expfmod (used to be hard to sample)

Percentage Accurate: 6.7% → 98.7%

Time: 23.6s

Alternatives: 12

Speedup: 5.0×

• could not determine a ground truth

  1. x = -1.8931153154322546e+191

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: 6.7% 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: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right) + \log \left(\sqrt[3]{e}\right)}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (/
    (fmod
     (exp x)
     (sqrt (+ (log (pow (cbrt (exp (cos x))) 2.0)) (log (cbrt E)))))
    (exp x))
   (/ (fmod x 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fmod(exp(x), sqrt((log(pow(cbrt(exp(cos(x))), 2.0)) + log(cbrt(((double) M_E)))))) / exp(x);
	} else {
		tmp = fmod(x, 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(rem(exp(x), sqrt(Float64(log((cbrt(exp(cos(x))) ^ 2.0)) + log(cbrt(exp(1)))))) / exp(x));
	else
		tmp = Float64(rem(x, 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[Log[N[Power[N[Power[N[Exp[N[Cos[x], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $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 < -1.000000000000002e-309

   1. Initial program 8.3%

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

      1. /-rgt-identity 8.3%

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

      2. associate-/r/ 8.3%

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

      3. exp-neg 8.3%

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

      4. remove-double-neg 8.3%

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

   3. Simplified 8.3%

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

      1. add-log-exp 8.3%

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

      2. add-cube-cbrt 100.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \color{blue}{\left(\left(\sqrt[3]{e^{\cos x}} \cdot \sqrt[3]{e^{\cos x}}\right) \cdot \sqrt[3]{e^{\cos x}}\right)}}\right)\right)}{e^{x}} \]

      3. log-prod 100.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\log \left(\sqrt[3]{e^{\cos x}} \cdot \sqrt[3]{e^{\cos x}}\right) + \log \left(\sqrt[3]{e^{\cos x}}\right)}}\right)\right)}{e^{x}} \]

      4. pow2 100.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \color{blue}{\left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\cos x}}\right)}\right)\right)}{e^{x}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\cos x}}\right)}}\right)\right)}{e^{x}} \]

   7. Taylor expanded in x around 0 100.0%

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

      1. exp-1-e 100.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{e}}\right)}\right)\right)}{e^{x}} \]

   9. Simplified 100.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right) + \log \color{blue}{\left(\sqrt[3]{e}\right)}}\right)\right)}{e^{x}} \]

   if -1.000000000000002e-309 < x

   1. Initial program 5.5%

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

      1. /-rgt-identity 5.5%

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

      2. associate-/r/ 5.5%

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

      3. exp-neg 5.5%

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

      4. remove-double-neg 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in x around 0 39.3%

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

      1. +-commutative 39.3%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in x around inf 97.8%

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

   9. Taylor expanded in x around 0 97.8%

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

Alternative 2: 76.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ t_1 := \sqrt[3]{-1 + \frac{1}{x}}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\left(\left(x \cdot \frac{\frac{-1 + {x}^{-2}}{t\_1}}{{t\_1}^{2}}\right) \bmod t\_0\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\left(x \cdot {\left(\sqrt[3]{1 + \frac{1}{x}}\right)}^{3}\right) \bmod t\_0\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))) (t_1 (cbrt (+ -1.0 (/ 1.0 x)))))
   (if (<= x -7.5e-104)
     (/
      (fmod (* x (/ (/ (+ -1.0 (pow x -2.0)) t_1) (pow t_1 2.0))) t_0)
      (exp x))
     (if (<= x -5e-309)
       (/ (fmod (* x (pow (cbrt (+ 1.0 (/ 1.0 x))) 3.0)) t_0) (exp x))
       (/ (fmod x 1.0) (exp x))))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double t_1 = cbrt((-1.0 + (1.0 / x)));
	double tmp;
	if (x <= -7.5e-104) {
		tmp = fmod((x * (((-1.0 + pow(x, -2.0)) / t_1) / pow(t_1, 2.0))), t_0) / exp(x);
	} else if (x <= -5e-309) {
		tmp = fmod((x * pow(cbrt((1.0 + (1.0 / x))), 3.0)), t_0) / exp(x);
	} else {
		tmp = fmod(x, 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	t_1 = cbrt(Float64(-1.0 + Float64(1.0 / x)))
	tmp = 0.0
	if (x <= -7.5e-104)
		tmp = Float64(rem(Float64(x * Float64(Float64(Float64(-1.0 + (x ^ -2.0)) / t_1) / (t_1 ^ 2.0))), t_0) / exp(x));
	elseif (x <= -5e-309)
		tmp = Float64(rem(Float64(x * (cbrt(Float64(1.0 + Float64(1.0 / x))) ^ 3.0)), t_0) / exp(x));
	else
		tmp = Float64(rem(x, 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, -7.5e-104], N[(N[With[{TMP1 = N[(x * N[(N[(N[(-1.0 + N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-309], N[(N[With[{TMP1 = N[(x * N[Power[N[Power[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $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 3 regimes

