expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 64.6%

Time: 13.2s

Alternatives: 8

Speedup: 1.9×

• could not determine a ground truth

  1. x = -1.0690968743855742e+279

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: 6.9% 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: 64.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left({\left(-x\right)}^{3} \cdot \left(\frac{\frac{\frac{-1}{x} - 1}{x} - 0.5}{x} - 0.16666666666666666\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{-2} - {\left(e^{x}\right)}^{2}}{4 \cdot \sinh \left(-x\right)}, 1, \sinh x\right)\right) \bmod t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod t\_1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (fma (* x x) -0.25 1.0)))
   (if (<= x -1.8e-103)
     (*
      (fmod
       (*
        (pow (- x) 3.0)
        (- (/ (- (/ (- (/ -1.0 x) 1.0) x) 0.5) x) 0.16666666666666666))
       (fma (- (* -0.010416666666666666 (* x x)) 0.25) (* x x) 1.0))
      t_0)
     (if (<= x 1.12e-16)
       (fmod
        (fma
         (/ (- (pow (exp x) -2.0) (pow (exp x) 2.0)) (* 4.0 (sinh (- x))))
         1.0
         (sinh x))
        t_1)
       (* (fmod (+ 1.0 x) t_1) t_0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fma((x * x), -0.25, 1.0);
	double tmp;
	if (x <= -1.8e-103) {
		tmp = fmod((pow(-x, 3.0) * ((((((-1.0 / x) - 1.0) / x) - 0.5) / x) - 0.16666666666666666)), fma(((-0.010416666666666666 * (x * x)) - 0.25), (x * x), 1.0)) * t_0;
	} else if (x <= 1.12e-16) {
		tmp = fmod(fma(((pow(exp(x), -2.0) - pow(exp(x), 2.0)) / (4.0 * sinh(-x))), 1.0, sinh(x)), t_1);
	} else {
		tmp = fmod((1.0 + x), t_1) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = fma(Float64(x * x), -0.25, 1.0)
	tmp = 0.0
	if (x <= -1.8e-103)
		tmp = Float64(rem(Float64((Float64(-x) ^ 3.0) * Float64(Float64(Float64(Float64(Float64(Float64(-1.0 / x) - 1.0) / x) - 0.5) / x) - 0.16666666666666666)), fma(Float64(Float64(-0.010416666666666666 * Float64(x * x)) - 0.25), Float64(x * x), 1.0)) * t_0);
	elseif (x <= 1.12e-16)
		tmp = rem(fma(Float64(Float64((exp(x) ^ -2.0) - (exp(x) ^ 2.0)) / Float64(4.0 * sinh(Float64(-x)))), 1.0, sinh(x)), t_1);
	else
		tmp = Float64(rem(Float64(1.0 + x), t_1) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.8e-103], N[(N[With[{TMP1 = N[(N[Power[(-x), 3.0], $MachinePrecision] * N[(N[(N[(N[(N[(N[(-1.0 / x), $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision], TMP2 = N[(N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 1.12e-16], N[With[{TMP1 = N[(N[(N[(N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision] - N[Power[N[Exp[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[Sinh[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + N[Sinh[x], $MachinePrecision]), $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[With[{TMP1 = N[(1.0 + x), $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999999e-103

   1. Initial program 20.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\color{blue}{\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2}} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, 1\right)\right)}\right) \cdot e^{-x} \]

      4. lower--.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2}} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      6. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      8. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

      9. lower-*.f64 4.0

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 4.0%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x} + 1\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       4. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       5. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       7. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       8. lower-fma.f64 16.3

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   11. Applied rewrites 16.3%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around -inf

       \[\leadsto \left(\left(-1 \cdot \color{blue}{\left({x}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   14. Applied rewrites 34.4%

       \[\leadsto \left(\left({\left(-x\right)}^{3} \cdot \color{blue}{\left(\frac{-\left(\frac{\frac{1}{x} + 1}{x} + 0.5\right)}{x} - 0.16666666666666666\right)}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if -1.7999999999999999e-103 < x < 1.12e-16

