expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 98.9%

Time: 12.3s

Alternatives: 5

Speedup: 5.0×

• could not determine a ground truth

  1. x = -4.731269156424536e+115

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: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{0.5}\right)\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{{\left({e}^{\left(x \cdot x - {t\_0}^{2}\right)}\right)}^{\left(\frac{1}{x + t\_0}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (fmod (exp x) (pow (cos x) 0.5)))))
   (if (<= x -2e-310)
     (/ 1.0 (pow (pow E (- (* x x) (pow t_0 2.0))) (/ 1.0 (+ x t_0))))
     (/ (fmod x 1.0) (exp x)))))

C

double code(double x) {
	double t_0 = log(fmod(exp(x), pow(cos(x), 0.5)));
	double tmp;
	if (x <= -2e-310) {
		tmp = 1.0 / pow(pow(((double) M_E), ((x * x) - pow(t_0, 2.0))), (1.0 / (x + t_0)));
	} else {
		tmp = fmod(x, 1.0) / exp(x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.log(math.fmod(math.exp(x), math.pow(math.cos(x), 0.5)))
	tmp = 0
	if x <= -2e-310:
		tmp = 1.0 / math.pow(math.pow(math.e, ((x * x) - math.pow(t_0, 2.0))), (1.0 / (x + t_0)))
	else:
		tmp = math.fmod(x, 1.0) / math.exp(x)
	return tmp

Julia

function code(x)
	t_0 = log(rem(exp(x), (cos(x) ^ 0.5)))
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(1.0 / ((exp(1) ^ Float64(Float64(x * x) - (t_0 ^ 2.0))) ^ Float64(1.0 / Float64(x + t_0))));
	else
		tmp = Float64(rem(x, 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Power[N[Cos[x], $MachinePrecision], 0.5], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2e-310], N[(1.0 / N[Power[N[Power[E, N[(N[(x * x), $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(x + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 6.2%

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

      1. exp-neg N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\sqrt{\cos x}\right)\right)\right)\right) \]

      7. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{fmod.f64}\left(\left(e^{x}\right), \color{blue}{\left(\sqrt{\cos x}\right)}\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\color{blue}{\cos x}}\right)\right)\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left({\cos x}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\cos x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      11. cos-lowering-cos.f64 6.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), \frac{1}{2}\right)\right)\right)\right) \]

   4. Applied egg-rr 6.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left({\cos x}^{0.5}\right)\right)}}} \]
   5. Step-by-step derivation

      1. rem-exp-log N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(e^{\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}\right)}\right)\right) \]

      2. unpow1 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(e^{\log \left({\left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}\right)}^{1}\right)}\right)\right) \]

      3. log-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(e^{1 \cdot \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}\right)}\right)\right) \]

      4. exp-prod N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(e^{1}\right)}^{\color{blue}{\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}\right)}}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\left(e^{1}\right), \color{blue}{\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}\right)}\right)\right) \]

      6. exp-1-e N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{E}\left(\right), \log \color{blue}{\left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}\right)}\right)\right) \]

      7. E-lowering-E.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \log \color{blue}{\left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}\right)}\right)\right) \]

      8. log-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \left(\log \left(e^{x}\right) - \color{blue}{\log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}\right)\right)\right) \]

      9. rem-log-exp N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \left(x - \log \color{blue}{\left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \mathsf{\_.f64}\left(x, \color{blue}{\log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}\right)\right)\right) \]

      11. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      13. fmod-lowering-fmod.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      14. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      15. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left({\cos x}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\cos x, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      17. cos-lowering-cos.f64 6.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 6.2%

      \[\leadsto \frac{1}{\color{blue}{{e}^{\left(x - \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{0.5}\right)\right)\right)}}} \]
   7. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\mathsf{E}\left(\right)}^{\left(\frac{x \cdot x - \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right) \cdot \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}{\color{blue}{x + \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}}\right)}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\mathsf{E}\left(\right)}^{\left(\left(x \cdot x - \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right) \cdot \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)\right) \cdot \color{blue}{\frac{1}{x + \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}}\right)}\right)\right) \]