2. if x < -7.5e-104

   1. Initial program 23.3%

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

      1. /-rgt-identity 23.3%

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

      2. associate-/r/ 23.3%

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

      3. exp-neg 23.3%

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

      4. remove-double-neg 23.3%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in x around 0 19.3%

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

      1. +-commutative 19.3%

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

   7. Simplified 19.3%

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

   8. Taylor expanded in x around inf 27.3%

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

      1. +-commutative 27.3%

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

      2. flip-+ 27.3%

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

      3. inv-pow 27.3%

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

      4. inv-pow 27.3%

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

      5. pow-prod-up 32.5%

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

      6. metadata-eval 32.5%

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

      7. metadata-eval 32.5%

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

   10. Applied egg-rr 32.5%

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

       1. *-un-lft-identity 32.5%

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

       2. add-cube-cbrt 71.1%

          \[\leadsto \frac{\left(\left(x \cdot \frac{1 \cdot \left({x}^{-2} - 1\right)}{\color{blue}{\left(\sqrt[3]{\frac{1}{x} - 1} \cdot \sqrt[3]{\frac{1}{x} - 1}\right) \cdot \sqrt[3]{\frac{1}{x} - 1}}}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       3. times-frac 70.7%

          \[\leadsto \frac{\left(\left(x \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\frac{1}{x} - 1} \cdot \sqrt[3]{\frac{1}{x} - 1}} \cdot \frac{{x}^{-2} - 1}{\sqrt[3]{\frac{1}{x} - 1}}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       4. pow2 70.7%

          \[\leadsto \frac{\left(\left(x \cdot \left(\frac{1}{\color{blue}{{\left(\sqrt[3]{\frac{1}{x} - 1}\right)}^{2}}} \cdot \frac{{x}^{-2} - 1}{\sqrt[3]{\frac{1}{x} - 1}}\right)\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       5. sub-neg 70.7%

          \[\leadsto \frac{\left(\left(x \cdot \left(\frac{1}{{\left(\sqrt[3]{\color{blue}{\frac{1}{x} + \left(-1\right)}}\right)}^{2}} \cdot \frac{{x}^{-2} - 1}{\sqrt[3]{\frac{1}{x} - 1}}\right)\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       6. metadata-eval 70.7%

          \[\leadsto \frac{\left(\left(x \cdot \left(\frac{1}{{\left(\sqrt[3]{\frac{1}{x} + \color{blue}{-1}}\right)}^{2}} \cdot \frac{{x}^{-2} - 1}{\sqrt[3]{\frac{1}{x} - 1}}\right)\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       7. sub-neg 70.7%

          \[\leadsto \frac{\left(\left(x \cdot \left(\frac{1}{{\left(\sqrt[3]{\frac{1}{x} + -1}\right)}^{2}} \cdot \frac{\color{blue}{{x}^{-2} + \left(-1\right)}}{\sqrt[3]{\frac{1}{x} - 1}}\right)\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       8. metadata-eval 70.7%

          \[\leadsto \frac{\left(\left(x \cdot \left(\frac{1}{{\left(\sqrt[3]{\frac{1}{x} + -1}\right)}^{2}} \cdot \frac{{x}^{-2} + \color{blue}{-1}}{\sqrt[3]{\frac{1}{x} - 1}}\right)\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       9. sub-neg 70.7%

          \[\leadsto \frac{\left(\left(x \cdot \left(\frac{1}{{\left(\sqrt[3]{\frac{1}{x} + -1}\right)}^{2}} \cdot \frac{{x}^{-2} + -1}{\sqrt[3]{\color{blue}{\frac{1}{x} + \left(-1\right)}}}\right)\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       10. metadata-eval 70.7%

           \[\leadsto \frac{\left(\left(x \cdot \left(\frac{1}{{\left(\sqrt[3]{\frac{1}{x} + -1}\right)}^{2}} \cdot \frac{{x}^{-2} + -1}{\sqrt[3]{\frac{1}{x} + \color{blue}{-1}}}\right)\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

   12. Applied egg-rr 70.7%

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

       1. associate-*l/ 71.1%

          \[\leadsto \frac{\left(\left(x \cdot \color{blue}{\frac{1 \cdot \frac{{x}^{-2} + -1}{\sqrt[3]{\frac{1}{x} + -1}}}{{\left(\sqrt[3]{\frac{1}{x} + -1}\right)}^{2}}}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       2. *-lft-identity 71.1%

          \[\leadsto \frac{\left(\left(x \cdot \frac{\color{blue}{\frac{{x}^{-2} + -1}{\sqrt[3]{\frac{1}{x} + -1}}}}{{\left(\sqrt[3]{\frac{1}{x} + -1}\right)}^{2}}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       3. +-commutative 71.1%

          \[\leadsto \frac{\left(\left(x \cdot \frac{\frac{\color{blue}{-1 + {x}^{-2}}}{\sqrt[3]{\frac{1}{x} + -1}}}{{\left(\sqrt[3]{\frac{1}{x} + -1}\right)}^{2}}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       4. +-commutative 71.1%

          \[\leadsto \frac{\left(\left(x \cdot \frac{\frac{-1 + {x}^{-2}}{\sqrt[3]{\color{blue}{-1 + \frac{1}{x}}}}}{{\left(\sqrt[3]{\frac{1}{x} + -1}\right)}^{2}}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       5. +-commutative 71.1%

          \[\leadsto \frac{\left(\left(x \cdot \frac{\frac{-1 + {x}^{-2}}{\sqrt[3]{-1 + \frac{1}{x}}}}{{\left(\sqrt[3]{\color{blue}{-1 + \frac{1}{x}}}\right)}^{2}}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

