expfmod (used to be hard to sample)

Percentage Accurate: 7.1% → 97.5%

Time: 10.7s

Alternatives: 11

Speedup: 3.9×

• could not determine a ground truth

  1. x = -1764466277580753.0

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

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

Python

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

Julia

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

Wolfram

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

Initial Program: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    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: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;1 \cdot \left(1 \bmod \left(\left(\left({\left(\cos \left(2 \cdot x\right) + 1\right)}^{0.25} \cdot {0.5}^{0.125}\right) \cdot {0.5}^{0.0625}\right) \cdot {0.5}^{0.0625}\right)\right)\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310)
   (*
    1.0
    (fmod
     1.0
     (*
      (*
       (* (pow (+ (cos (* 2.0 x)) 1.0) 0.25) (pow 0.5 0.125))
       (pow 0.5 0.0625))
      (pow 0.5 0.0625))))
   (if (<= x 0.04)
     (* (fma (fma 0.5 x -1.0) x 1.0) (fmod (* (fma 0.5 x 1.0) x) 1.0))
     (* (fmod 1.0 1.0) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = 1.0 * fmod(1.0, (((pow((cos((2.0 * x)) + 1.0), 0.25) * pow(0.5, 0.125)) * pow(0.5, 0.0625)) * pow(0.5, 0.0625)));
	} else if (x <= 0.04) {
		tmp = fma(fma(0.5, x, -1.0), x, 1.0) * fmod((fma(0.5, x, 1.0) * x), 1.0);
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(1.0 * rem(1.0, Float64(Float64(Float64((Float64(cos(Float64(2.0 * x)) + 1.0) ^ 0.25) * (0.5 ^ 0.125)) * (0.5 ^ 0.0625)) * (0.5 ^ 0.0625))));
	elseif (x <= 0.04)
		tmp = Float64(fma(fma(0.5, x, -1.0), x, 1.0) * rem(Float64(fma(0.5, x, 1.0) * x), 1.0));
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4e-310], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = N[(N[(N[(N[Power[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[0.5, 0.125], $MachinePrecision]), $MachinePrecision] * N[Power[0.5, 0.0625], $MachinePrecision]), $MachinePrecision] * N[Power[0.5, 0.0625], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.04], N[(N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. sqr-cos-a N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. distribute-lft-in N/A

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

      13. lift-+.f64 N/A

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

      14. pow-prod-down N/A

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

      15. sqr-pow N/A

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

      16. lift-pow.f64 N/A

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

      17. associate-*l* N/A

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

      18. sqr-pow N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

   7. Applied rewrites 96.1%

      \[\leadsto \left(1 \bmod \color{blue}{\left({0.5}^{0.0625} \cdot \left({0.5}^{0.0625} \cdot \left({0.5}^{0.125} \cdot {\left(\cos \left(2 \cdot x\right) + 1\right)}^{0.25}\right)\right)\right)}\right) \cdot e^{-x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left({\frac{1}{2}}^{\frac{1}{16}} \cdot \left({\frac{1}{2}}^{\frac{1}{16}} \cdot \left({\frac{1}{2}}^{\frac{1}{8}} \cdot {\left(\cos \left(2 \cdot x\right) + 1\right)}^{\frac{1}{4}}\right)\right)\right)\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.5%

       \[\leadsto \left(1 \bmod \left({0.5}^{0.0625} \cdot \left({0.5}^{0.0625} \cdot \left({0.5}^{0.125} \cdot {\left(\cos \left(2 \cdot x\right) + 1\right)}^{0.25}\right)\right)\right)\right) \cdot \color{blue}{1} \]

