expfmod (used to be hard to sample)

Percentage Accurate: 9.2% → 40.2%

Time: 13.9s

Alternatives: 16

Speedup: 1.5×

• could not determine a ground truth

  1. x = -1.3906677794648327e-214

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: 9.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 40.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (* (fmod (exp x) (/ 1.0 (pow (cos x) -0.5))) t_0)
     (*
      (fmod 1.0 (fma (fma -0.010416666666666666 (* x x) -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), (1.0 / pow(cos(x), -0.5))) * t_0;
	} else {
		tmp = fmod(1.0, fma(fma(-0.010416666666666666, (x * x), -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), Float64(1.0 / (cos(x) ^ -0.5))) * t_0);
	else
		tmp = Float64(rem(1.0, fma(fma(-0.010416666666666666, Float64(x * x), -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[(1.0 / N[Power[N[Cos[x], $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, 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]]]

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 9.2%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-cos.f64 9.2

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

   3. Applied rewrites 9.2%

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

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

   1. Initial program 9.2%

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

   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%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 40.2% accurate, 0.5× speedup

Math

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

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 (fma -0.010416666666666666 (* x x) -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(fma(-0.010416666666666666, (x * x), -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(fma(-0.010416666666666666, Float64(x * x), -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[(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$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 9.2%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-fmod.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-fmod.f64 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-exp.f64 9.2

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

   3. Applied rewrites 9.2%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-fmod.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-fmod.f64 N/A

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

      10. lift-exp.f64 9.2

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

   5. Applied rewrites 9.2%

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

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

   1. Initial program 9.2%

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

   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%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 40.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right) \cdot x, x, -0.5\right) \cdot x, x, 1\right)}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (/
    (fmod
     (exp x)
     (sqrt
      (fma
       (*
        (fma
         (* (fma (* x x) -0.001388888888888889 0.041666666666666664) x)
         x
         -0.5)
        x)
       x
       1.0)))
    (exp x))
   (* (fmod (- x -1.0) (sqrt (cos x))) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = fmod(exp(x), sqrt(fma((fma((fma((x * x), -0.001388888888888889, 0.041666666666666664) * x), x, -0.5) * x), x, 1.0))) / exp(x);
	} else {
		tmp = fmod((x - -1.0), sqrt(cos(x))) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64(rem(exp(x), sqrt(fma(Float64(fma(Float64(fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664) * x), x, -0.5) * x), x, 1.0))) / exp(x));
	else
		tmp = Float64(rem(Float64(x - -1.0), sqrt(cos(x))) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 9.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1}\right)\right) \cdot e^{-x} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1}\right)\right) \cdot e^{-x} \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot x\right) \cdot x + 1}\right)\right) \cdot e^{-x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot x, \color{blue}{x}, 1\right)}\right)\right) \cdot e^{-x} \]

   4. Applied rewrites 8.8%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x\right) \cdot x - 0.5\right) \cdot x, x, 1\right)}}\right)\right) \cdot e^{-x} \]

   6. Applied rewrites 8.8%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right) \cdot x, x, -0.5\right) \cdot x, x, 1\right)}\right)\right) \cdot 1}{e^{x}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right) \cdot x, x, \frac{-1}{2}\right) \cdot x, x, 1\right)}\right)\right) \cdot 1}}{e^{x}} \]

      2. *-rgt-identity 8.8

         \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right) \cdot x, x, -0.5\right) \cdot x, x, 1\right)}\right)\right)}}{e^{x}} \]

   8. Applied rewrites 8.8%

      \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right) \cdot x, x, -0.5\right) \cdot x, x, 1\right)}\right)\right)}}{e^{x}} \]

   if 0.5 < x

   1. Initial program 9.2%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 38.5

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

   4. Applied rewrites 38.5%

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

Alternative 4: 40.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right) \cdot x, x, -0.5\right) \cdot x, x, 1\right)}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (/
    (fmod
     (exp x)
     (sqrt
      (fma
       (*
        (fma
         (* (fma (* x x) -0.001388888888888889 0.041666666666666664) x)
         x
         -0.5)
        x)
       x
       1.0)))
    (exp x))
   (* (fmod 1.0 (sqrt (cos x))) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = fmod(exp(x), sqrt(fma((fma((fma((x * x), -0.001388888888888889, 0.041666666666666664) * x), x, -0.5) * x), x, 1.0))) / exp(x);
	} else {
		tmp = fmod(1.0, sqrt(cos(x))) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64(rem(exp(x), sqrt(fma(Float64(fma(Float64(fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664) * x), x, -0.5) * x), x, 1.0))) / exp(x));
	else
		tmp = Float64(rem(1.0, sqrt(cos(x))) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 9.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1}\right)\right) \cdot e^{-x} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1}\right)\right) \cdot e^{-x} \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot x\right) \cdot x + 1}\right)\right) \cdot e^{-x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot x, \color{blue}{x}, 1\right)}\right)\right) \cdot e^{-x} \]

   4. Applied rewrites 8.8%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x\right) \cdot x - 0.5\right) \cdot x, x, 1\right)}}\right)\right) \cdot e^{-x} \]

   6. Applied rewrites 8.8%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right) \cdot x, x, -0.5\right) \cdot x, x, 1\right)}\right)\right) \cdot 1}{e^{x}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right) \cdot x, x, \frac{-1}{2}\right) \cdot x, x, 1\right)}\right)\right) \cdot 1}}{e^{x}} \]

      2. *-rgt-identity 8.8

         \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right) \cdot x, x, -0.5\right) \cdot x, x, 1\right)}\right)\right)}}{e^{x}} \]

   8. Applied rewrites 8.8%

      \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right) \cdot x, x, -0.5\right) \cdot x, x, 1\right)}\right)\right)}}{e^{x}} \]

   if 0.5 < x

   1. Initial program 9.2%

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

   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%

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

Alternative 5: 40.0% accurate, 0.5× speedup

Math

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

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 (fma -0.010416666666666666 (* x x) -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(fma(-0.010416666666666666, (x * x), -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(fma(-0.010416666666666666, Float64(x * x), -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[(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]]]

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 9.2%

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

   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 9.0%

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

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

   1. Initial program 9.2%

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

   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%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 39.9% accurate, 0.6× speedup

Math

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

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 (fma -0.010416666666666666 (* x x) -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(fma(-0.010416666666666666, (x * x), -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(fma(-0.010416666666666666, Float64(x * x), -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[(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]]]

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 9.2%

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

   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 9.0

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

   4. Applied rewrites 9.0%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}}\right)\right) \cdot e^{-x} \]
   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 9.0

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

   6. Applied rewrites 9.0%

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

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

   1. Initial program 9.2%

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

   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%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 39.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod t\_1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod t\_1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))
        (t_1 (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0)))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (* (fmod (exp x) t_1) t_0)
     (* (fmod 1.0 t_1) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = fmod(exp(x), t_1) * t_0;
	} else {
		tmp = fmod(1.0, t_1) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(rem(exp(x), t_1) * t_0);
	else
		tmp = Float64(rem(1.0, t_1) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $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 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = t$95$1}, 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 9.2%

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

   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 9.0

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

   4. Applied rewrites 9.0%

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

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

   1. Initial program 9.2%

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

   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%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 39.8% accurate, 0.6× speedup

Math

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

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)) (exp x))
     (*
      (fmod 1.0 (fma (fma -0.010416666666666666 (* x x) -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)) / exp(x);
	} else {
		tmp = fmod(1.0, fma(fma(-0.010416666666666666, (x * x), -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)) / exp(x));
	else
		tmp = Float64(rem(1.0, fma(fma(-0.010416666666666666, Float64(x * x), -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[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, 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]]]

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 9.2%

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

   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 8.8

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

   4. Applied rewrites 8.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-exp.f64 8.9

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

   6. Applied rewrites 8.9%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 8.9

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

   8. Applied rewrites 8.9%

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

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

   1. Initial program 9.2%

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

   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%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 39.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (* (fmod (exp x) (fma (* x x) -0.25 1.0)) (fma (fma 0.5 x -1.0) x 1.0))
   (*
    (fmod 1.0 (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
    (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = fmod(exp(x), fma((x * x), -0.25, 1.0)) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 9.2%

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

   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 8.8

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 8.2

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

   7. Applied rewrites 8.2%

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

   if 0.5 < x

   1. Initial program 9.2%

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

   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%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 38.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\frac{1}{x} - -1\right) \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fmod (* (- (/ 1.0 x) -1.0) x) (sqrt (cos x))) (exp (- x))))

C

double code(double x) {
	return fmod((((1.0 / x) - -1.0) * 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((((1.0d0 / x) - (-1.0d0)) * x), sqrt(cos(x))) * exp(-x)
end function

Python

def code(x):
	return math.fmod((((1.0 / x) - -1.0) * x), math.sqrt(math.cos(x))) * math.exp(-x)

Julia

function code(x)
	return Float64(rem(Float64(Float64(Float64(1.0 / x) - -1.0) * x), sqrt(cos(x))) * exp(Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 9.2%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 38.5

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

4. Applied rewrites 38.5%

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

   1. lift--.f64 N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

   5. *-rgt-identity N/A

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

   6. metadata-eval N/A

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

   7. rgt-mult-inverse N/A

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

   8. distribute-lft-in N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. fp-cancel-sub-sign N/A

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

   15. metadata-eval N/A

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

   16. lower--.f64 N/A

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

   17. lower-/.f64 38.6

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

6. Applied rewrites 38.6%

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

Alternative 11: 38.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \left(\left(\frac{1}{x} - 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (* (fmod (exp x) (fma (* x x) -0.25 1.0)) (* (- (/ 1.0 x) 1.0) x))
   (*
    (fmod 1.0 (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
    (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = fmod(exp(x), fma((x * x), -0.25, 1.0)) * (((1.0 / x) - 1.0) * x);
	} else {
		tmp = fmod(1.0, fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) * Float64(Float64(Float64(1.0 / x) - 1.0) * x));
	else
		tmp = Float64(rem(1.0, fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 9.2%

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

   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 8.8

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 7.8

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

   7. Applied rewrites 7.8%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sign-sub N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-/.f64 7.8

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

   10. Applied rewrites 7.8%

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

   if 0.5 < x

   1. Initial program 9.2%

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

   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%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 38.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (* (fmod (exp x) (fma (* x x) -0.25 1.0)) (- 1.0 x))
   (*
    (fmod 1.0 (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
    (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = fmod(exp(x), fma((x * x), -0.25, 1.0)) * (1.0 - x);
	} else {
		tmp = fmod(1.0, fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(1.0, fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 9.2%

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

   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 8.8

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 7.8

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

   7. Applied rewrites 7.8%

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

   if 0.5 < x

   1. Initial program 9.2%

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

   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%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 7.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

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 8.8

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

4. Applied rewrites 8.8%

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

5. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f64 7.8

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

7. Applied rewrites 7.8%

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

Alternative 14: 4.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(1 \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fmod 1.0 (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
  (- 1.0 x)))

C

double code(double x) {
	return fmod(1.0, fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * (1.0 - x);
}

Julia

function code(x)
	return Float64(rem(1.0, fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * Float64(1.0 - x))
end

Wolfram

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

Derivation

1. Initial program 9.2%

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

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

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f64 4.7

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

7. Applied rewrites 4.7%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 4.7

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

10. Applied rewrites 4.7%

    \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot \left(1 - x\right) \]
11. Add Preprocessing

Alternative 15: 3.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

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

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f64 4.7

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

7. Applied rewrites 4.7%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 3.9

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

10. Applied rewrites 3.9%

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

Alternative 16: 0.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

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

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f64 4.7

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

7. Applied rewrites 4.7%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 3.9

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

10. Applied rewrites 3.9%

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

11. Taylor expanded in x around inf

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

    1. pow2 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 N/A

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

    4. lift-*.f64 0.0

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

13. Applied rewrites 0.0%

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

Reproduce

herbie shell --seed 2025131 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))