   14. Simplified 71.1%

       \[\leadsto \frac{\left(\left(x \cdot \color{blue}{\frac{\frac{-1 + {x}^{-2}}{\sqrt[3]{-1 + \frac{1}{x}}}}{{\left(\sqrt[3]{-1 + \frac{1}{x}}\right)}^{2}}}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

   if -7.5e-104 < x < -4.9999999999999995e-309

   1. Initial program 3.1%

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

      1. /-rgt-identity 3.1%

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

      2. associate-/r/ 3.1%

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

      3. exp-neg 3.1%

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

      4. remove-double-neg 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around 0 3.1%

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

      1. +-commutative 3.1%

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

   7. Simplified 3.1%

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

   8. Taylor expanded in x around inf 19.7%

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

      1. add-cube-cbrt 40.3%

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

      2. pow3 41.7%

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

   10. Applied egg-rr 41.7%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 5.5%

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

      1. /-rgt-identity 5.5%

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

      2. associate-/r/ 5.5%

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

      3. exp-neg 5.5%

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

      4. remove-double-neg 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in x around 0 39.3%

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

      1. +-commutative 39.3%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in x around inf 97.8%

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

   9. Taylor expanded in x around 0 97.8%

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

Alternative 3: 76.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{-141}:\\ \;\;\;\;\frac{\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(x + {x}^{-2}\right)\right)}{1 + \left({x}^{-2} + \frac{-1}{x}\right)}\right) \bmod t\_0\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\left(x \cdot {\left(\sqrt[3]{1 + \frac{1}{x}}\right)}^{3}\right) \bmod t\_0\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))))
   (if (<= x -7.6e-141)
     (/
      (fmod
       (/
        (expm1 (log1p (+ x (pow x -2.0))))
        (+ 1.0 (+ (pow x -2.0) (/ -1.0 x))))
       t_0)
      (exp x))
     (if (<= x -5e-309)
       (/ (fmod (* x (pow (cbrt (+ 1.0 (/ 1.0 x))) 3.0)) t_0) (exp x))
       (/ (fmod x 1.0) (exp x))))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double tmp;
	if (x <= -7.6e-141) {
		tmp = fmod((expm1(log1p((x + pow(x, -2.0)))) / (1.0 + (pow(x, -2.0) + (-1.0 / x)))), t_0) / exp(x);
	} else if (x <= -5e-309) {
		tmp = fmod((x * pow(cbrt((1.0 + (1.0 / x))), 3.0)), t_0) / exp(x);
	} else {
		tmp = fmod(x, 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	tmp = 0.0
	if (x <= -7.6e-141)
		tmp = Float64(rem(Float64(expm1(log1p(Float64(x + (x ^ -2.0)))) / Float64(1.0 + Float64((x ^ -2.0) + Float64(-1.0 / x)))), t_0) / exp(x));
	elseif (x <= -5e-309)
		tmp = Float64(rem(Float64(x * (cbrt(Float64(1.0 + Float64(1.0 / x))) ^ 3.0)), t_0) / exp(x));
	else
		tmp = Float64(rem(x, 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -7.6e-141], N[(N[With[{TMP1 = N[(N[(Exp[N[Log[1 + N[(x + N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] / N[(1.0 + N[(N[Power[x, -2.0], $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-309], N[(N[With[{TMP1 = N[(x * N[Power[N[Power[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $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 3 regimes

2. if x < -7.59999999999999973e-141

   1. Initial program 15.5%

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

      1. /-rgt-identity 15.5%

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

      2. associate-/r/ 15.5%

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

      3. exp-neg 15.5%

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

      4. remove-double-neg 15.5%

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

   3. Simplified 15.5%

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

   5. Taylor expanded in x around 0 13.1%

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

      1. +-commutative 13.1%

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

   7. Simplified 13.1%

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

   8. Taylor expanded in x around inf 18.0%

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

      1. *-commutative 18.0%

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

      2. flip3-+ 15.3%

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

      3. associate-*l/ 17.3%

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

      4. metadata-eval 17.3%

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

      5. inv-pow 17.3%

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

      6. pow-pow 20.0%

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

      7. metadata-eval 20.0%

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

      8. metadata-eval 20.0%

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

      9. inv-pow 20.0%

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

      10. inv-pow 20.0%

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

      11. pow-prod-up 22.1%

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

      12. metadata-eval 22.1%

          \[\leadsto \frac{\left(\left(\frac{\left(1 + {x}^{-3}\right) \cdot x}{1 + \left({x}^{\color{blue}{-2}} - 1 \cdot \frac{1}{x}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