   1. Initial program 4.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 4.3%

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

   10. Applied rewrites 36.7%

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

   12. Applied rewrites 61.5%

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

   if 1.12e-16 < x

   1. Initial program 5.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 91.2

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

   8. Applied rewrites 91.2%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 94.2

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

   11. Applied rewrites 94.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left({\left(-x\right)}^{3} \cdot \left(\frac{\frac{\frac{-1}{x} - 1}{x} - 0.5}{x} - 0.16666666666666666\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{-2} - {\left(e^{x}\right)}^{2}}{4 \cdot \sinh \left(-x\right)}, 1, \sinh x\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 46.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left({\left(-x\right)}^{3} \cdot \left(\frac{\frac{\frac{-1}{x} - 1}{x} - 0.5}{x} - 0.16666666666666666\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot t\_1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left({\left(e^{x}\right)}^{-2} - {\left(e^{x}\right)}^{2}, 2, \left(\left(-2.6666666666666665 \cdot \left(x \cdot x\right) - 8\right) \cdot x\right) \cdot x\right)}{\left(\left(2 \cdot \sinh \left(-x\right)\right) \cdot 2\right) \cdot 2}\right) \bmod t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod t\_0\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)) (t_1 (exp (- x))))
   (if (<= x -1.8e-103)
     (*
      (fmod
       (*
        (pow (- x) 3.0)
        (- (/ (- (/ (- (/ -1.0 x) 1.0) x) 0.5) x) 0.16666666666666666))
       (fma (- (* -0.010416666666666666 (* x x)) 0.25) (* x x) 1.0))
      t_1)
     (if (<= x 1.12e-16)
       (fmod
        (/
         (fma
          (- (pow (exp x) -2.0) (pow (exp x) 2.0))
          2.0
          (* (* (- (* -2.6666666666666665 (* x x)) 8.0) x) x))
         (* (* (* 2.0 (sinh (- x))) 2.0) 2.0))
        t_0)
       (* (fmod (+ 1.0 x) t_0) t_1)))))

C

double code(double x) {
	double t_0 = fma((x * x), -0.25, 1.0);
	double t_1 = exp(-x);
	double tmp;
	if (x <= -1.8e-103) {
		tmp = fmod((pow(-x, 3.0) * ((((((-1.0 / x) - 1.0) / x) - 0.5) / x) - 0.16666666666666666)), fma(((-0.010416666666666666 * (x * x)) - 0.25), (x * x), 1.0)) * t_1;
	} else if (x <= 1.12e-16) {
		tmp = fmod((fma((pow(exp(x), -2.0) - pow(exp(x), 2.0)), 2.0, ((((-2.6666666666666665 * (x * x)) - 8.0) * x) * x)) / (((2.0 * sinh(-x)) * 2.0) * 2.0)), t_0);
	} else {
		tmp = fmod((1.0 + x), t_0) * t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), -0.25, 1.0)
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1.8e-103)
		tmp = Float64(rem(Float64((Float64(-x) ^ 3.0) * Float64(Float64(Float64(Float64(Float64(Float64(-1.0 / x) - 1.0) / x) - 0.5) / x) - 0.16666666666666666)), fma(Float64(Float64(-0.010416666666666666 * Float64(x * x)) - 0.25), Float64(x * x), 1.0)) * t_1);
	elseif (x <= 1.12e-16)
		tmp = rem(Float64(fma(Float64((exp(x) ^ -2.0) - (exp(x) ^ 2.0)), 2.0, Float64(Float64(Float64(Float64(-2.6666666666666665 * Float64(x * x)) - 8.0) * x) * x)) / Float64(Float64(Float64(2.0 * sinh(Float64(-x))) * 2.0) * 2.0)), t_0);
	else
		tmp = Float64(rem(Float64(1.0 + x), t_0) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1.8e-103], N[(N[With[{TMP1 = N[(N[Power[(-x), 3.0], $MachinePrecision] * N[(N[(N[(N[(N[(N[(-1.0 / x), $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision], TMP2 = N[(N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x, 1.12e-16], N[With[{TMP1 = N[(N[(N[(N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision] - N[Power[N[Exp[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(-2.6666666666666665 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * N[Sinh[(-x)], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[With[{TMP1 = N[(1.0 + x), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999999e-103