      3. pow-unpow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left({\mathsf{E}\left(\right)}^{\left(x \cdot x - \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right) \cdot \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)\right)}\right)}^{\color{blue}{\left(\frac{1}{x + \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}\right)}}\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\left({\mathsf{E}\left(\right)}^{\left(x \cdot x - \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right) \cdot \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)\right)}\right), \color{blue}{\left(\frac{1}{x + \log \left(\left(e^{x}\right) \bmod \left({\cos x}^{\frac{1}{2}}\right)\right)}\right)}\right)\right) \]

   8. Applied egg-rr 100.0%

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

   if -1.999999999999994e-310 < x

   1. Initial program 5.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\color{blue}{\left(1 + x\right)}, \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\left(x + 1\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x}\right)\right)\right) \]

      2. +-lowering-+.f64 38.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x}\right)\right)\right) \]

   5. Simplified 38.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\color{blue}{x}, \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 98.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(x, \color{blue}{1}\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 98.6%

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

       1. exp-neg N/A

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

       2. un-div-inv N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x \bmod 1\right), \color{blue}{\left(e^{x}\right)}\right) \]

       4. fmod-lowering-fmod.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(x, 1\right), \left(e^{\color{blue}{x}}\right)\right) \]

       5. exp-lowering-exp.f64 98.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(x, 1\right), \mathsf{exp.f64}\left(x\right)\right) \]

   13. Applied egg-rr 98.6%

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

Alternative 2: 62.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-103}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(0.16666666666666666 + \frac{\frac{1}{x} + \left(0.5 + \frac{1}{x \cdot x}\right)}{x}\right)\right)\right) \bmod 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.75e-103)
   (*
    (fmod
     (*
      (* x x)
      (*
       x
       (+ 0.16666666666666666 (/ (+ (/ 1.0 x) (+ 0.5 (/ 1.0 (* x x)))) x))))
     1.0)
    (exp (- 0.0 x)))
   (/ (fmod x 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.75e-103) {
		tmp = fmod(((x * x) * (x * (0.16666666666666666 + (((1.0 / x) + (0.5 + (1.0 / (x * x)))) / x)))), 1.0) * exp((0.0 - x));
	} else {
		tmp = fmod(x, 1.0) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.75d-103)) then
        tmp = mod(((x * x) * (x * (0.16666666666666666d0 + (((1.0d0 / x) + (0.5d0 + (1.0d0 / (x * x)))) / x)))), 1.0d0) * exp((0.0d0 - x))
    else
        tmp = mod(x, 1.0d0) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1.75e-103:
		tmp = math.fmod(((x * x) * (x * (0.16666666666666666 + (((1.0 / x) + (0.5 + (1.0 / (x * x)))) / x)))), 1.0) * math.exp((0.0 - x))
	else:
		tmp = math.fmod(x, 1.0) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.75e-103)
		tmp = Float64(rem(Float64(Float64(x * x) * Float64(x * Float64(0.16666666666666666 + Float64(Float64(Float64(1.0 / x) + Float64(0.5 + Float64(1.0 / Float64(x * x)))) / x)))), 1.0) * exp(Float64(0.0 - x)));
	else
		tmp = Float64(rem(x, 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.75e-103], N[(N[With[{TMP1 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.16666666666666666 + N[(N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75000000000000008e-103

   1. Initial program 11.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{1}\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 11.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)}, 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot x\right)\right)\right)\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

      7. *-lowering-*.f64 8.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right)\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

   8. Simplified 8.8%

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

   9. Taylor expanded in x around -inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\color{blue}{\left(-1 \cdot \left({x}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right)}, 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\left(\mathsf{neg}\left({x}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x}\right)\right)\right) \]