      13. *-un-lft-identity 22.1%

          \[\leadsto \frac{\left(\left(\frac{\left(1 + {x}^{-3}\right) \cdot x}{1 + \left({x}^{-2} - \color{blue}{\frac{1}{x}}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

   10. Applied egg-rr 22.1%

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

       1. expm1-log1p-u 37.1%

          \[\leadsto \frac{\left(\left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + {x}^{-3}\right) \cdot x\right)\right)}}{1 + \left({x}^{-2} - \frac{1}{x}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       2. expm1-undefine 37.1%

          \[\leadsto \frac{\left(\left(\frac{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + {x}^{-3}\right) \cdot x\right)} - 1}}{1 + \left({x}^{-2} - \frac{1}{x}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       3. *-commutative 37.1%

          \[\leadsto \frac{\left(\left(\frac{e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + {x}^{-3}\right)}\right)} - 1}{1 + \left({x}^{-2} - \frac{1}{x}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

   12. Applied egg-rr 37.1%

       \[\leadsto \frac{\left(\left(\frac{\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(1 + {x}^{-3}\right)\right)} - 1}}{1 + \left({x}^{-2} - \frac{1}{x}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]
   13. Step-by-step derivation

       1. expm1-define 37.1%

          \[\leadsto \frac{\left(\left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(1 + {x}^{-3}\right)\right)\right)}}{1 + \left({x}^{-2} - \frac{1}{x}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       2. distribute-rgt-in 37.1%

          \[\leadsto \frac{\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1 \cdot x + {x}^{-3} \cdot x}\right)\right)}{1 + \left({x}^{-2} - \frac{1}{x}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       3. *-lft-identity 37.1%

          \[\leadsto \frac{\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + {x}^{-3} \cdot x\right)\right)}{1 + \left({x}^{-2} - \frac{1}{x}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       4. pow-plus 59.2%

          \[\leadsto \frac{\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \color{blue}{{x}^{\left(-3 + 1\right)}}\right)\right)}{1 + \left({x}^{-2} - \frac{1}{x}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

       5. metadata-eval 59.2%

          \[\leadsto \frac{\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(x + {x}^{\color{blue}{-2}}\right)\right)}{1 + \left({x}^{-2} - \frac{1}{x}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

   14. Simplified 59.2%

       \[\leadsto \frac{\left(\left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + {x}^{-2}\right)\right)}}{1 + \left({x}^{-2} - \frac{1}{x}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

   if -7.59999999999999973e-141 < x < -4.9999999999999995e-309

   1. Initial program 3.1%

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

      1. /-rgt-identity 3.1%

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

      2. associate-/r/ 3.1%

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

      3. exp-neg 3.1%

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

      4. remove-double-neg 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around 0 3.1%

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

      1. +-commutative 3.1%

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

   7. Simplified 3.1%

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

   8. Taylor expanded in x around inf 24.3%

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

      1. add-cube-cbrt 38.0%

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

      2. pow3 39.8%

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

   10. Applied egg-rr 39.8%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 5.5%

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

      1. /-rgt-identity 5.5%

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

      2. associate-/r/ 5.5%

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

      3. exp-neg 5.5%

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

      4. remove-double-neg 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in x around 0 39.3%

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

      1. +-commutative 39.3%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in x around inf 97.8%

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

   9. Taylor expanded in x around 0 97.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-141}:\\ \;\;\;\;\frac{\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(x + {x}^{-2}\right)\right)}{1 + \left({x}^{-2} + \frac{-1}{x}\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\left(x \cdot {\left(\sqrt[3]{1 + \frac{1}{x}}\right)}^{3}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-135}:\\ \;\;\;\;\frac{\left(\left(\left(1 - {x}^{-2}\right) \cdot \frac{x}{1 + \frac{-1}{x}}\right) \bmod t\_0\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-296}:\\ \;\;\;\;\frac{\log \left(e^{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod t\_0\right)}\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(x \bmod 1\right)\right|}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))))
   (if (<= x -2e-135)
     (/ (fmod (* (- 1.0 (pow x -2.0)) (/ x (+ 1.0 (/ -1.0 x)))) t_0) (exp x))
     (if (<= x -8.5e-296)
       (/ (log (exp (fmod (* x (+ 1.0 (/ 1.0 x))) t_0))) (exp x))
       (/ (fabs (fmod x 1.0)) (exp x))))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double tmp;
	if (x <= -2e-135) {
		tmp = fmod(((1.0 - pow(x, -2.0)) * (x / (1.0 + (-1.0 / x)))), t_0) / exp(x);
	} else if (x <= -8.5e-296) {
		tmp = log(exp(fmod((x * (1.0 + (1.0 / x))), t_0))) / exp(x);
	} else {
		tmp = fabs(fmod(x, 1.0)) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(cos(x))
    if (x <= (-2d-135)) then
        tmp = mod(((1.0d0 - (x ** (-2.0d0))) * (x / (1.0d0 + ((-1.0d0) / x)))), t_0) / exp(x)
    else if (x <= (-8.5d-296)) then
        tmp = log(exp(mod((x * (1.0d0 + (1.0d0 / x))), t_0))) / exp(x)
    else
        tmp = abs(mod(x, 1.0d0)) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.sqrt(math.cos(x))
	tmp = 0
	if x <= -2e-135:
		tmp = math.fmod(((1.0 - math.pow(x, -2.0)) * (x / (1.0 + (-1.0 / x)))), t_0) / math.exp(x)
	elif x <= -8.5e-296:
		tmp = math.log(math.exp(math.fmod((x * (1.0 + (1.0 / x))), t_0))) / math.exp(x)
	else:
		tmp = math.fabs(math.fmod(x, 1.0)) / math.exp(x)
	return tmp