   1. Initial program 20.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\color{blue}{\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2}} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, 1\right)\right)}\right) \cdot e^{-x} \]

      4. lower--.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2}} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      6. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      8. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

      9. lower-*.f64 4.0

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 4.0%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x} + 1\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       4. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       5. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       7. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       8. lower-fma.f64 16.3

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   11. Applied rewrites 16.3%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around -inf

       \[\leadsto \left(\left(-1 \cdot \color{blue}{\left({x}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   14. Applied rewrites 34.4%

       \[\leadsto \left(\left({\left(-x\right)}^{3} \cdot \color{blue}{\left(\frac{-\left(\frac{\frac{1}{x} + 1}{x} + 0.5\right)}{x} - 0.16666666666666666\right)}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if -1.7999999999999999e-103 < x < 1.12e-16

   1. Initial program 4.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 4.3%

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

   10. Applied rewrites 36.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(\left(\frac{\mathsf{fma}\left({\left(e^{x}\right)}^{-2} - {\left(e^{x}\right)}^{2}, 2, {x}^{2} \cdot \left(\frac{-8}{3} \cdot {x}^{2} - 8\right)\right)}{\left(\left(2 \cdot \sinh \left(-x\right)\right) \cdot 2\right) \cdot 2}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 36.7%

       \[\leadsto \left(\left(\frac{\mathsf{fma}\left({\left(e^{x}\right)}^{-2} - {\left(e^{x}\right)}^{2}, 2, \left(\left(-2.6666666666666665 \cdot \left(x \cdot x\right) - 8\right) \cdot x\right) \cdot x\right)}{\left(\left(2 \cdot \sinh \left(-x\right)\right) \cdot 2\right) \cdot 2}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \]

