expfmod (used to be hard to sample)

Percentage Accurate: 8.9% → 39.8%

Time: 16.9s

Alternatives: 12

Speedup: 1.9×

• could not determine a ground truth

  1. x = -7.615422295738604e+66

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: 39.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))) (t_1 (exp (- x))))
   (if (<= (* t_0 t_1) 2.0)
     (/ t_0 (exp x))
     (* (fmod 1.0 (fma -0.25 (* x x) 1.0)) 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, fma(-0.25, (x * x), 1.0)) * 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, fma(-0.25, Float64(x * x), 1.0)) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, 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[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $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 34.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.9

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \]
   9. 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. rec-exp N/A

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

      3. rec-exp N/A

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

      4. associate-*r/ N/A

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

      5. frac-2neg N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

   10. 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 34.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.9

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 34.9%

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

Alternative 2: 39.6% accurate, 0.5× speedup

Math

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

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
          (- (* -0.003298611111111111 (* x x)) 0.010416666666666666)
          (* x x)
          -0.25)
         x)
        x
        1.0))
      t_0)
     (* (fmod 1.0 (fma -0.25 (* x x) 1.0)) 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(((-0.003298611111111111 * (x * x)) - 0.010416666666666666), (x * x), -0.25) * x), x, 1.0)) * t_0;
	} else {
		tmp = fmod(1.0, fma(-0.25, (x * x), 1.0)) * 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(Float64(fma(Float64(Float64(-0.003298611111111111 * Float64(x * x)) - 0.010416666666666666), Float64(x * x), -0.25) * x), x, 1.0)) * t_0);
	else
		tmp = Float64(rem(1.0, fma(-0.25, Float64(x * x), 1.0)) * 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[(N[(-0.003298611111111111 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $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%

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

   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 34.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.9

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 34.9%

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

Alternative 3: 39.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (/
      (*
       1.0
       (fmod
        (exp x)
        (sqrt (fma (fma (* x x) 0.041666666666666664 -0.5) (* x x) 1.0))))
      (exp x))
     (* (fmod 1.0 (fma -0.25 (* x x) 1.0)) 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 = (1.0 * fmod(exp(x), sqrt(fma(fma((x * x), 0.041666666666666664, -0.5), (x * x), 1.0)))) / exp(x);
	} else {
		tmp = fmod(1.0, fma(-0.25, (x * x), 1.0)) * 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(Float64(1.0 * rem(exp(x), sqrt(fma(fma(Float64(x * x), 0.041666666666666664, -0.5), Float64(x * x), 1.0)))) / exp(x));
	else
		tmp = Float64(rem(1.0, fma(-0.25, Float64(x * x), 1.0)) * 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[(1.0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 8.7

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

   4. Applied rewrites 8.7%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. rec-exp N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 8.7%

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

   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 34.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.9

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 34.9%

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

Alternative 4: 39.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (*
      (fmod
       (exp x)
       (sqrt (fma (fma (* x x) 0.041666666666666664 -0.5) (* x x) 1.0)))
      t_0)
     (* (fmod 1.0 (fma -0.25 (* x x) 1.0)) 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), sqrt(fma(fma((x * x), 0.041666666666666664, -0.5), (x * x), 1.0))) * t_0;
	} else {
		tmp = fmod(1.0, fma(-0.25, (x * x), 1.0)) * 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), sqrt(fma(fma(Float64(x * x), 0.041666666666666664, -0.5), Float64(x * x), 1.0))) * t_0);
	else
		tmp = Float64(rem(1.0, fma(-0.25, Float64(x * x), 1.0)) * 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[Sqrt[N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 8.7

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

   4. Applied rewrites 8.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 8.7

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

   6. Applied rewrites 8.7%

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

   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 34.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.9

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 34.9%

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

Alternative 5: 39.5% accurate, 0.6× speedup

Math

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

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 (fma -0.25 (* x x) 1.0)) 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, fma(-0.25, (x * x), 1.0)) * 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, fma(-0.25, Float64(x * x), 1.0)) * 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[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $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 8.6

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

   4. Applied rewrites 8.6%

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

   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 34.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.9

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 34.9%

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

Alternative 6: 39.4% accurate, 0.6× speedup

Math

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

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 (* x x) -0.25 1.0)) (/ 1.0 (exp x)))
     (* (fmod 1.0 (fma -0.25 (* x x) 1.0)) 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((x * x), -0.25, 1.0)) * (1.0 / exp(x));
	} else {
		tmp = fmod(1.0, fma(-0.25, (x * x), 1.0)) * 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(Float64(x * x), -0.25, 1.0)) * Float64(1.0 / exp(x)));
	else
		tmp = Float64(rem(1.0, fma(-0.25, Float64(x * x), 1.0)) * 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[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $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