Julia

function code(x)
	t_0 = sqrt(cos(x))
	tmp = 0.0
	if (x <= -2e-135)
		tmp = Float64(rem(Float64(Float64(1.0 - (x ^ -2.0)) * Float64(x / Float64(1.0 + Float64(-1.0 / x)))), t_0) / exp(x));
	elseif (x <= -8.5e-296)
		tmp = Float64(log(exp(rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), t_0))) / exp(x));
	else
		tmp = Float64(abs(rem(x, 1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2e-135], N[(N[With[{TMP1 = N[(N[(1.0 - N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] * N[(x / N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.5e-296], N[(N[Log[N[Exp[N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0000000000000001e-135

   1. Initial program 15.9%

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

      1. /-rgt-identity 15.9%

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

      2. associate-/r/ 15.9%

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

      3. exp-neg 15.9%

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

      4. remove-double-neg 15.9%

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

   3. Simplified 15.9%

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

   5. Taylor expanded in x around 0 13.4%

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

      1. +-commutative 13.4%

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

   7. Simplified 13.4%

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

   8. Taylor expanded in x around inf 18.4%

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

      1. flip-+ 18.4%

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

      2. associate-*r/ 18.8%

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

      3. metadata-eval 18.8%

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

      4. inv-pow 18.8%

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

      5. inv-pow 18.8%

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

      6. pow-prod-up 21.5%

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

      7. metadata-eval 21.5%

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

   10. Applied egg-rr 21.5%

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

       1. *-commutative 21.5%

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

       2. associate-/l* 39.1%

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

   12. Simplified 39.1%

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

   if -2.0000000000000001e-135 < x < -8.50000000000000018e-296

   1. Initial program 3.1%

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

      1. /-rgt-identity 3.1%

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

      2. associate-/r/ 3.1%

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

      3. exp-neg 3.1%

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

      4. remove-double-neg 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around 0 3.1%

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

      1. +-commutative 3.1%

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

   7. Simplified 3.1%

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

   8. Taylor expanded in x around inf 24.3%

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

      1. +-commutative 24.3%

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

      2. flip-+ 0.1%

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

      3. inv-pow 0.1%

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

      4. inv-pow 0.1%

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

      5. pow-prod-up 1.9%

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

      6. metadata-eval 1.9%

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

      7. metadata-eval 1.9%

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

   10. Applied egg-rr 1.9%

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

       1. add-log-exp 1.9%

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

       2. metadata-eval 1.9%

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

       3. pow-prod-up 0.1%

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

       4. inv-pow 0.1%

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

       5. inv-pow 0.1%

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

       6. metadata-eval 0.1%

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

       7. flip-+ 24.7%

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

   12. Applied egg-rr 24.7%

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

   if -8.50000000000000018e-296 < x

   1. Initial program 5.5%

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

      1. /-rgt-identity 5.5%

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

      2. associate-/r/ 5.5%

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

      3. exp-neg 5.5%

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

      4. remove-double-neg 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in x around 0 39.0%

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

      1. +-commutative 39.0%

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

   7. Simplified 39.0%

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

   8. Taylor expanded in x around inf 97.2%

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

      1. add-sqr-sqrt 96.6%

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

      2. sqrt-unprod 65.5%

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

      3. pow2 65.5%

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

   10. Applied egg-rr 65.5%

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

       1. unpow2 65.5%

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

       2. rem-sqrt-square 97.2%

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

   12. Simplified 97.2%

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

   13. Taylor expanded in x around 0 97.2%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-135}:\\ \;\;\;\;\frac{\left(\left(\left(1 - {x}^{-2}\right) \cdot \frac{x}{1 + \frac{-1}{x}}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-296}:\\ \;\;\;\;\frac{\log \left(e^{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(x \bmod 1\right)\right|}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\left(x \cdot {\left(\sqrt[3]{1 + \frac{1}{x}}\right)}^{3}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-309)
   (/ (fmod (* x (pow (cbrt (+ 1.0 (/ 1.0 x))) 3.0)) (sqrt (cos x))) (exp x))
   (/ (fmod x 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-309) {
		tmp = fmod((x * pow(cbrt((1.0 + (1.0 / x))), 3.0)), sqrt(cos(x))) / exp(x);
	} else {
		tmp = fmod(x, 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-309)
		tmp = Float64(rem(Float64(x * (cbrt(Float64(1.0 + Float64(1.0 / x))) ^ 3.0)), sqrt(cos(x))) / exp(x));
	else
		tmp = Float64(rem(x, 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-309], N[(N[With[{TMP1 = N[(x * N[Power[N[Power[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $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 < -4.9999999999999995e-309