       2. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\left(\mathsf{neg}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\left(\mathsf{neg}\left(\left({x}^{2} \cdot x\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\left(\mathsf{neg}\left({x}^{2} \cdot \left(x \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\left({x}^{2} \cdot \left(\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\left({x}^{2} \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right) \cdot x\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right) \cdot x\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right) \cdot x\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right) \cdot x\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

       11. distribute-rgt-neg-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right)\right)\right), 1\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

   11. Simplified 17.3%

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

   if -1.75000000000000008e-103 < x

   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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\color{blue}{\left(1 + x\right)}, \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\left(x + 1\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x}\right)\right)\right) \]

      2. +-lowering-+.f64 29.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x}\right)\right)\right) \]

   5. Simplified 29.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\color{blue}{x}, \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 72.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(x, \color{blue}{1}\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 72.6%

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

       1. exp-neg N/A

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

       2. un-div-inv N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x \bmod 1\right), \color{blue}{\left(e^{x}\right)}\right) \]

       4. fmod-lowering-fmod.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(x, 1\right), \left(e^{\color{blue}{x}}\right)\right) \]

       5. exp-lowering-exp.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(x, 1\right), \mathsf{exp.f64}\left(x\right)\right) \]

   13. Applied egg-rr 72.6%

       \[\leadsto \color{blue}{\frac{\left(x \bmod 1\right)}{e^{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-103}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(0.16666666666666666 + \frac{\frac{1}{x} + \left(0.5 + \frac{1}{x \cdot x}\right)}{x}\right)\right)\right) \bmod 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.2% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (fmod x 1.0) (exp x)))

C

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

Fortran

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

Python

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

Julia

function code(x)
	return Float64(rem(x, 1.0) / exp(x))
end

Wolfram

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

Derivation

1. Initial program 5.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\color{blue}{\left(1 + x\right)}, \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\left(x + 1\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x}\right)\right)\right) \]

   2. +-lowering-+.f64 26.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x}\right)\right)\right) \]

5. Simplified 26.5%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\color{blue}{x}, \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
7. Step-by-step derivation

8. Simplified 63.2%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(x, \color{blue}{1}\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
10. Step-by-step derivation

11. Simplified 63.2%

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

    1. exp-neg N/A

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

    2. un-div-inv N/A

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

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(x \bmod 1\right), \color{blue}{\left(e^{x}\right)}\right) \]

    4. fmod-lowering-fmod.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(x, 1\right), \left(e^{\color{blue}{x}}\right)\right) \]

    5. exp-lowering-exp.f64 63.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(x, 1\right), \mathsf{exp.f64}\left(x\right)\right) \]

13. Applied egg-rr 63.2%

    \[\leadsto \color{blue}{\frac{\left(x \bmod 1\right)}{e^{x}}} \]
14. Add Preprocessing

Alternative 4: 58.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\color{blue}{\left(1 + x\right)}, \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\left(x + 1\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x}\right)\right)\right) \]

   2. +-lowering-+.f64 26.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x}\right)\right)\right) \]

5. Simplified 26.5%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(\color{blue}{x}, \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
7. Step-by-step derivation

8. Simplified 63.2%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{fmod.f64}\left(x, \color{blue}{1}\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]
10. Step-by-step derivation

11. Simplified 63.2%

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

12. Taylor expanded in x around 0

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

    1. fmod-lowering-fmod.f64 62.1%

       \[\leadsto \mathsf{fmod.f64}\left(x, \color{blue}{1}\right) \]

14. Simplified 62.1%

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

Alternative 5: 23.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.8%

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

3. Taylor expanded in x around 0

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

   1. fmod-lowering-fmod.f64 N/A

      \[\leadsto \mathsf{fmod.f64}\left(\left(e^{x}\right), \color{blue}{\left(\sqrt{\cos x}\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\color{blue}{\cos x}}\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right) \]

   4. cos-lowering-cos.f64 4.8%

      \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right) \]

5. Simplified 4.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{1}\right) \]
7. Step-by-step derivation

8. Simplified 4.8%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{1}, 1\right) \]
10. Step-by-step derivation

11. Simplified 24.8%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))