   if -3.999999999999988e-310 < x < 0.0400000000000000008

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 6.5

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

   8. Applied rewrites 6.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 6.5

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

   11. Applied rewrites 6.5%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 99.7%

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

   if 0.0400000000000000008 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;1 \cdot \left(1 \bmod \left(\left(\left({\left(\cos \left(2 \cdot x\right) + 1\right)}^{0.25} \cdot {0.5}^{0.125}\right) \cdot {0.5}^{0.0625}\right) \cdot {0.5}^{0.0625}\right)\right)\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2e-10)
   (* (fma (fma 0.5 x -1.0) x 1.0) (fmod (* (fma 0.5 x 1.0) x) 1.0))
   (* (- 1.0 x) (fmod (+ 1.0 x) 1.0))))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2e-10) {
		tmp = fma(fma(0.5, x, -1.0), x, 1.0) * fmod((fma(0.5, x, 1.0) * x), 1.0);
	} else {
		tmp = (1.0 - x) * fmod((1.0 + x), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2e-10)
		tmp = Float64(fma(fma(0.5, x, -1.0), x, 1.0) * rem(Float64(fma(0.5, x, 1.0) * x), 1.0));
	else
		tmp = Float64(Float64(1.0 - x) * rem(Float64(1.0 + x), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := 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] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2e-10], N[(N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * N[With[{TMP1 = N[(1.0 + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $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))) < 2.00000000000000007e-10

   1. Initial program 4.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 4.9

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

   8. Applied rewrites 4.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 4.9

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

   11. Applied rewrites 4.9%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 53.7%

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

   if 2.00000000000000007e-10 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 10.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in x around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 6.8

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

   8. Applied rewrites 6.8%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 91.6

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

   11. Applied rewrites 91.6%

       \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod 1\right) \cdot \left(1 - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.4%

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

Alternative 3: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left({0.5}^{0.25} \cdot {2}^{0.25}\right)\right)\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310)
   (* (exp (- x)) (fmod (exp x) (* (pow 0.5 0.25) (pow 2.0 0.25))))
   (if (<= x 0.04)
     (* (fma (fma 0.5 x -1.0) x 1.0) (fmod (* (fma 0.5 x 1.0) x) 1.0))
     (* (fmod 1.0 1.0) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = exp(-x) * fmod(exp(x), (pow(0.5, 0.25) * pow(2.0, 0.25)));
	} else if (x <= 0.04) {
		tmp = fma(fma(0.5, x, -1.0), x, 1.0) * fmod((fma(0.5, x, 1.0) * x), 1.0);
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(exp(Float64(-x)) * rem(exp(x), Float64((0.5 ^ 0.25) * (2.0 ^ 0.25))));
	elseif (x <= 0.04)
		tmp = Float64(fma(fma(0.5, x, -1.0), x, 1.0) * rem(Float64(fma(0.5, x, 1.0) * x), 1.0));
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4e-310], N[(N[Exp[(-x)], $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[Power[0.5, 0.25], $MachinePrecision] * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.04], N[(N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 9.0%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. cos-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. unpow-prod-down N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 15.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({\left(1 + \cos \left(2 \cdot x\right)\right)}^{0.25} \cdot {0.5}^{0.25}\right)}\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 15.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({\color{blue}{2}}^{0.25} \cdot {0.5}^{0.25}\right)\right) \cdot e^{-x} \]