   1. Initial program 8.3%

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

      1. /-rgt-identity 8.3%

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

      2. associate-/r/ 8.3%

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

      3. exp-neg 8.3%

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

      4. remove-double-neg 8.3%

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

   3. Simplified 8.3%

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

   5. Taylor expanded in x around 0 7.3%

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

      1. +-commutative 7.3%

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

   7. Simplified 7.3%

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

   8. Taylor expanded in x around inf 21.7%

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

      1. add-cube-cbrt 39.6%

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

      2. pow3 40.7%

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

   10. Applied egg-rr 40.7%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 5.5%

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

      1. /-rgt-identity 5.5%

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

      2. associate-/r/ 5.5%

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

      3. exp-neg 5.5%

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

      4. remove-double-neg 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in x around 0 39.3%

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

      1. +-commutative 39.3%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in x around inf 97.8%

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

   9. Taylor expanded in x around 0 97.8%

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

Alternative 6: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(\left(\left(1 - {x}^{-2}\right) \cdot \frac{x}{1 + \frac{-1}{x}}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-296}:\\ \;\;\;\;\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(x \bmod 1\right)\right|}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-142)
   (/
    (fmod (* (- 1.0 (pow x -2.0)) (/ x (+ 1.0 (/ -1.0 x)))) (sqrt (cos x)))
    (exp x))
   (if (<= x -8.5e-296)
     (fmod (* x (+ 1.0 (/ 1.0 x))) 1.0)
     (/ (fabs (fmod x 1.0)) (exp x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-142) {
		tmp = fmod(((1.0 - pow(x, -2.0)) * (x / (1.0 + (-1.0 / x)))), sqrt(cos(x))) / exp(x);
	} else if (x <= -8.5e-296) {
		tmp = fmod((x * (1.0 + (1.0 / x))), 1.0);
	} else {
		tmp = fabs(fmod(x, 1.0)) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-142)) then
        tmp = mod(((1.0d0 - (x ** (-2.0d0))) * (x / (1.0d0 + ((-1.0d0) / x)))), sqrt(cos(x))) / exp(x)
    else if (x <= (-8.5d-296)) then
        tmp = mod((x * (1.0d0 + (1.0d0 / x))), 1.0d0)
    else
        tmp = abs(mod(x, 1.0d0)) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-142:
		tmp = math.fmod(((1.0 - math.pow(x, -2.0)) * (x / (1.0 + (-1.0 / x)))), math.sqrt(math.cos(x))) / math.exp(x)
	elif x <= -8.5e-296:
		tmp = math.fmod((x * (1.0 + (1.0 / x))), 1.0)
	else:
		tmp = math.fabs(math.fmod(x, 1.0)) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-142)
		tmp = Float64(rem(Float64(Float64(1.0 - (x ^ -2.0)) * Float64(x / Float64(1.0 + Float64(-1.0 / x)))), sqrt(cos(x))) / exp(x));
	elseif (x <= -8.5e-296)
		tmp = rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), 1.0);
	else
		tmp = Float64(abs(rem(x, 1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-142], N[(N[With[{TMP1 = N[(N[(1.0 - N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] * N[(x / N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.5e-296], N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[Abs[N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e-142

   1. Initial program 15.2%

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

      1. /-rgt-identity 15.2%

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

      2. associate-/r/ 15.2%

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

      3. exp-neg 15.2%

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

      4. remove-double-neg 15.2%

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

   3. Simplified 15.2%

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

   5. Taylor expanded in x around 0 12.8%

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

      1. +-commutative 12.8%

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

   7. Simplified 12.8%

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

   8. Taylor expanded in x around inf 17.6%

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

      1. flip-+ 17.6%

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

      2. associate-*r/ 18.0%

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

      3. metadata-eval 18.0%

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

      4. inv-pow 18.0%

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

      5. inv-pow 18.0%

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

      6. pow-prod-up 22.9%

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

      7. metadata-eval 22.9%

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

   10. Applied egg-rr 22.9%

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

       1. *-commutative 22.9%

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

       2. associate-/l* 37.3%

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

   12. Simplified 37.3%

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

   if -1e-142 < x < -8.50000000000000018e-296

   1. Initial program 3.1%

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

      1. /-rgt-identity 3.1%

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

      2. associate-/r/ 3.1%

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

      3. exp-neg 3.1%

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

      4. remove-double-neg 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 3.1%

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

   7. Taylor expanded in x around 0 3.1%

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

      1. +-commutative 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in x around inf 25.1%

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

   if -8.50000000000000018e-296 < x

   1. Initial program 5.5%

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

      1. /-rgt-identity 5.5%

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

      2. associate-/r/ 5.5%

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

      3. exp-neg 5.5%

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

      4. remove-double-neg 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in x around 0 39.0%