   if 1.12e-16 < x

   1. Initial program 5.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 91.2

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

   8. Applied rewrites 91.2%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 94.2

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

   11. Applied rewrites 94.2%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left({\left(-x\right)}^{3} \cdot \left(\frac{\frac{\frac{-1}{x} - 1}{x} - 0.5}{x} - 0.16666666666666666\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left({\left(e^{x}\right)}^{-2} - {\left(e^{x}\right)}^{2}, 2, \left(\left(-2.6666666666666665 \cdot \left(x \cdot x\right) - 8\right) \cdot x\right) \cdot x\right)}{\left(\left(2 \cdot \sinh \left(-x\right)\right) \cdot 2\right) \cdot 2}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left({\left(-x\right)}^{3} \cdot \left(\frac{\frac{\frac{-1}{x} - 1}{x} - 0.5}{x} - 0.16666666666666666\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left({\left(e^{x}\right)}^{-2} - {\left(e^{x}\right)}^{2}, 2, -8 \cdot \left(x \cdot x\right)\right)}{\left(\left(2 \cdot \sinh \left(-x\right)\right) \cdot 2\right) \cdot 2}\right) \bmod t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod t\_1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (fma (* x x) -0.25 1.0)))
   (if (<= x -1.8e-103)
     (*
      (fmod
       (*
        (pow (- x) 3.0)
        (- (/ (- (/ (- (/ -1.0 x) 1.0) x) 0.5) x) 0.16666666666666666))
       (fma (- (* -0.010416666666666666 (* x x)) 0.25) (* x x) 1.0))
      t_0)
     (if (<= x 1.12e-16)
       (fmod
        (/
         (fma (- (pow (exp x) -2.0) (pow (exp x) 2.0)) 2.0 (* -8.0 (* x x)))
         (* (* (* 2.0 (sinh (- x))) 2.0) 2.0))
        t_1)
       (* (fmod (+ 1.0 x) t_1) t_0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fma((x * x), -0.25, 1.0);
	double tmp;
	if (x <= -1.8e-103) {
		tmp = fmod((pow(-x, 3.0) * ((((((-1.0 / x) - 1.0) / x) - 0.5) / x) - 0.16666666666666666)), fma(((-0.010416666666666666 * (x * x)) - 0.25), (x * x), 1.0)) * t_0;
	} else if (x <= 1.12e-16) {
		tmp = fmod((fma((pow(exp(x), -2.0) - pow(exp(x), 2.0)), 2.0, (-8.0 * (x * x))) / (((2.0 * sinh(-x)) * 2.0) * 2.0)), t_1);
	} else {
		tmp = fmod((1.0 + x), t_1) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = fma(Float64(x * x), -0.25, 1.0)
	tmp = 0.0
	if (x <= -1.8e-103)
		tmp = Float64(rem(Float64((Float64(-x) ^ 3.0) * Float64(Float64(Float64(Float64(Float64(Float64(-1.0 / x) - 1.0) / x) - 0.5) / x) - 0.16666666666666666)), fma(Float64(Float64(-0.010416666666666666 * Float64(x * x)) - 0.25), Float64(x * x), 1.0)) * t_0);
	elseif (x <= 1.12e-16)
		tmp = rem(Float64(fma(Float64((exp(x) ^ -2.0) - (exp(x) ^ 2.0)), 2.0, Float64(-8.0 * Float64(x * x))) / Float64(Float64(Float64(2.0 * sinh(Float64(-x))) * 2.0) * 2.0)), t_1);
	else
		tmp = Float64(rem(Float64(1.0 + x), t_1) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.8e-103], N[(N[With[{TMP1 = N[(N[Power[(-x), 3.0], $MachinePrecision] * N[(N[(N[(N[(N[(N[(-1.0 / x), $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision], TMP2 = N[(N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 1.12e-16], N[With[{TMP1 = N[(N[(N[(N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision] - N[Power[N[Exp[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(-8.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * N[Sinh[(-x)], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[With[{TMP1 = N[(1.0 + x), $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999999e-103

   1. Initial program 20.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\color{blue}{\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2}} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, 1\right)\right)}\right) \cdot e^{-x} \]

      4. lower--.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2}} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      6. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      8. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

      9. lower-*.f64 4.0

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 4.0%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x} + 1\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       4. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       5. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       7. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       8. lower-fma.f64 16.3

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   11. Applied rewrites 16.3%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around -inf

       \[\leadsto \left(\left(-1 \cdot \color{blue}{\left({x}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   14. Applied rewrites 34.4%

       \[\leadsto \left(\left({\left(-x\right)}^{3} \cdot \color{blue}{\left(\frac{-\left(\frac{\frac{1}{x} + 1}{x} + 0.5\right)}{x} - 0.16666666666666666\right)}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if -1.7999999999999999e-103 < x < 1.12e-16

   1. Initial program 4.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 4.3%

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

   10. Applied rewrites 36.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 36.6%

       \[\leadsto \left(\left(\frac{\mathsf{fma}\left({\left(e^{x}\right)}^{-2} - {\left(e^{x}\right)}^{2}, 2, -8 \cdot \left(x \cdot x\right)\right)}{\left(\left(2 \cdot \sinh \left(-x\right)\right) \cdot 2\right) \cdot 2}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \]

