((fmod(exp(x), (t_1 * t_1)) * fma(0.5, x, -1.0)), x, fmod((((pow(x, -1.0) + 0.5) + pow(x, -2.0)) * (x * x)), t_0));
	} else if (x <= -6e-309) {
		tmp = fmod(((pow(x, -1.0) + 1.0) * x), t_0) * 1.0;
	} else {
		tmp = fmod(x, t_0) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	t_1 = (cos(x) ^ -0.25) ^ -1.0
	tmp = 0.0
	if (x <= -1e-90)
		tmp = fma(Float64(rem(exp(x), Float64(t_1 * t_1)) * fma(0.5, x, -1.0)), x, rem(Float64(Float64(Float64((x ^ -1.0) + 0.5) + (x ^ -2.0)) * Float64(x * x)), t_0));
	elseif (x <= -6e-309)
		tmp = Float64(rem(Float64(Float64((x ^ -1.0) + 1.0) * x), t_0) * 1.0);
	else
		tmp = Float64(rem(x, t_0) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[N[Cos[x], $MachinePrecision], -0.25], $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[x, -1e-90], N[(N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision] * x + N[With[{TMP1 = N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] + 0.5), $MachinePrecision] + N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6e-309], N[(N[With[{TMP1 = N[(N[(N[Power[x, -1.0], $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999995e-91

   1. Initial program 20.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 17.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. sqr-pow N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-cos.f64 17.8

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

   7. Applied rewrites 17.8%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 19.0

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

   10. Applied rewrites 19.0%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-+r+ N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. inv-pow N/A

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

       8. lower-pow.f64 N/A

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

       9. pow-flip N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-pow.f64 N/A

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

       13. metadata-eval N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 32.9

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

   13. Applied rewrites 32.9%

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

   if -9.99999999999999995e-91 < x < -6.000000000000001e-309

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 2.5%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. inv-pow N/A

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

       6. lower-pow.f64 16.7

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

   14. Applied rewrites 16.7%

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

   if -6.000000000000001e-309 < x

   1. Initial program 5.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.7%

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

Alternative 2: 65.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ t_1 := {\left({\cos x}^{-0.25}\right)}^{-1}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(e^{x}\right) \bmod \left(t\_1 \cdot t\_1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -1\right), x, \left(\left(\left(\left({x}^{-1} + 0.5\right) + {x}^{-2}\right) \cdot \left(x \cdot x\right)\right) \bmod t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod t\_0\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))) (t_1 (pow (pow (cos x) -0.25) -1.0)))
   (if (<= x -7.5e-155)
     (fma
      (* (fmod (exp x) (* t_1 t_1)) (fma 0.5 x -1.0))
      x
      (fmod (* (+ (+ (pow x -1.0) 0.5) (pow x -2.0)) (* x x)) t_0))
     (* (fmod x t_0) (exp (- x))))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double t_1 = pow(pow(cos(x), -0.25), -1.0);
	double tmp;
	if (x <= -7.5e-155) {
		tmp = fma((fmod(exp(x), (t_1 * t_1)) * fma(0.5, x, -1.0)), x, fmod((((pow(x, -1.0) + 0.5) + pow(x, -2.0)) * (x * x)), t_0));
	} else {
		tmp = fmod(x, t_0) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	t_1 = (cos(x) ^ -0.25) ^ -1.0
	tmp = 0.0
	if (x <= -7.5e-155)
		tmp = fma(Float64(rem(exp(x), Float64(t_1 * t_1)) * fma(0.5, x, -1.0)), x, rem(Float64(Float64(Float64((x ^ -1.0) + 0.5) + (x ^ -2.0)) * Float64(x * x)), t_0));
	else
		tmp = Float64(rem(x, t_0) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[N[Cos[x], $MachinePrecision], -0.25], $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[x, -7.5e-155], N[(N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision] * x + N[With[{TMP1 = N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] + 0.5), $MachinePrecision] + N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000006e-155

   1. Initial program 13.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 12.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. sqr-pow N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-cos.f64 12.3

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

   7. Applied rewrites 12.3%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 13.1

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

   10. Applied rewrites 13.1%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-+r+ N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. inv-pow N/A

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

       8. lower-pow.f64 N/A

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

       9. pow-flip N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-pow.f64 N/A

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

       13. metadata-eval N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 23.8

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

   13. Applied rewrites 23.8%

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

   if -7.5000000000000006e-155 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 74.9%