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

      1. +-commutative 39.0%

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

   7. Simplified 39.0%

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

   8. Taylor expanded in x around inf 97.2%

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

      1. add-sqr-sqrt 96.6%

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

      2. sqrt-unprod 65.5%

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

      3. pow2 65.5%

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

   10. Applied egg-rr 65.5%

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

       1. unpow2 65.5%

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

       2. rem-sqrt-square 97.2%

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

   12. Simplified 97.2%

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

   13. Taylor expanded in x around 0 97.2%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(\left(\left(1 - {x}^{-2}\right) \cdot \frac{x}{1 + \frac{-1}{x}}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-296}:\\ \;\;\;\;\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(x \bmod 1\right)\right|}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(\left(x \cdot \frac{-1 + {x}^{-2}}{-1 + \frac{1}{x}}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-296}:\\ \;\;\;\;\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(x \bmod 1\right)\right|}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-142)
   (/ (fmod (* x (/ (+ -1.0 (pow x -2.0)) (+ -1.0 (/ 1.0 x)))) 1.0) (exp x))
   (if (<= x -8.5e-296)
     (fmod (* x (+ 1.0 (/ 1.0 x))) 1.0)
     (/ (fabs (fmod x 1.0)) (exp x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-142) {
		tmp = fmod((x * ((-1.0 + pow(x, -2.0)) / (-1.0 + (1.0 / x)))), 1.0) / exp(x);
	} else if (x <= -8.5e-296) {
		tmp = fmod((x * (1.0 + (1.0 / x))), 1.0);
	} else {
		tmp = fabs(fmod(x, 1.0)) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-142)) then
        tmp = mod((x * (((-1.0d0) + (x ** (-2.0d0))) / ((-1.0d0) + (1.0d0 / x)))), 1.0d0) / exp(x)
    else if (x <= (-8.5d-296)) then
        tmp = mod((x * (1.0d0 + (1.0d0 / x))), 1.0d0)
    else
        tmp = abs(mod(x, 1.0d0)) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-142:
		tmp = math.fmod((x * ((-1.0 + math.pow(x, -2.0)) / (-1.0 + (1.0 / x)))), 1.0) / math.exp(x)
	elif x <= -8.5e-296:
		tmp = math.fmod((x * (1.0 + (1.0 / x))), 1.0)
	else:
		tmp = math.fabs(math.fmod(x, 1.0)) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-142)
		tmp = Float64(rem(Float64(x * Float64(Float64(-1.0 + (x ^ -2.0)) / Float64(-1.0 + Float64(1.0 / x)))), 1.0) / exp(x));
	elseif (x <= -8.5e-296)
		tmp = rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), 1.0);
	else
		tmp = Float64(abs(rem(x, 1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-142], N[(N[With[{TMP1 = N[(x * N[(N[(-1.0 + N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.5e-296], N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[Abs[N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e-142

   1. Initial program 15.2%

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

      1. /-rgt-identity 15.2%

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

      2. associate-/r/ 15.2%

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

      3. exp-neg 15.2%

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

      4. remove-double-neg 15.2%

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

   3. Simplified 15.2%

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

   5. Taylor expanded in x around 0 12.8%

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

      1. +-commutative 12.8%

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

   7. Simplified 12.8%

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

   8. Taylor expanded in x around inf 17.6%

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

      1. +-commutative 17.6%

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

      2. flip-+ 17.6%

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

      3. inv-pow 17.6%

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

      4. inv-pow 17.6%

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

      5. pow-prod-up 32.8%

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

      6. metadata-eval 32.8%

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

      7. metadata-eval 32.8%

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

   10. Applied egg-rr 32.8%

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

   11. Taylor expanded in x around 0 32.8%

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

   if -1e-142 < x < -8.50000000000000018e-296

   1. Initial program 3.1%

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

      1. /-rgt-identity 3.1%

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

      2. associate-/r/ 3.1%

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

      3. exp-neg 3.1%

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

      4. remove-double-neg 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 3.1%

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

   7. Taylor expanded in x around 0 3.1%

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

      1. +-commutative 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in x around inf 25.1%

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

   if -8.50000000000000018e-296 < x

   1. Initial program 5.5%

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

      1. /-rgt-identity 5.5%

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

      2. associate-/r/ 5.5%

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

      3. exp-neg 5.5%

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

      4. remove-double-neg 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in x around 0 39.0%

