expfmod (used to be hard to sample)

Percentage Accurate: 7.1% → 62.6%

Time: 12.6s

Alternatives: 15

Speedup: 1.9×

• could not determine a ground truth

  1. x = -9.90232606563866e+171

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Initial Program: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 10^{-11}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{t\_0}{-1} \cdot \frac{-1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x))))
        (t_1 (exp (- x)))
        (t_2 (* t_0 t_1)))
   (if (<= t_2 1e-11)
     (*
      (fmod
       (* (fma 0.5 x 1.0) x)
       (sqrt (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0)))
      t_1)
     (if (<= t_2 2.0)
       (* (/ t_0 -1.0) (/ -1.0 (exp x)))
       (* (fmod 1.0 (fma (* x x) -0.25 1.0)) t_1)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = exp(-x);
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= 1e-11) {
		tmp = fmod((fma(0.5, x, 1.0) * x), sqrt(fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0))) * t_1;
	} else if (t_2 <= 2.0) {
		tmp = (t_0 / -1.0) * (-1.0 / exp(x));
	} else {
		tmp = fmod(1.0, fma((x * x), -0.25, 1.0)) * t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = exp(Float64(-x))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_2 <= 1e-11)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), sqrt(fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0))) * t_1);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(t_0 / -1.0) * Float64(-1.0 / exp(x)));
	else
		tmp = Float64(rem(1.0, fma(Float64(x * x), -0.25, 1.0)) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-11], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(t$95$0 / -1.0), $MachinePrecision] * N[(-1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 9.99999999999999939e-12

   1. Initial program 4.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 4.7

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

   8. Applied rewrites 4.7%

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

   10. Applied rewrites 4.7%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 48.9%

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

   if 9.99999999999999939e-12 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 92.7%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-exp.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. exp-neg N/A

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

      11. lift-neg.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. associate-/r/ N/A

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

      17. metadata-eval N/A

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

      18. times-frac N/A

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

   4. Applied rewrites 92.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 10^{-11}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{-1} \cdot \frac{-1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 10^{-11}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{t\_0}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x))))
        (t_1 (exp (- x)))
        (t_2 (* t_0 t_1)))
   (if (<= t_2 1e-11)
     (*
      (fmod
       (* (fma 0.5 x 1.0) x)
       (sqrt (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0)))
      t_1)
     (if (<= t_2 2.0)
       (/ t_0 (exp x))
       (* (fmod 1.0 (fma (* x x) -0.25 1.0)) t_1)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = exp(-x);
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= 1e-11) {
		tmp = fmod((fma(0.5, x, 1.0) * x), sqrt(fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0))) * t_1;
	} else if (t_2 <= 2.0) {
		tmp = t_0 / exp(x);
	} else {
		tmp = fmod(1.0, fma((x * x), -0.25, 1.0)) * t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = exp(Float64(-x))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_2 <= 1e-11)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), sqrt(fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0))) * t_1);
	elseif (t_2 <= 2.0)
		tmp = Float64(t_0 / exp(x));
	else
		tmp = Float64(rem(1.0, fma(Float64(x * x), -0.25, 1.0)) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-11], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 9.99999999999999939e-12

   1. Initial program 4.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 4.7

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

   8. Applied rewrites 4.7%

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

   10. Applied rewrites 4.7%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 48.9%

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

   if 9.99999999999999939e-12 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 92.7%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 92.9

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

   4. Applied rewrites 92.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 10^{-11}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 10^{-11}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* (fmod (exp x) (sqrt (cos x))) t_0)))
   (if (<= t_1 1e-11)
     (*
      (fmod
       (* (fma 0.5 x 1.0) x)
       (sqrt (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0)))
      t_0)
     (if (<= t_1 2.0) t_1 (* (fmod 1.0 (fma (* x x) -0.25 1.0)) t_0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x))) * t_0;
	double tmp;
	if (t_1 <= 1e-11) {
		tmp = fmod((fma(0.5, x, 1.0) * x), sqrt(fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0))) * t_0;
	} else if (t_1 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = fmod(1.0, fma((x * x), -0.25, 1.0)) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(rem(exp(x), sqrt(cos(x))) * t_0)
	tmp = 0.0
	if (t_1 <= 1e-11)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), sqrt(fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0))) * t_0);
	elseif (t_1 <= 2.0)
		tmp = t_1;
	else
		tmp = Float64(rem(1.0, fma(Float64(x * x), -0.25, 1.0)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-11], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], t$95$1, N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 9.99999999999999939e-12

   1. Initial program 4.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 4.7

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

   8. Applied rewrites 4.7%

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

   10. Applied rewrites 4.7%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 48.9%

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

   if 9.99999999999999939e-12 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 92.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 10^{-11}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{--1}{e^{x}} \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.003298611111111111 \cdot x, x, -0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, -1\right), 1\right)\right)}^{-1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (*
    (/ (- -1.0) (exp x))
    (fmod
     (exp x)
     (fma
      (-
       (* (* (fma (* -0.003298611111111111 x) x -0.010416666666666666) x) x)
       0.25)
      (* x x)
      1.0)))
   (if (<= x 2e+154)
     (* (fmod (* (fma 0.5 x 1.0) x) (sqrt (cos x))) (exp (- x)))
     (fmod (pow (fma x (fma 0.5 x -1.0) 1.0) -1.0) (fma (* x x) -0.25 1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (-(-1.0) / exp(x)) * fmod(exp(x), fma((((fma((-0.003298611111111111 * x), x, -0.010416666666666666) * x) * x) - 0.25), (x * x), 1.0));
	} else if (x <= 2e+154) {
		tmp = fmod((fma(0.5, x, 1.0) * x), sqrt(cos(x))) * exp(-x);
	} else {
		tmp = fmod(pow(fma(x, fma(0.5, x, -1.0), 1.0), -1.0), fma((x * x), -0.25, 1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(Float64(-(-1.0)) / exp(x)) * rem(exp(x), fma(Float64(Float64(Float64(fma(Float64(-0.003298611111111111 * x), x, -0.010416666666666666) * x) * x) - 0.25), Float64(x * x), 1.0)));
	elseif (x <= 2e+154)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), sqrt(cos(x))) * exp(Float64(-x)));
	else
		tmp = rem((fma(x, fma(0.5, x, -1.0), 1.0) ^ -1.0), fma(Float64(x * x), -0.25, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[((--1.0) / N[Exp[x], $MachinePrecision]), $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(N[(N[(-0.003298611111111111 * x), $MachinePrecision] * x + -0.010416666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+154], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = N[Power[N[(x * N[(0.5 * x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 11.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-exp.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. exp-neg N/A

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

      11. lift-neg.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. associate-/r/ N/A

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

      17. metadata-eval N/A

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

      18. times-frac N/A

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

   4. Applied rewrites 11.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right)}{-1} \cdot \frac{-1}{e^{x}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) + 1\right)}\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot {x}^{2}} + 1\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)}\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right) \cdot x} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right) \cdot x} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) \cdot x\right)} \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) \cdot x\right)} \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\color{blue}{\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)} \cdot x\right) \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\color{blue}{\frac{-19}{5760} \cdot {x}^{2}} - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, \color{blue}{x \cdot x}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      16. lower-*.f64 11.0

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, \color{blue}{x \cdot x}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

   7. Applied rewrites 11.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)}\right)}{-1} \cdot \frac{-1}{e^{x}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)}{-1}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{e^{x}} \cdot \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)}{-1}} \]

      4. frac-2neg N/A

         \[\leadsto \frac{-1}{e^{x}} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right)}{\mathsf{neg}\left(-1\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-1}{e^{x}} \cdot \frac{\mathsf{neg}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right)}{\color{blue}{1}} \]

      6. /-rgt-identity N/A

         \[\leadsto \frac{-1}{e^{x}} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{e^{x}} \cdot \left(\mathsf{neg}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right)\right)} \]

   9. Applied rewrites 11.0%

      \[\leadsto \color{blue}{\frac{-1}{e^{x}} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)\right)\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{-1}{e^{x}} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

   12. Applied rewrites 11.0%

       \[\leadsto \frac{-1}{e^{x}} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.003298611111111111 \cdot x, x, -0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)\right)\right) \]

   if -1.999999999999994e-310 < x < 2.00000000000000007e154

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 23.7

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

   5. Applied rewrites 23.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.6%

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

   if 2.00000000000000007e154 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \]
   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 0.0

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

   5. Applied rewrites 0.0%

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

   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 0.0%

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

       \[\leadsto \left(\left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, -1\right), 1\right)}\right) \bmod \left(\mathsf{fma}\left(x \cdot \color{blue}{x}, -0.25, 1\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{--1}{e^{x}} \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.003298611111111111 \cdot x, x, -0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, -1\right), 1\right)\right)}^{-1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod t\_0\right) \cdot t\_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, -1\right), 1\right)\right)}^{-1}\right) \bmod t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)) (t_1 (exp (- x))))
   (if (<= x -2e-310)
     (* (fmod (exp x) t_0) t_1)
     (if (<= x 2e+154)
       (* (fmod (* (fma 0.5 x 1.0) x) (sqrt (cos x))) t_1)
       (fmod (pow (fma x (fma 0.5 x -1.0) 1.0) -1.0) t_0)))))

C

double code(double x) {
	double t_0 = fma((x * x), -0.25, 1.0);
	double t_1 = exp(-x);
	double tmp;
	if (x <= -2e-310) {
		tmp = fmod(exp(x), t_0) * t_1;
	} else if (x <= 2e+154) {
		tmp = fmod((fma(0.5, x, 1.0) * x), sqrt(cos(x))) * t_1;
	} else {
		tmp = fmod(pow(fma(x, fma(0.5, x, -1.0), 1.0), -1.0), t_0);
	}
	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 <= -2e-310)
		tmp = Float64(rem(exp(x), t_0) * t_1);
	elseif (x <= 2e+154)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), sqrt(cos(x))) * t_1);
	else
		tmp = rem((fma(x, fma(0.5, x, -1.0), 1.0) ^ -1.0), t_0);
	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, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $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[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], N[With[{TMP1 = N[Power[N[(x * N[(0.5 * x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 11.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 11.0

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

   5. Applied rewrites 11.0%

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

   if -1.999999999999994e-310 < x < 2.00000000000000007e154

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 23.7

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

   5. Applied rewrites 23.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.6%

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

   if 2.00000000000000007e154 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \]
   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 0.0

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

   5. Applied rewrites 0.0%

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

   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 0.0%

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

       \[\leadsto \left(\left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, -1\right), 1\right)}\right) \bmod \left(\mathsf{fma}\left(x \cdot \color{blue}{x}, -0.25, 1\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, -1\right), 1\right)\right)}^{-1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -2e-310)
     (* (fmod (exp x) (fma (* x x) -0.25 1.0)) t_0)
     (if (<= x 0.5)
       (*
        (fmod
         (* (fma 0.5 x 1.0) x)
         (sqrt (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* 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 <= -2e-310) {
		tmp = fmod(exp(x), fma((x * x), -0.25, 1.0)) * t_0;
	} else if (x <= 0.5) {
		tmp = fmod((fma(0.5, x, 1.0) * x), sqrt(fma(((0.041666666666666664 * (x * x)) - 0.5), (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 <= -2e-310)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) * t_0);
	elseif (x <= 0.5)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), sqrt(fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), 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, -2e-310], N[(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] * t$95$0), $MachinePrecision], If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $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 < -1.999999999999994e-310

   1. Initial program 11.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 11.0

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

   5. Applied rewrites 11.0%

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

   if -1.999999999999994e-310 < x < 0.5

   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

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 8.0

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

   5. Applied rewrites 8.0%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 8.0

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

   8. Applied rewrites 8.0%

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

   10. Applied rewrites 8.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 97.9%

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

   if 0.5 < 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 98.3%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(x - \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x, x, 1\right)\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -2e-310)
     (*
      (- x (fma (* (fma -0.16666666666666666 x 0.5) x) x 1.0))
      (-
       (fmod
        (exp x)
        (fma
         (-
          (*
           (* (- (* -0.003298611111111111 (* x x)) 0.010416666666666666) x)
           x)
          0.25)
         (* x x)
         1.0))))
     (if (<= x 0.5)
       (*
        (fmod
         (* (fma 0.5 x 1.0) x)
         (sqrt (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* 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 <= -2e-310) {
		tmp = (x - fma((fma(-0.16666666666666666, x, 0.5) * x), x, 1.0)) * -fmod(exp(x), fma((((((-0.003298611111111111 * (x * x)) - 0.010416666666666666) * x) * x) - 0.25), (x * x), 1.0));
	} else if (x <= 0.5) {
		tmp = fmod((fma(0.5, x, 1.0) * x), sqrt(fma(((0.041666666666666664 * (x * x)) - 0.5), (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 <= -2e-310)
		tmp = Float64(Float64(x - fma(Float64(fma(-0.16666666666666666, x, 0.5) * x), x, 1.0)) * Float64(-rem(exp(x), fma(Float64(Float64(Float64(Float64(Float64(-0.003298611111111111 * Float64(x * x)) - 0.010416666666666666) * x) * x) - 0.25), Float64(x * x), 1.0))));
	elseif (x <= 0.5)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), sqrt(fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), 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, -2e-310], N[(N[(x - N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] * (-N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(N[(N[(-0.003298611111111111 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $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 < -1.999999999999994e-310

   1. Initial program 11.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-exp.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. exp-neg N/A

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

      11. lift-neg.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. associate-/r/ N/A

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

      17. metadata-eval N/A

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

      18. times-frac N/A

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

   4. Applied rewrites 11.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right)}{-1} \cdot \frac{-1}{e^{x}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) + 1\right)}\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot {x}^{2}} + 1\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)}\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right) \cdot x} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right) \cdot x} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) \cdot x\right)} \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) \cdot x\right)} \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\color{blue}{\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)} \cdot x\right) \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\color{blue}{\frac{-19}{5760} \cdot {x}^{2}} - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, \color{blue}{x \cdot x}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      16. lower-*.f64 11.0

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, \color{blue}{x \cdot x}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

   7. Applied rewrites 11.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)}\right)}{-1} \cdot \frac{-1}{e^{x}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)}{-1}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{e^{x}} \cdot \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)}{-1}} \]

      4. frac-2neg N/A

         \[\leadsto \frac{-1}{e^{x}} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right)}{\mathsf{neg}\left(-1\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-1}{e^{x}} \cdot \frac{\mathsf{neg}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right)}{\color{blue}{1}} \]

      6. /-rgt-identity N/A

         \[\leadsto \frac{-1}{e^{x}} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{e^{x}} \cdot \left(\mathsf{neg}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right)\right)} \]

   9. Applied rewrites 11.0%

      \[\leadsto \color{blue}{\frac{-1}{e^{x}} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)\right)\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right) - 1\right)} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right) \cdot x} - 1\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right) + 1\right)} \cdot x - 1\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       3. distribute-rgt1-in N/A

          \[\leadsto \left(\color{blue}{\left(x + \left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right) \cdot x\right)} - 1\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(x + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right)}\right) - 1\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

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

          \[\leadsto \left(\color{blue}{\left(x - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right)\right)} - 1\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       6. associate--l- N/A

          \[\leadsto \color{blue}{\left(x - \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right) + 1\right)\right)} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       7. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(x - \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right) + 1\right)\right)} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \left(x - \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot x - \frac{1}{2}\right) \cdot x\right)} + 1\right)\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(x - \left(\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right) \cdot x} + 1\right)\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \left(x - \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right), x, 1\right)}\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

   12. Applied rewrites 9.2%

       \[\leadsto \color{blue}{\left(x - \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x, x, 1\right)\right)} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)\right)\right) \]

   if -1.999999999999994e-310 < x < 0.5

   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

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 8.0

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

   5. Applied rewrites 8.0%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 8.0

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

   8. Applied rewrites 8.0%

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

   10. Applied rewrites 8.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 97.9%

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

   if 0.5 < 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 98.3%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(x - \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x, x, 1\right)\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(x - \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x, x, 1\right)\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -2e-310)
     (*
      (- x (fma (* (fma -0.16666666666666666 x 0.5) x) x 1.0))
      (-
       (fmod
        (exp x)
        (fma
         (-
          (*
           (* (- (* -0.003298611111111111 (* x x)) 0.010416666666666666) x)
           x)
          0.25)
         (* x x)
         1.0))))
     (if (<= x 0.5)
       (*
        (fmod
         (* (fma 0.5 x 1.0) x)
         (sqrt (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0)))
        t_0)
       (* (fmod 1.0 (fma (* x x) -0.25 1.0)) t_0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -2e-310) {
		tmp = (x - fma((fma(-0.16666666666666666, x, 0.5) * x), x, 1.0)) * -fmod(exp(x), fma((((((-0.003298611111111111 * (x * x)) - 0.010416666666666666) * x) * x) - 0.25), (x * x), 1.0));
	} else if (x <= 0.5) {
		tmp = fmod((fma(0.5, x, 1.0) * x), sqrt(fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0))) * t_0;
	} else {
		tmp = fmod(1.0, fma((x * x), -0.25, 1.0)) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(x - fma(Float64(fma(-0.16666666666666666, x, 0.5) * x), x, 1.0)) * Float64(-rem(exp(x), fma(Float64(Float64(Float64(Float64(Float64(-0.003298611111111111 * Float64(x * x)) - 0.010416666666666666) * x) * x) - 0.25), Float64(x * x), 1.0))));
	elseif (x <= 0.5)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), sqrt(fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0))) * t_0);
	else
		tmp = Float64(rem(1.0, fma(Float64(x * x), -0.25, 1.0)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -2e-310], N[(N[(x - N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] * (-N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(N[(N[(-0.003298611111111111 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 11.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-exp.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. exp-neg N/A

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

      11. lift-neg.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. associate-/r/ N/A

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

      17. metadata-eval N/A

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

      18. times-frac N/A

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

   4. Applied rewrites 11.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right)}{-1} \cdot \frac{-1}{e^{x}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) + 1\right)}\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot {x}^{2}} + 1\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)}\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right)} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right) \cdot x} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right) \cdot x} - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) \cdot x\right)} \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) \cdot x\right)} \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\color{blue}{\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)} \cdot x\right) \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\color{blue}{\frac{-19}{5760} \cdot {x}^{2}} - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, {x}^{2}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, \color{blue}{x \cdot x}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

      16. lower-*.f64 11.0

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, \color{blue}{x \cdot x}, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}} \]

   7. Applied rewrites 11.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)}\right)}{-1} \cdot \frac{-1}{e^{x}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)}{-1} \cdot \frac{-1}{e^{x}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)}{-1}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{e^{x}} \cdot \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)}{-1}} \]

      4. frac-2neg N/A

         \[\leadsto \frac{-1}{e^{x}} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right)}{\mathsf{neg}\left(-1\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-1}{e^{x}} \cdot \frac{\mathsf{neg}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right)}{\color{blue}{1}} \]

      6. /-rgt-identity N/A

         \[\leadsto \frac{-1}{e^{x}} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{e^{x}} \cdot \left(\mathsf{neg}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right)\right)} \]

   9. Applied rewrites 11.0%

      \[\leadsto \color{blue}{\frac{-1}{e^{x}} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)\right)\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right) - 1\right)} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right) \cdot x} - 1\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right) + 1\right)} \cdot x - 1\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       3. distribute-rgt1-in N/A

          \[\leadsto \left(\color{blue}{\left(x + \left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right) \cdot x\right)} - 1\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(x + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right)}\right) - 1\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

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

          \[\leadsto \left(\color{blue}{\left(x - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right)\right)} - 1\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       6. associate--l- N/A

          \[\leadsto \color{blue}{\left(x - \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right) + 1\right)\right)} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       7. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(x - \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right) + 1\right)\right)} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \left(x - \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot x - \frac{1}{2}\right) \cdot x\right)} + 1\right)\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(x - \left(\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right) \cdot x} + 1\right)\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \left(x - \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right), x, 1\right)}\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}\right) \cdot x\right) \cdot x - \frac{1}{4}, x \cdot x, 1\right)\right)\right)\right) \]

   12. Applied rewrites 9.2%

       \[\leadsto \color{blue}{\left(x - \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x, x, 1\right)\right)} \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)\right)\right) \]

   if -1.999999999999994e-310 < x < 0.5

   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

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 8.0

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

   5. Applied rewrites 8.0%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 8.0

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

   8. Applied rewrites 8.0%

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

   10. Applied rewrites 8.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 97.9%

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

   if 0.5 < 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 98.3%

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

   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 98.3

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

   8. Applied rewrites 98.3%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(x - \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x, x, 1\right)\right) \cdot \left(-\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666\right) \cdot x\right) \cdot x - 0.25, x \cdot x, 1\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod t\_0\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod t\_0\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)) (t_1 (exp (- x))))
   (if (<= x -2e-310)
     (*
      (fmod (exp x) t_0)
      (fma (- (* (fma -0.16666666666666666 x 0.5) x) 1.0) x 1.0))
     (if (<= x 0.5)
       (*
        (fmod
         (* (fma 0.5 x 1.0) x)
         (sqrt (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0)))
        t_1)
       (* (fmod 1.0 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 <= -2e-310) {
		tmp = fmod(exp(x), t_0) * fma(((fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0);
	} else if (x <= 0.5) {
		tmp = fmod((fma(0.5, x, 1.0) * x), sqrt(fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0))) * t_1;
	} else {
		tmp = fmod(1.0, 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 <= -2e-310)
		tmp = Float64(rem(exp(x), t_0) * fma(Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0));
	elseif (x <= 0.5)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), sqrt(fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0))) * t_1);
	else
		tmp = Float64(rem(1.0, 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, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[With[{TMP1 = 1.0, 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 < -1.999999999999994e-310

   1. Initial program 11.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 11.0

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

   5. Applied rewrites 11.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 9.2%

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

   if -1.999999999999994e-310 < x < 0.5

   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

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 8.0

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

   5. Applied rewrites 8.0%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 8.0

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

   8. Applied rewrites 8.0%

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

   10. Applied rewrites 8.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 97.9%

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

   if 0.5 < 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 98.3%

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

   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 98.3

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

   8. Applied rewrites 98.3%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod t\_0\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod t\_0\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)))
   (if (<= x 0.5)
     (*
      (fmod (exp x) t_0)
      (fma (- (* (fma -0.16666666666666666 x 0.5) x) 1.0) x 1.0))
     (* (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 <= 0.5) {
		tmp = fmod(exp(x), t_0) * fma(((fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0);
	} 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 <= 0.5)
		tmp = Float64(rem(exp(x), t_0) * fma(Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0));
	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, 0.5], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $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 < 0.5

   1. Initial program 9.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 9.6

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

   5. Applied rewrites 9.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 8.7%

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

   if 0.5 < 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 98.3%

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

   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 98.3

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

   8. Applied rewrites 98.3%

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

Alternative 11: 18.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\right)}^{-1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fmod
  (pow (fma (- (* (fma -0.16666666666666666 x 0.5) x) 1.0) x 1.0) -1.0)
  (fma (* x x) -0.25 1.0)))

C

double code(double x) {
	return fmod(pow(fma(((fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0), -1.0), fma((x * x), -0.25, 1.0));
}

Julia

function code(x)
	return rem((fma(Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0) ^ -1.0), fma(Float64(x * x), -0.25, 1.0))
end

Wolfram

code[x_] := N[With[{TMP1 = N[Power[N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], -1.0], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]

Derivation

1. Initial program 8.3%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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 6.0

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

5. Applied rewrites 6.0%

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

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 6.0%

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

10. Applied rewrites 6.0%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 19.5%

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

14. Final simplification 19.5%

    \[\leadsto \left(\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\right)}^{-1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \]
15. Add Preprocessing

Alternative 12: 26.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (*
    (fmod
     (+ (fma (* 0.5 x) x 1.0) x)
     (sqrt (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0)))
    (fma x (fma 0.5 x -1.0) 1.0))
   (* (fmod 1.0 (fma (* x x) -0.25 1.0)) (exp (- x)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 9.8%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 8.5

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

   5. Applied rewrites 8.5%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 8.5

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

   8. Applied rewrites 8.5%

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

   10. Applied rewrites 8.5%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2} \cdot x, x, 1\right) + x\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right)}\right)\right) \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} \cdot x - 1\right)\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

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

   13. Applied rewrites 8.6%

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

   if 0.5 < 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 98.3%

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

   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 98.3

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

   8. Applied rewrites 98.3%

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

Alternative 13: 15.7% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fmod (pow (fma x (fma 0.5 x -1.0) 1.0) -1.0) (fma (* x x) -0.25 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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 6.0

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

5. Applied rewrites 6.0%

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

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 6.0%

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

10. Applied rewrites 6.0%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 17.1%

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

14. Final simplification 17.1%

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

Alternative 14: 6.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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 6.0

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

5. Applied rewrites 6.0%

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

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 6.0%

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

10. Applied rewrites 6.0%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 7.0%

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

14. Final simplification 7.0%

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

Alternative 15: 4.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 22.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   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 22.0

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

8. Applied rewrites 22.0%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

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

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

    3. *-commutative N/A

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

    4. mul-1-neg N/A

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

    5. remove-double-neg N/A

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

    6. lower--.f64 4.0

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

11. Applied rewrites 4.0%

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

Reproduce

herbie shell --seed 2024360 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))