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

Alternative 3: 46.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ t_1 := {\left({\cos x}^{-0.25}\right)}^{-1}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(e^{x}\right) \bmod \left(t\_1 \cdot t\_1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -1\right), x, \left(\left(\left(\left({x}^{-1} + 0.5\right) + {x}^{-2}\right) \cdot \left(x \cdot x\right)\right) \bmod t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod t\_0\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))) (t_1 (pow (pow (cos x) -0.25) -1.0)))
   (if (<= x -7.5e-155)
     (fma
      (* (fmod (exp x) (* t_1 t_1)) (fma 0.5 x -1.0))
      x
      (fmod (* (+ (+ (pow x -1.0) 0.5) (pow x -2.0)) (* x x)) t_0))
     (* (fmod x t_0) 1.0))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double t_1 = pow(pow(cos(x), -0.25), -1.0);
	double tmp;
	if (x <= -7.5e-155) {
		tmp = fma((fmod(exp(x), (t_1 * t_1)) * fma(0.5, x, -1.0)), x, fmod((((pow(x, -1.0) + 0.5) + pow(x, -2.0)) * (x * x)), t_0));
	} else {
		tmp = fmod(x, t_0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	t_1 = (cos(x) ^ -0.25) ^ -1.0
	tmp = 0.0
	if (x <= -7.5e-155)
		tmp = fma(Float64(rem(exp(x), Float64(t_1 * t_1)) * fma(0.5, x, -1.0)), x, rem(Float64(Float64(Float64((x ^ -1.0) + 0.5) + (x ^ -2.0)) * Float64(x * x)), t_0));
	else
		tmp = Float64(rem(x, t_0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[N[Cos[x], $MachinePrecision], -0.25], $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[x, -7.5e-155], N[(N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision] * x + N[With[{TMP1 = N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] + 0.5), $MachinePrecision] + N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000006e-155

   1. Initial program 13.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 12.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. sqr-pow N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-cos.f64 12.3

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

   7. Applied rewrites 12.3%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 13.1

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

   10. Applied rewrites 13.1%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-+r+ N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. inv-pow N/A

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

       8. lower-pow.f64 N/A

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

       9. pow-flip N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-pow.f64 N/A

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

       13. metadata-eval N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 23.8

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

   13. Applied rewrites 23.8%

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

   if -7.5000000000000006e-155 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 74.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 50.0%

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

Alternative 4: 27.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{-1} + 0.5\\ t_1 := \sqrt{\cos x}\\ t_2 := {\left({\cos x}^{-0.25}\right)}^{-1}\\ t_3 := \left(\left(e^{x}\right) \bmod \left(t\_2 \cdot t\_2\right)\right) \cdot \mathsf{fma}\left(0.5, x, -1\right)\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, x, \left(\left(\left(t\_0 + {x}^{-2}\right) \cdot \left(x \cdot x\right)\right) \bmod t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, x, \left(\left(t\_0 \cdot \left(x \cdot x\right)\right) \bmod t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (pow x -1.0) 0.5))
        (t_1 (sqrt (cos x)))
        (t_2 (pow (pow (cos x) -0.25) -1.0))
        (t_3 (* (fmod (exp x) (* t_2 t_2)) (fma 0.5 x -1.0))))
   (if (<= x -7.5e-155)
     (fma t_3 x (fmod (* (+ t_0 (pow x -2.0)) (* x x)) t_1))
     (fma t_3 x (fmod (* t_0 (* x x)) t_1)))))

C

double code(double x) {
	double t_0 = pow(x, -1.0) + 0.5;
	double t_1 = sqrt(cos(x));
	double t_2 = pow(pow(cos(x), -0.25), -1.0);
	double t_3 = fmod(exp(x), (t_2 * t_2)) * fma(0.5, x, -1.0);
	double tmp;
	if (x <= -7.5e-155) {
		tmp = fma(t_3, x, fmod(((t_0 + pow(x, -2.0)) * (x * x)), t_1));
	} else {
		tmp = fma(t_3, x, fmod((t_0 * (x * x)), t_1));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64((x ^ -1.0) + 0.5)
	t_1 = sqrt(cos(x))
	t_2 = (cos(x) ^ -0.25) ^ -1.0
	t_3 = Float64(rem(exp(x), Float64(t_2 * t_2)) * fma(0.5, x, -1.0))
	tmp = 0.0
	if (x <= -7.5e-155)
		tmp = fma(t_3, x, rem(Float64(Float64(t_0 + (x ^ -2.0)) * Float64(x * x)), t_1));
	else
		tmp = fma(t_3, x, rem(Float64(t_0 * Float64(x * x)), t_1));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, -1.0], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[N[Cos[x], $MachinePrecision], -0.25], $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.5e-155], N[(t$95$3 * x + N[With[{TMP1 = N[(N[(t$95$0 + N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 * x + N[With[{TMP1 = N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000006e-155