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 8.5

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

   4. Applied rewrites 8.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. rec-exp N/A

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

      4. lower-/.f64 N/A

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

      5. lift-exp.f64 8.5

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

   6. Applied rewrites 8.5%

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

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

   1. Initial program 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 34.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.9

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 34.9%

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

Alternative 7: 39.4% accurate, 0.6× speedup

Math

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

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 (* x x) -0.25 1.0)) t_0)
     (* (fmod 1.0 (fma -0.25 (* x x) 1.0)) 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((x * x), -0.25, 1.0)) * t_0;
	} else {
		tmp = fmod(1.0, fma(-0.25, (x * x), 1.0)) * 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(Float64(x * x), -0.25, 1.0)) * t_0);
	else
		tmp = Float64(rem(1.0, fma(-0.25, Float64(x * x), 1.0)) * 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[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $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

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 8.5

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

   4. Applied rewrites 8.5%

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

   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 34.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.9

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 34.9%

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

Alternative 8: 38.0% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (* (fmod (- x -1.0) (sqrt (fma -0.5 (* x x) 1.0))) (- (- x) -1.0))
     (* (fmod 1.0 (fma -0.25 (* x x) 1.0)) 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((x - -1.0), sqrt(fma(-0.5, (x * x), 1.0))) * (-x - -1.0);
	} else {
		tmp = fmod(1.0, fma(-0.25, (x * x), 1.0)) * 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(Float64(x - -1.0), sqrt(fma(-0.5, Float64(x * x), 1.0))) * Float64(Float64(-x) - -1.0));
	else
		tmp = Float64(rem(1.0, fma(-0.25, Float64(x * x), 1.0)) * 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[(x - -1.0), $MachinePrecision], TMP2 = N[Sqrt[N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[((-x) - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $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

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 3.9

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in x around 0

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

      1. rec-exp N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 3.9

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

   10. Applied rewrites 3.9%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

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

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. lower--.f64 7.1

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

   13. Applied rewrites 7.1%

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

   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 34.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.9

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 34.9%

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

Alternative 9: 38.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = N[(x - -1.0), $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]

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 34.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 34.9

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

7. Applied rewrites 34.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 34.9%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

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

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

    4. metadata-eval N/A

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

    5. metadata-eval N/A

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

    6. lower--.f64 38.0

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

13. Applied rewrites 38.0%

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

Alternative 10: 7.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[Sqrt[N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[((-x) - -1.0), $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 34.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 3.9

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

7. Applied rewrites 3.9%

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

8. Taylor expanded in x around 0

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

   1. rec-exp N/A

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

   2. mul-1-neg N/A

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

   3. lift-neg.f64 N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

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

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower--.f64 3.9

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

10. Applied rewrites 3.9%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

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

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

    4. metadata-eval N/A

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

    5. metadata-eval N/A

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

    6. lower--.f64 7.1

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

13. Applied rewrites 7.1%

    \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]
14. Add Preprocessing

Alternative 11: 3.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[((-x) - -1.0), $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 34.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 3.9

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

7. Applied rewrites 3.9%

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

8. Taylor expanded in x around 0

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

   1. rec-exp N/A

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

   2. mul-1-neg N/A

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

   3. lift-neg.f64 N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

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

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower--.f64 3.9

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

10. Applied rewrites 3.9%

    \[\leadsto \left(1 \bmod \left(\sqrt{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)}\right)\right) \cdot \color{blue}{\left(\left(-x\right) - -1\right)} \]
11. Add Preprocessing

Alternative 12: 0.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = mod(1.0d0, sqrt(((-0.5d0) * (x * x)))) * (-x - (-1.0d0))
end function

Python

def code(x):
	return math.fmod(1.0, math.sqrt((-0.5 * (x * x)))) * (-x - -1.0)

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[((-x) - -1.0), $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 34.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 3.9

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

7. Applied rewrites 3.9%

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

8. Taylor expanded in x around 0

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

   1. rec-exp N/A

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

   2. mul-1-neg N/A

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

   3. lift-neg.f64 N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

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

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower--.f64 3.9

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

10. Applied rewrites 3.9%

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

11. Taylor expanded in x around inf

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

    1. pow2 N/A

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

    2. lift-*.f64 N/A

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

    3. lower-*.f64 0.0

       \[\leadsto \left(1 \bmod \left(\sqrt{-0.5 \cdot \left(x \cdot \color{blue}{x}\right)}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

13. Applied rewrites 0.0%

    \[\leadsto \left(1 \bmod \left(\sqrt{-0.5 \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))