   if -3.999999999999988e-310 < x < 0.0400000000000000008

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 6.5

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

   8. Applied rewrites 6.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 6.5

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

   11. Applied rewrites 6.5%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 99.7%

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

   if 0.0400000000000000008 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left({0.5}^{0.25} \cdot {2}^{0.25}\right)\right)\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left({0.5}^{0.25} \cdot {2}^{0.25}\right)\right) \cdot 1\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310)
   (* (fmod (exp x) (* (pow 0.5 0.25) (pow 2.0 0.25))) 1.0)
   (if (<= x 0.04)
     (* (fma (fma 0.5 x -1.0) x 1.0) (fmod (* (fma 0.5 x 1.0) x) 1.0))
     (* (fmod 1.0 1.0) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = fmod(exp(x), (pow(0.5, 0.25) * pow(2.0, 0.25))) * 1.0;
	} else if (x <= 0.04) {
		tmp = fma(fma(0.5, x, -1.0), x, 1.0) * fmod((fma(0.5, x, 1.0) * x), 1.0);
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(rem(exp(x), Float64((0.5 ^ 0.25) * (2.0 ^ 0.25))) * 1.0);
	elseif (x <= 0.04)
		tmp = Float64(fma(fma(0.5, x, -1.0), x, 1.0) * rem(Float64(fma(0.5, x, 1.0) * x), 1.0));
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[Power[0.5, 0.25], $MachinePrecision] * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, 0.04], N[(N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 9.0%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. cos-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. unpow-prod-down N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 15.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({\left(1 + \cos \left(2 \cdot x\right)\right)}^{0.25} \cdot {0.5}^{0.25}\right)}\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 15.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({\color{blue}{2}}^{0.25} \cdot {0.5}^{0.25}\right)\right) \cdot e^{-x} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 12.2%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left({2}^{0.25} \cdot {0.5}^{0.25}\right)\right) \cdot \color{blue}{1} \]

   if -3.999999999999988e-310 < x < 0.0400000000000000008

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 6.5

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

   8. Applied rewrites 6.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 6.5

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

   11. Applied rewrites 6.5%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 99.7%

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

   if 0.0400000000000000008 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left({0.5}^{0.25} \cdot {2}^{0.25}\right)\right) \cdot 1\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310)
   (* (fmod (exp x) 1.0) (exp (- x)))
   (if (<= x 0.04)
     (* (fma (fma 0.5 x -1.0) x 1.0) (fmod (* (fma 0.5 x 1.0) x) 1.0))
     (* (fmod 1.0 1.0) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = fmod(exp(x), 1.0) * exp(-x);
	} else if (x <= 0.04) {
		tmp = fma(fma(0.5, x, -1.0), x, 1.0) * fmod((fma(0.5, x, 1.0) * x), 1.0);
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(rem(exp(x), 1.0) * exp(Float64(-x)));
	elseif (x <= 0.04)
		tmp = Float64(fma(fma(0.5, x, -1.0), x, 1.0) * rem(Float64(fma(0.5, x, 1.0) * x), 1.0));
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.04], N[(N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 9.0%

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

   if -3.999999999999988e-310 < x < 0.0400000000000000008

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 6.5

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

   8. Applied rewrites 6.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 6.5

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

   11. Applied rewrites 6.5%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 99.7%

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

   if 0.0400000000000000008 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \cdot \left(\left(e^{x}\right) \bmod 1\right)\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310)
   (*
    (fma (fma (fma -0.16666666666666666 x 0.5) x -1.0) x 1.0)
    (fmod (exp x) 1.0))
   (if (<= x 0.04)
     (* (fma (fma 0.5 x -1.0) x 1.0) (fmod (* (fma 0.5 x 1.0) x) 1.0))
     (* (fmod 1.0 1.0) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0) * fmod(exp(x), 1.0);
	} else if (x <= 0.04) {
		tmp = fma(fma(0.5, x, -1.0), x, 1.0) * fmod((fma(0.5, x, 1.0) * x), 1.0);
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0) * rem(exp(x), 1.0));
	elseif (x <= 0.04)
		tmp = Float64(fma(fma(0.5, x, -1.0), x, 1.0) * rem(Float64(fma(0.5, x, 1.0) * x), 1.0));
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4e-310], N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.04], N[(N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 9.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 8.2

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

   8. Applied rewrites 8.2%

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

   if -3.999999999999988e-310 < x < 0.0400000000000000008

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 6.5

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

   8. Applied rewrites 6.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 6.5

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

   11. Applied rewrites 6.5%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 99.7%

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

   if 0.0400000000000000008 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \cdot \left(\left(e^{x}\right) \bmod 1\right)\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310)
   (*
    (fmod (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0) 1.0)
    (fma (fma (fma -0.16666666666666666 x 0.5) x -1.0) x 1.0))
   (if (<= x 0.04)
     (* (fma (fma 0.5 x -1.0) x 1.0) (fmod (* (fma 0.5 x 1.0) x) 1.0))
     (* (fmod 1.0 1.0) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = fmod(fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0), 1.0) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0);
	} else if (x <= 0.04) {
		tmp = fma(fma(0.5, x, -1.0), x, 1.0) * fmod((fma(0.5, x, 1.0) * x), 1.0);
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(rem(fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0), 1.0) * fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0));
	elseif (x <= 0.04)
		tmp = Float64(fma(fma(0.5, x, -1.0), x, 1.0) * rem(Float64(fma(0.5, x, 1.0) * x), 1.0));
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4e-310], N[(N[With[{TMP1 = N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.04], N[(N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 9.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 8.2

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

   8. Applied rewrites 8.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 8.1

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

   11. Applied rewrites 8.1%

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

   if -3.999999999999988e-310 < x < 0.0400000000000000008

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 6.5

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

   8. Applied rewrites 6.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 6.5

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

   11. Applied rewrites 6.5%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 99.7%

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

   if 0.0400000000000000008 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right)\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod 1\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;t\_0 \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 x -1.0) x 1.0)))
   (if (<= x -4e-310)
     (* (fmod (fma (fma 0.5 x 1.0) x 1.0) 1.0) t_0)
     (if (<= x 0.04)
       (* t_0 (fmod (* (fma 0.5 x 1.0) x) 1.0))
       (* (fmod 1.0 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = fma(fma(0.5, x, -1.0), x, 1.0);
	double tmp;
	if (x <= -4e-310) {
		tmp = fmod(fma(fma(0.5, x, 1.0), x, 1.0), 1.0) * t_0;
	} else if (x <= 0.04) {
		tmp = t_0 * fmod((fma(0.5, x, 1.0) * x), 1.0);
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(fma(0.5, x, -1.0), x, 1.0)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(rem(fma(fma(0.5, x, 1.0), x, 1.0), 1.0) * t_0);
	elseif (x <= 0.04)
		tmp = Float64(t_0 * rem(Float64(fma(0.5, x, 1.0) * x), 1.0));
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]}, If[LessEqual[x, -4e-310], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 0.04], N[(t$95$0 * N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 9.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 7.8

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

   8. Applied rewrites 7.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 8.1

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

   11. Applied rewrites 8.1%

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

   if -3.999999999999988e-310 < x < 0.0400000000000000008

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 6.5

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

   8. Applied rewrites 6.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 6.5

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

   11. Applied rewrites 6.5%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 99.7%

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

   if 0.0400000000000000008 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 24.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - x) * mod((1.0d0 + x), 1.0d0)
end function

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 6.1%

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

6. Taylor expanded in x around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 5.3

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

8. Applied rewrites 5.3%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 24.9

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

11. Applied rewrites 24.9%

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

12. Final simplification 24.9%

    \[\leadsto \left(1 - x\right) \cdot \left(\left(1 + x\right) \bmod 1\right) \]
13. Add Preprocessing

Alternative 10: 24.3% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (fmod (+ 1.0 x) 1.0) 1.0))

C

double code(double x) {
	return fmod((1.0 + x), 1.0) * 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod((1.0d0 + x), 1.0d0) * 1.0d0
end function

Python

def code(x):
	return math.fmod((1.0 + x), 1.0) * 1.0

Julia

function code(x)
	return Float64(rem(Float64(1.0 + x), 1.0) * 1.0)
end

Wolfram

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

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 6.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 4.9%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 24.5

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

11. Applied rewrites 24.5%

    \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod 1\right) \cdot 1 \]
12. Add Preprocessing

Alternative 11: 23.2% accurate, 3.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (fmod 1.0 1.0) 1.0))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(1.0d0, 1.0d0) * 1.0d0
end function

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 6.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 4.9%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
10. Step-by-step derivation

11. Applied rewrites 24.2%

    \[\leadsto \left(\color{blue}{1} \bmod 1\right) \cdot 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024242 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))