   if 1.12e-16 < x

   1. Initial program 5.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 91.2

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

   8. Applied rewrites 91.2%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 94.2

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

   11. Applied rewrites 94.2%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left({\left(-x\right)}^{3} \cdot \left(\frac{\frac{\frac{-1}{x} - 1}{x} - 0.5}{x} - 0.16666666666666666\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left({\left(e^{x}\right)}^{-2} - {\left(e^{x}\right)}^{2}, 2, -8 \cdot \left(x \cdot x\right)\right)}{\left(\left(2 \cdot \sinh \left(-x\right)\right) \cdot 2\right) \cdot 2}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 40.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-103}:\\ \;\;\;\;\left(\left({\left(-x\right)}^{3} \cdot \left(\frac{\frac{\frac{-1}{x} - 1}{x} - 0.5}{x} - 0.16666666666666666\right)\right) \bmod t\_0\right) \cdot t\_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(\left(\frac{0.5}{x} + 0.16666666666666666\right) \cdot {x}^{3}\right) \bmod t\_0\right) \cdot t\_1\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;\left(\left(\left(\frac{{x}^{-1} + 0.5}{x} + 0.16666666666666666\right) \cdot {x}^{3}\right) \bmod t\_0\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (- (* -0.010416666666666666 (* x x)) 0.25) (* x x) 1.0))
        (t_1 (exp (- x))))
   (if (<= x -1.85e-103)
     (*
      (fmod
       (*
        (pow (- x) 3.0)
        (- (/ (- (/ (- (/ -1.0 x) 1.0) x) 0.5) x) 0.16666666666666666))
       t_0)
      t_1)
     (if (<= x 2e-154)
       (* (fmod (* (+ (/ 0.5 x) 0.16666666666666666) (pow x 3.0)) t_0) t_1)
       (if (<= x 0.6)
         (*
          (fmod
           (* (+ (/ (+ (pow x -1.0) 0.5) x) 0.16666666666666666) (pow x 3.0))
           t_0)
          t_1)
         (* (fmod 1.0 (sqrt (cos x))) t_1))))))

C

double code(double x) {
	double t_0 = fma(((-0.010416666666666666 * (x * x)) - 0.25), (x * x), 1.0);
	double t_1 = exp(-x);
	double tmp;
	if (x <= -1.85e-103) {
		tmp = fmod((pow(-x, 3.0) * ((((((-1.0 / x) - 1.0) / x) - 0.5) / x) - 0.16666666666666666)), t_0) * t_1;
	} else if (x <= 2e-154) {
		tmp = fmod((((0.5 / x) + 0.16666666666666666) * pow(x, 3.0)), t_0) * t_1;
	} else if (x <= 0.6) {
		tmp = fmod(((((pow(x, -1.0) + 0.5) / x) + 0.16666666666666666) * pow(x, 3.0)), t_0) * t_1;
	} else {
		tmp = fmod(1.0, sqrt(cos(x))) * t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(Float64(-0.010416666666666666 * Float64(x * x)) - 0.25), Float64(x * x), 1.0)
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1.85e-103)
		tmp = Float64(rem(Float64((Float64(-x) ^ 3.0) * Float64(Float64(Float64(Float64(Float64(Float64(-1.0 / x) - 1.0) / x) - 0.5) / x) - 0.16666666666666666)), t_0) * t_1);
	elseif (x <= 2e-154)
		tmp = Float64(rem(Float64(Float64(Float64(0.5 / x) + 0.16666666666666666) * (x ^ 3.0)), t_0) * t_1);
	elseif (x <= 0.6)
		tmp = Float64(rem(Float64(Float64(Float64(Float64((x ^ -1.0) + 0.5) / x) + 0.16666666666666666) * (x ^ 3.0)), t_0) * t_1);
	else
		tmp = Float64(rem(1.0, sqrt(cos(x))) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1.85e-103], N[(N[With[{TMP1 = N[(N[Power[(-x), 3.0], $MachinePrecision] * N[(N[(N[(N[(N[(N[(-1.0 / x), $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x, 2e-154], N[(N[With[{TMP1 = N[(N[(N[(0.5 / x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x, 0.6], N[(N[With[{TMP1 = N[(N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.85e-103

   1. Initial program 20.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\color{blue}{\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2}} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, 1\right)\right)}\right) \cdot e^{-x} \]

      4. lower--.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2}} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      6. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      8. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

      9. lower-*.f64 4.0

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 4.0%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x} + 1\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       4. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       5. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       7. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       8. lower-fma.f64 16.3

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   11. Applied rewrites 16.3%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around -inf

       \[\leadsto \left(\left(-1 \cdot \color{blue}{\left({x}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   14. Applied rewrites 34.4%

       \[\leadsto \left(\left({\left(-x\right)}^{3} \cdot \color{blue}{\left(\frac{-\left(\frac{\frac{1}{x} + 1}{x} + 0.5\right)}{x} - 0.16666666666666666\right)}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if -1.85e-103 < x < 1.9999999999999999e-154

   1. Initial program 4.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\color{blue}{\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2}} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, 1\right)\right)}\right) \cdot e^{-x} \]

      4. lower--.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2}} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      6. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      8. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

      9. lower-*.f64 4.5

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 4.5%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x} + 1\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       4. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       5. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       7. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       8. lower-fma.f64 4.5

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   11. Applied rewrites 4.5%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around inf

       \[\leadsto \left(\left({x}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]
   13. Step-by-step derivation

   14. Applied rewrites 4.5%

       \[\leadsto \left(\left(\left(\frac{0.5}{x} + 0.16666666666666666\right) \cdot \color{blue}{{x}^{3}}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 1.9999999999999999e-154 < x < 0.599999999999999978

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\color{blue}{\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2}} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, 1\right)\right)}\right) \cdot e^{-x} \]

      4. lower--.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2}} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      6. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      8. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

      9. lower-*.f64 4.0

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 4.0%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x} + 1\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       4. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       5. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       7. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       8. lower-fma.f64 8.1

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   11. Applied rewrites 8.1%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around inf

       \[\leadsto \left(\left({x}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]
   13. Step-by-step derivation

   14. Applied rewrites 65.8%

       \[\leadsto \left(\left(\left(\frac{\frac{1}{x} + 0.5}{x} + 0.16666666666666666\right) \cdot \color{blue}{{x}^{3}}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 0.599999999999999978 < x

   1. Initial program 1.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.8%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-103}:\\ \;\;\;\;\left(\left({\left(-x\right)}^{3} \cdot \left(\frac{\frac{\frac{-1}{x} - 1}{x} - 0.5}{x} - 0.16666666666666666\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(\left(\frac{0.5}{x} + 0.16666666666666666\right) \cdot {x}^{3}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;\left(\left(\left(\frac{{x}^{-1} + 0.5}{x} + 0.16666666666666666\right) \cdot {x}^{3}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq 10^{-138}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664 - 0.5, x \cdot x, 1\right)}\right)\right)}{e^{x}}\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;\left(\left(\left(\frac{{x}^{-1} + 0.5}{x} + 0.16666666666666666\right) \cdot {x}^{3}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x 1e-138)
     (/
      (fmod
       (exp x)
       (sqrt (fma (- (* (* x x) 0.041666666666666664) 0.5) (* x x) 1.0)))
      (exp x))
     (if (<= x 0.6)
       (*
        (fmod
         (* (+ (/ (+ (pow x -1.0) 0.5) x) 0.16666666666666666) (pow x 3.0))
         (fma (- (* -0.010416666666666666 (* x x)) 0.25) (* x x) 1.0))
        t_0)
       (* (fmod 1.0 (sqrt (cos x))) t_0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= 1e-138) {
		tmp = fmod(exp(x), sqrt(fma((((x * x) * 0.041666666666666664) - 0.5), (x * x), 1.0))) / exp(x);
	} else if (x <= 0.6) {
		tmp = fmod(((((pow(x, -1.0) + 0.5) / x) + 0.16666666666666666) * pow(x, 3.0)), fma(((-0.010416666666666666 * (x * x)) - 0.25), (x * x), 1.0)) * t_0;
	} else {
		tmp = fmod(1.0, sqrt(cos(x))) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= 1e-138)
		tmp = Float64(rem(exp(x), sqrt(fma(Float64(Float64(Float64(x * x) * 0.041666666666666664) - 0.5), Float64(x * x), 1.0))) / exp(x));
	elseif (x <= 0.6)
		tmp = Float64(rem(Float64(Float64(Float64(Float64((x ^ -1.0) + 0.5) / x) + 0.16666666666666666) * (x ^ 3.0)), fma(Float64(Float64(-0.010416666666666666 * Float64(x * x)) - 0.25), Float64(x * x), 1.0)) * t_0);
	else
		tmp = Float64(rem(1.0, sqrt(cos(x))) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, 1e-138], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.6], N[(N[With[{TMP1 = N[(N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], TMP2 = N[(N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.00000000000000007e-138

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 8.1

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

   5. Applied rewrites 8.1%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 8.1%

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

   if 1.00000000000000007e-138 < x < 0.599999999999999978

   1. Initial program 8.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\color{blue}{\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2}} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, 1\right)\right)}\right) \cdot e^{-x} \]

      4. lower--.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{96} \cdot {x}^{2}} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      6. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      8. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

      9. lower-*.f64 3.9

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 3.9%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x} + 1\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       4. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       5. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       7. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

       8. lower-fma.f64 8.6

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right), x, 1\right)\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   11. Applied rewrites 8.6%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around inf

       \[\leadsto \left(\left({x}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]
   13. Step-by-step derivation

   14. Applied rewrites 74.2%

       \[\leadsto \left(\left(\left(\frac{\frac{1}{x} + 0.5}{x} + 0.16666666666666666\right) \cdot \color{blue}{{x}^{3}}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 0.599999999999999978 < x

   1. Initial program 1.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.8%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-138}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664 - 0.5, x \cdot x, 1\right)}\right)\right)}{e^{x}}\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;\left(\left(\left(\frac{{x}^{-1} + 0.5}{x} + 0.16666666666666666\right) \cdot {x}^{3}\right) \bmod \left(\mathsf{fma}\left(-0.010416666666666666 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 25.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod t\_0\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)))
   (if (<= x 0.62) (fmod (exp x) t_0) (* (fmod 1.0 t_0) (exp (- x))))))

C

double code(double x) {
	double t_0 = fma((x * x), -0.25, 1.0);
	double tmp;
	if (x <= 0.62) {
		tmp = fmod(exp(x), t_0);
	} else {
		tmp = fmod(1.0, t_0) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), -0.25, 1.0)
	tmp = 0.0
	if (x <= 0.62)
		tmp = rem(exp(x), t_0);
	else
		tmp = Float64(rem(1.0, t_0) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, If[LessEqual[x, 0.62], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[With[{TMP1 = 1.0, 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 < 0.619999999999999996

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 6.4

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.4%

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

   if 0.619999999999999996 < x

   1. Initial program 1.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.7

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

   8. Applied rewrites 98.7%

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

Alternative 7: 26.0% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (fmod (+ 1.0 x) (fma (* x x) -0.25 1.0)) (exp (- x))))

C

double code(double x) {
	return fmod((1.0 + x), fma((x * x), -0.25, 1.0)) * exp(-x);
}

Julia

function code(x)
	return Float64(rem(Float64(1.0 + x), fma(Float64(x * x), -0.25, 1.0)) * exp(Float64(-x)))
end

Wolfram

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

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

5. Applied rewrites 25.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 25.3

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

8. Applied rewrites 25.3%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 27.3

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

11. Applied rewrites 27.3%

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

Alternative 8: 5.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fmod (exp x) (fma (* x x) -0.25 1.0)))

C

double code(double x) {
	return fmod(exp(x), fma((x * x), -0.25, 1.0));
}

Julia

function code(x)
	return rem(exp(x), fma(Float64(x * x), -0.25, 1.0))
end

Wolfram

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

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-cos.f64 5.1

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

5. Applied rewrites 5.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 5.1%

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

Reproduce

herbie shell --seed 2024347 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))