   1. Initial program 13.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 12.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. sqr-pow N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-cos.f64 12.3

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

   7. Applied rewrites 12.3%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 13.1

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

   10. Applied rewrites 13.1%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-+r+ N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. inv-pow N/A

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

       8. lower-pow.f64 N/A

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

       9. pow-flip N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-pow.f64 N/A

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

       13. metadata-eval N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 23.8

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

   13. Applied rewrites 23.8%

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

   if -7.5000000000000006e-155 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 4.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. sqr-pow N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-cos.f64 4.5

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

   7. Applied rewrites 4.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 4.4

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

   10. Applied rewrites 4.4%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. inv-pow N/A

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

       6. lower-pow.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 30.8

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

   13. Applied rewrites 30.8%

       \[\leadsto \mathsf{fma}\left(\left(\left(e^{x}\right) \bmod \left({\left({\cos x}^{-0.25}\right)}^{-1} \cdot {\left({\cos x}^{-0.25}\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(0.5, x, -1\right), x, \left(\left(\left({x}^{-1} + 0.5\right) \cdot \left(x \cdot x\right)\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 9.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ t_1 := {\left({\cos x}^{-0.25}\right)}^{-1}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(e^{x}\right) \bmod \left(t\_1 \cdot t\_1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -1\right), x, \left(\left(\left(\left({x}^{-1} + 0.5\right) + {x}^{-2}\right) \cdot \left(x \cdot x\right)\right) \bmod t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left(t\_1 \cdot {\left(\mathsf{fma}\left(0.125, x \cdot x, 1\right)\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(0.5, x, -1\right), x, \left(\left(e^{x}\right) \bmod t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))) (t_1 (pow (pow (cos x) -0.25) -1.0)))
   (if (<= x -5e-144)
     (fma
      (* (fmod (exp x) (* t_1 t_1)) (fma 0.5 x -1.0))
      x
      (fmod (* (+ (+ (pow x -1.0) 0.5) (pow x -2.0)) (* x x)) t_0))
     (fma
      (*
       (fmod
        (fma (fma 0.5 x 1.0) x 1.0)
        (* t_1 (pow (fma 0.125 (* x x) 1.0) -1.0)))
       (fma 0.5 x -1.0))
      x
      (fmod (exp x) t_0)))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double t_1 = pow(pow(cos(x), -0.25), -1.0);
	double tmp;
	if (x <= -5e-144) {
		tmp = fma((fmod(exp(x), (t_1 * t_1)) * fma(0.5, x, -1.0)), x, fmod((((pow(x, -1.0) + 0.5) + pow(x, -2.0)) * (x * x)), t_0));
	} else {
		tmp = fma((fmod(fma(fma(0.5, x, 1.0), x, 1.0), (t_1 * pow(fma(0.125, (x * x), 1.0), -1.0))) * fma(0.5, x, -1.0)), x, fmod(exp(x), t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	t_1 = (cos(x) ^ -0.25) ^ -1.0
	tmp = 0.0
	if (x <= -5e-144)
		tmp = fma(Float64(rem(exp(x), Float64(t_1 * t_1)) * fma(0.5, x, -1.0)), x, rem(Float64(Float64(Float64((x ^ -1.0) + 0.5) + (x ^ -2.0)) * Float64(x * x)), t_0));
	else
		tmp = fma(Float64(rem(fma(fma(0.5, x, 1.0), x, 1.0), Float64(t_1 * (fma(0.125, Float64(x * x), 1.0) ^ -1.0))) * fma(0.5, x, -1.0)), x, rem(exp(x), t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[N[Cos[x], $MachinePrecision], -0.25], $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[x, -5e-144], N[(N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision] * x + N[With[{TMP1 = N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] + 0.5), $MachinePrecision] + N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = N[(t$95$1 * N[Power[N[(0.125 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision] * x + N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-144

   1. Initial program 14.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 12.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. sqr-pow N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-cos.f64 12.7

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

   7. Applied rewrites 12.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 13.5

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

   10. Applied rewrites 13.5%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-+r+ N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. inv-pow N/A

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

       8. lower-pow.f64 N/A

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

       9. pow-flip N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-pow.f64 N/A

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

       13. metadata-eval N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 24.6

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

   13. Applied rewrites 24.6%

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

   if -4.9999999999999998e-144 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 4.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. sqr-pow N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-cos.f64 4.5

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

   7. Applied rewrites 4.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 4.5

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

   10. Applied rewrites 4.5%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{\frac{-1}{4}}\right)}^{-1} \cdot {\left(1 + \frac{1}{8} \cdot {x}^{2}\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{\frac{-1}{4}}\right)}^{-1} \cdot {\left(\frac{1}{8} \cdot {x}^{2} + 1\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{\frac{-1}{4}}\right)}^{-1} \cdot {\left(\mathsf{fma}\left(\frac{1}{8}, {x}^{2}, 1\right)\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{\frac{-1}{4}}\right)}^{-1} \cdot {\left(\mathsf{fma}\left(\frac{1}{8}, x \cdot x, 1\right)\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]

       4. lower-*.f64 4.5

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{-0.25}\right)}^{-1} \cdot {\left(\mathsf{fma}\left(0.125, x \cdot x, 1\right)\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(0.5, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]

   13. Applied rewrites 4.5%

       \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{-0.25}\right)}^{-1} \cdot {\left(\mathsf{fma}\left(0.125, x \cdot x, 1\right)\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(0.5, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 6.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{-0.25}\right)}^{-1} \cdot {\left(\mathsf{fma}\left(0.125, x \cdot x, 1\right)\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(0.5, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (*
   (fmod
    (fma (fma 0.5 x 1.0) x 1.0)
    (* (pow (pow (cos x) -0.25) -1.0) (pow (fma 0.125 (* x x) 1.0) -1.0)))
   (fma 0.5 x -1.0))
  x
  (fmod (exp x) (sqrt (cos x)))))

C

double code(double x) {
	return fma((fmod(fma(fma(0.5, x, 1.0), x, 1.0), (pow(pow(cos(x), -0.25), -1.0) * pow(fma(0.125, (x * x), 1.0), -1.0))) * fma(0.5, x, -1.0)), x, fmod(exp(x), sqrt(cos(x))));
}

Julia

function code(x)
	return fma(Float64(rem(fma(fma(0.5, x, 1.0), x, 1.0), Float64(((cos(x) ^ -0.25) ^ -1.0) * (fma(0.125, Float64(x * x), 1.0) ^ -1.0))) * fma(0.5, x, -1.0)), x, rem(exp(x), sqrt(cos(x))))
end

Wolfram

code[x_] := N[(N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = N[(N[Power[N[Power[N[Cos[x], $MachinePrecision], -0.25], $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(0.125 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision] * x + N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

5. Applied rewrites 6.1%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

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

   4. metadata-eval N/A

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

   5. pow-pow N/A

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

   6. sqr-pow N/A

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

   7. unpow-prod-down N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-cos.f64 6.1

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

7. Applied rewrites 6.1%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 6.2

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

10. Applied rewrites 6.2%

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

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{\frac{-1}{4}}\right)}^{-1} \cdot {\left(1 + \frac{1}{8} \cdot {x}^{2}\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{\frac{-1}{4}}\right)}^{-1} \cdot {\left(\frac{1}{8} \cdot {x}^{2} + 1\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]

    2. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{\frac{-1}{4}}\right)}^{-1} \cdot {\left(\mathsf{fma}\left(\frac{1}{8}, {x}^{2}, 1\right)\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{\frac{-1}{4}}\right)}^{-1} \cdot {\left(\mathsf{fma}\left(\frac{1}{8}, x \cdot x, 1\right)\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]

    4. lower-*.f64 6.2

       \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{-0.25}\right)}^{-1} \cdot {\left(\mathsf{fma}\left(0.125, x \cdot x, 1\right)\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(0.5, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]

13. Applied rewrites 6.2%

    \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left({\left({\cos x}^{-0.25}\right)}^{-1} \cdot {\left(\mathsf{fma}\left(0.125, x \cdot x, 1\right)\right)}^{-1}\right)\right) \cdot \mathsf{fma}\left(0.5, x, -1\right), x, \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))