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

      1. +-commutative 39.0%

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

   7. Simplified 39.0%

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

   8. Taylor expanded in x around inf 97.2%

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

      1. add-sqr-sqrt 96.6%

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

      2. sqrt-unprod 65.5%

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

      3. pow2 65.5%

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

   10. Applied egg-rr 65.5%

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

       1. unpow2 65.5%

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

       2. rem-sqrt-square 97.2%

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

   12. Simplified 97.2%

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

   13. Taylor expanded in x around 0 97.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(\left(x \cdot \frac{-1 + {x}^{-2}}{-1 + \frac{1}{x}}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-296}:\\ \;\;\;\;\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(x \bmod 1\right)\right|}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-296}:\\ \;\;\;\;\frac{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(x \bmod 1\right)\right|}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -8.5e-296)
   (/ (fmod (* x (+ 1.0 (/ 1.0 x))) 1.0) (exp x))
   (/ (fabs (fmod x 1.0)) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -8.5e-296) {
		tmp = fmod((x * (1.0 + (1.0 / x))), 1.0) / exp(x);
	} else {
		tmp = fabs(fmod(x, 1.0)) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-8.5d-296)) then
        tmp = mod((x * (1.0d0 + (1.0d0 / x))), 1.0d0) / exp(x)
    else
        tmp = abs(mod(x, 1.0d0)) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -8.5e-296:
		tmp = math.fmod((x * (1.0 + (1.0 / x))), 1.0) / math.exp(x)
	else:
		tmp = math.fabs(math.fmod(x, 1.0)) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -8.5e-296)
		tmp = Float64(rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), 1.0) / exp(x));
	else
		tmp = Float64(abs(rem(x, 1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -8.5e-296], N[(N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.50000000000000018e-296

   1. Initial program 8.4%

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

      1. /-rgt-identity 8.4%

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

      2. associate-/r/ 8.4%

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

      3. exp-neg 8.4%

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

      4. remove-double-neg 8.4%

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

   3. Simplified 8.4%

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

   5. Taylor expanded in x around 0 7.4%

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

      1. +-commutative 7.4%

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

   7. Simplified 7.4%

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

   8. Taylor expanded in x around inf 21.9%

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

   9. Taylor expanded in x around 0 21.9%

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

   if -8.50000000000000018e-296 < x

   1. Initial program 5.5%

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

      1. /-rgt-identity 5.5%

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

      2. associate-/r/ 5.5%

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

      3. exp-neg 5.5%

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

      4. remove-double-neg 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in x around 0 39.0%

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

      1. +-commutative 39.0%

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

   7. Simplified 39.0%

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

   8. Taylor expanded in x around inf 97.2%

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

      1. add-sqr-sqrt 96.6%

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

      2. sqrt-unprod 65.5%

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

      3. pow2 65.5%

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

   10. Applied egg-rr 65.5%

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

       1. unpow2 65.5%

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

       2. rem-sqrt-square 97.2%

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

   12. Simplified 97.2%

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

   13. Taylor expanded in x around 0 97.2%

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

Alternative 9: 67.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-309)
   (/ (fmod (* x (+ 1.0 (/ 1.0 x))) 1.0) (exp x))
   (/ (fmod x 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-309) {
		tmp = fmod((x * (1.0 + (1.0 / x))), 1.0) / exp(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 <= (-5d-309)) then
        tmp = mod((x * (1.0d0 + (1.0d0 / x))), 1.0d0) / exp(x)
    else
        tmp = mod(x, 1.0d0) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-309:
		tmp = math.fmod((x * (1.0 + (1.0 / x))), 1.0) / math.exp(x)
	else:
		tmp = math.fmod(x, 1.0) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-309)
		tmp = Float64(rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), 1.0) / exp(x));
	else
		tmp = Float64(rem(x, 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-309], N[(N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $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 < -4.9999999999999995e-309

   1. Initial program 8.3%

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

      1. /-rgt-identity 8.3%

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

      2. associate-/r/ 8.3%

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

      3. exp-neg 8.3%

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

      4. remove-double-neg 8.3%

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

   3. Simplified 8.3%

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

   5. Taylor expanded in x around 0 7.3%

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

      1. +-commutative 7.3%

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

   7. Simplified 7.3%

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

   8. Taylor expanded in x around inf 21.7%

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

   9. Taylor expanded in x around 0 21.7%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 5.5%

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

      1. /-rgt-identity 5.5%

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

      2. associate-/r/ 5.5%

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

      3. exp-neg 5.5%

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

      4. remove-double-neg 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in x around 0 39.3%

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

      1. +-commutative 39.3%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in x around inf 97.8%

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

   9. Taylor expanded in x around 0 97.8%

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

Alternative 10: 67.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-309)
   (fmod (* x (+ 1.0 (/ 1.0 x))) 1.0)
   (/ (fmod x 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-309) {
		tmp = fmod((x * (1.0 + (1.0 / x))), 1.0);
	} 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 <= (-5d-309)) then
        tmp = mod((x * (1.0d0 + (1.0d0 / x))), 1.0d0)
    else
        tmp = mod(x, 1.0d0) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-309:
		tmp = math.fmod((x * (1.0 + (1.0 / x))), 1.0)
	else:
		tmp = math.fmod(x, 1.0) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-309)
		tmp = rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), 1.0);
	else
		tmp = Float64(rem(x, 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-309], N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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 < -4.9999999999999995e-309

   1. Initial program 8.3%

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

      1. /-rgt-identity 8.3%

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

      2. associate-/r/ 8.3%

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

      3. exp-neg 8.3%

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

      4. remove-double-neg 8.3%

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

   3. Simplified 8.3%

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

   5. Taylor expanded in x around 0 8.3%

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

   6. Taylor expanded in x around 0 6.0%

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

   7. