expfmod (used to be hard to sample)

Percentage Accurate: 8.9% → 40.4%

Time: 16.2s

Alternatives: 10

Speedup: 1.6×

• could not determine a ground truth

  1. x = -1.5582933360678232e-46

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: 8.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 40.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))) (t_1 (exp (- x))))
   (if (<= (* t_0 t_1) 2.0)
     (/ t_0 (exp x))
     (* (fmod 1.0 (sqrt (* (* x x) (* (* x x) 0.041666666666666664)))) t_1))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = exp(-x);
	double tmp;
	if ((t_0 * t_1) <= 2.0) {
		tmp = t_0 / exp(x);
	} else {
		tmp = fmod(1.0, sqrt(((x * x) * ((x * x) * 0.041666666666666664)))) * t_1;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = mod(exp(x), sqrt(cos(x)))
    t_1 = exp(-x)
    if ((t_0 * t_1) <= 2.0d0) then
        tmp = t_0 / exp(x)
    else
        tmp = mod(1.0d0, sqrt(((x * x) * ((x * x) * 0.041666666666666664d0)))) * t_1
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	t_1 = math.exp(-x)
	tmp = 0
	if (t_0 * t_1) <= 2.0:
		tmp = t_0 / math.exp(x)
	else:
		tmp = math.fmod(1.0, math.sqrt(((x * x) * ((x * x) * 0.041666666666666664)))) * t_1
	return tmp

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * t_1) <= 2.0)
		tmp = Float64(t_0 / exp(x));
	else
		tmp = Float64(rem(1.0, sqrt(Float64(Float64(x * x) * Float64(Float64(x * x) * 0.041666666666666664)))) * 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]}, If[LessEqual[N[(t$95$0 * t$95$1), $MachinePrecision], 2.0], N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 35.4

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. pow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 33.3

         \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664}\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 33.3%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. exp-neg N/A

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

       3. associate-/l* N/A

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

       4. *-rgt-identity N/A

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

       5. lower-/.f64 N/A

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

       6. lower-fmod.f64 N/A

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

       7. lift-exp.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-cos.f64 N/A

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

       10. lift-exp.f64 8.9

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

   13. Applied rewrites 8.9%

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

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

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 35.4

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. pow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 33.3

         \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664}\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 33.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lower-*.f64 33.3

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 33.3

          \[\leadsto \left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}\right)\right) \cdot e^{-x} \]

   12. Applied rewrites 33.3%

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{0.041666666666666664}\right)}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 40.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (*
      (fmod
       (exp x)
       (fma
        (fma
         (* (fma -0.003298611111111111 (* x x) -0.010416666666666666) x)
         x
         -0.25)
        (* x x)
        1.0))
      t_0)
     (* (fmod 1.0 (sqrt (* (* x x) (* (* x x) 0.041666666666666664)))) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = fmod(exp(x), fma(fma((fma(-0.003298611111111111, (x * x), -0.010416666666666666) * x), x, -0.25), (x * x), 1.0)) * t_0;
	} else {
		tmp = fmod(1.0, sqrt(((x * x) * ((x * x) * 0.041666666666666664)))) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(rem(exp(x), fma(fma(Float64(fma(-0.003298611111111111, Float64(x * x), -0.010416666666666666) * x), x, -0.25), Float64(x * x), 1.0)) * t_0);
	else
		tmp = Float64(rem(1.0, sqrt(Float64(Float64(x * x) * Float64(Float64(x * x) * 0.041666666666666664)))) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(-0.003298611111111111 * N[(x * x), $MachinePrecision] + -0.010416666666666666), $MachinePrecision] * x), $MachinePrecision] * x + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   4. Applied rewrites 8.7%

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

   6. Applied rewrites 8.7%

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

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

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 35.4

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. pow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 33.3

         \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664}\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 33.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lower-*.f64 33.3

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 33.3

          \[\leadsto \left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}\right)\right) \cdot e^{-x} \]

   12. Applied rewrites 33.3%

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{0.041666666666666664}\right)}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 40.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (*
      (fmod
       (exp x)
       (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
      t_0)
     (* (fmod 1.0 (sqrt (* (* x x) (* (* x x) 0.041666666666666664)))) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = fmod(exp(x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else {
		tmp = fmod(1.0, sqrt(((x * x) * ((x * x) * 0.041666666666666664)))) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(rem(exp(x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	else
		tmp = Float64(rem(1.0, sqrt(Float64(Float64(x * x) * Float64(Float64(x * x) * 0.041666666666666664)))) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 8.7

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

   4. Applied rewrites 8.7%

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

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

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 35.4

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. pow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 33.3

         \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664}\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 33.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lower-*.f64 33.3

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 33.3

          \[\leadsto \left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}\right)\right) \cdot e^{-x} \]

   12. Applied rewrites 33.3%

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{0.041666666666666664}\right)}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 40.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.86:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.86)
   (/ (fmod (exp x) (fma -0.25 (* x x) 1.0)) (exp x))
   (*
    (fmod 1.0 (sqrt (* (* x x) (* (* x x) 0.041666666666666664))))
    (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.86) {
		tmp = fmod(exp(x), fma(-0.25, (x * x), 1.0)) / exp(x);
	} else {
		tmp = fmod(1.0, sqrt(((x * x) * ((x * x) * 0.041666666666666664)))) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.86)
		tmp = Float64(rem(exp(x), fma(-0.25, Float64(x * x), 1.0)) / exp(x));
	else
		tmp = Float64(rem(1.0, sqrt(Float64(Float64(x * x) * Float64(Float64(x * x) * 0.041666666666666664)))) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.86], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.859999999999999987

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 8.6

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

   4. Applied rewrites 8.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-exp.f64 8.6

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

   6. Applied rewrites 8.6%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 8.6

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 8.6

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

   8. Applied rewrites 8.6%

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

   if 0.859999999999999987 < x

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 35.4

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. pow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 33.3

         \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664}\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 33.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lower-*.f64 33.3

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 33.3

          \[\leadsto \left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}\right)\right) \cdot e^{-x} \]

   12. Applied rewrites 33.3%

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{0.041666666666666664}\right)}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 39.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.86:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.86)
   (* (fmod (exp x) (fma (* x x) -0.25 1.0)) (fma (fma 0.5 x -1.0) x 1.0))
   (*
    (fmod 1.0 (sqrt (* (* x x) (* (* x x) 0.041666666666666664))))
    (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.86) {
		tmp = fmod(exp(x), fma((x * x), -0.25, 1.0)) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, sqrt(((x * x) * ((x * x) * 0.041666666666666664)))) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.86)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, sqrt(Float64(Float64(x * x) * Float64(Float64(x * x) * 0.041666666666666664)))) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.86], 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] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.859999999999999987

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 8.6

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

   4. Applied rewrites 8.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 7.9

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

   7. Applied rewrites 7.9%

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

   if 0.859999999999999987 < x

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 35.4

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. pow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 33.3

         \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664}\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 33.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lower-*.f64 33.3

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 33.3

          \[\leadsto \left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}\right)\right) \cdot e^{-x} \]

   12. Applied rewrites 33.3%

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{0.041666666666666664}\right)}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 39.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.86:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.86)
   (* (fmod (exp x) (fma (* x x) -0.25 1.0)) (- 1.0 x))
   (*
    (fmod 1.0 (sqrt (* (* x x) (* (* x x) 0.041666666666666664))))
    (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.86) {
		tmp = fmod(exp(x), fma((x * x), -0.25, 1.0)) * (1.0 - x);
	} else {
		tmp = fmod(1.0, sqrt(((x * x) * ((x * x) * 0.041666666666666664)))) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.86)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(1.0, sqrt(Float64(Float64(x * x) * Float64(Float64(x * x) * 0.041666666666666664)))) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.86], 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] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.859999999999999987

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 8.6

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

   4. Applied rewrites 8.6%

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

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 7.5

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

   7. Applied rewrites 7.5%

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

   if 0.859999999999999987 < x

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 35.4

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. pow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 33.3

         \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664}\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 33.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lower-*.f64 33.3

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 33.3

          \[\leadsto \left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}\right)\right) \cdot e^{-x} \]

   12. Applied rewrites 33.3%

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{0.041666666666666664}\right)}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 7.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := 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] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 8.6

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

4. Applied rewrites 8.6%

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

5. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f64 7.5

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

7. Applied rewrites 7.5%

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

Alternative 8: 6.5% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fmod (exp x) (sqrt (fma -0.5 (* x x) 1.0))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.9%

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

2. Taylor expanded in x around 0

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

   1. lift-sqrt.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-fmod.f64 N/A

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

   4. lift-exp.f64 6.6

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

4. Applied rewrites 6.6%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 6.5

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

7. Applied rewrites 6.5%

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

Alternative 9: 2.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664}\right)\right) \cdot \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fmod 1.0 (sqrt (* (* (* x x) (* x x)) 0.041666666666666664))) (- 1.0 x)))

C

double code(double x) {
	return fmod(1.0, sqrt((((x * x) * (x * x)) * 0.041666666666666664))) * (1.0 - x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = mod(1.0d0, sqrt((((x * x) * (x * x)) * 0.041666666666666664d0))) * (1.0d0 - x)
end function

Python

def code(x):
	return math.fmod(1.0, math.sqrt((((x * x) * (x * x)) * 0.041666666666666664))) * (1.0 - x)

Julia

function code(x)
	return Float64(rem(1.0, sqrt(Float64(Float64(Float64(x * x) * Float64(x * x)) * 0.041666666666666664))) * Float64(1.0 - x))
end

Wolfram

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

Derivation

1. Initial program 8.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 35.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 35.4

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

7. Applied rewrites 35.4%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

      \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(1 \bmod \left(\sqrt{{x}^{4} \cdot \frac{1}{24}}\right)\right) \cdot e^{-x} \]

   3. metadata-eval N/A

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

   4. pow-prod-up N/A

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

   5. pow-prod-down N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lower-*.f64 33.3

      \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664}\right)\right) \cdot e^{-x} \]

10. Applied rewrites 33.3%

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

11. Taylor expanded in x around 0

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

    1. flip-+ N/A

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

    2. mul-1-neg N/A

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot x\right)}{1 - -1 \cdot x} \]

    3. mul-1-neg N/A

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}{1 - -1 \cdot x} \]

    4. sqr-neg-rev N/A

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{1 - -1 \cdot x} \]

    5. mul-1-neg N/A

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{1 - \left(\mathsf{neg}\left(x\right)\right)} \]

    6. lift-neg.f64 N/A

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{1 - \left(-x\right)} \]

    7. flip3-- N/A

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{\frac{{1}^{3} - {\left(-x\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}}} \]

    8. metadata-eval N/A

       \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{\frac{1 - {\left(-x\right)}^{3}}{\color{blue}{1} \cdot 1 + \left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} \]

    9. cube-mult N/A

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

    10. lift-neg.f64 N/A

        \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{\frac{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\left(-x\right) \cdot \left(-x\right)\right)}{1 \cdot 1 + \left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} \]

    11. lift-neg.f64 N/A

        \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{\frac{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(-x\right)\right)}{1 \cdot 1 + \left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} \]

    12. lift-neg.f64 N/A

        \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{\frac{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}{1 \cdot 1 + \left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} \]

    13. sqr-neg-rev N/A

        \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{\frac{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot x\right)}{1 \cdot 1 + \left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} \]

    14. lift-*.f64 N/A

        \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{\frac{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot x\right)}{1 \cdot 1 + \left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} \]

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

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

    16. metadata-eval N/A

        \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{\frac{{1}^{3} + x \cdot \left(x \cdot x\right)}{\color{blue}{1} \cdot 1 + \left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} \]

    17. lift-*.f64 N/A

        \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{\frac{{1}^{3} + x \cdot \left(x \cdot x\right)}{1 \cdot 1 + \left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} \]

    18. cube-mult N/A

        \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}}\right)\right) \cdot \frac{1 \cdot 1 - x \cdot x}{\frac{{1}^{3} + {x}^{3}}{1 \cdot \color{blue}{1} + \left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} \]

13. Applied rewrites 2.6%

    \[\leadsto \left(1 \bmod \left(\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664}\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]
14. Add Preprocessing

Alternative 10: 0.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(e^{x}\right) \bmod \left(\sqrt{-0.5 \cdot \left(x \cdot x\right)}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fmod (exp x) (sqrt (* -0.5 (* x x)))))

C

double code(double x) {
	return fmod(exp(x), sqrt((-0.5 * (x * 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(((-0.5d0) * (x * x))))
end function

Python

def code(x):
	return math.fmod(math.exp(x), math.sqrt((-0.5 * (x * x))))

Julia

function code(x)
	return rem(exp(x), sqrt(Float64(-0.5 * Float64(x * x))))
end

Wolfram

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

Derivation

1. Initial program 8.9%

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

2. Taylor expanded in x around 0

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

   1. lift-sqrt.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-fmod.f64 N/A

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

   4. lift-exp.f64 6.6

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

4. Applied rewrites 6.6%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 6.5

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

7. Applied rewrites 6.5%

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

8. Taylor expanded in x around inf

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

   1. pow2 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-*.f64 0.0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{-0.5 \cdot \left(x \cdot x\right)}\right)\right) \]

10. Applied rewrites 0.0%

    \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{-0.5 \cdot \left(x \cdot x\right)}\right)\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))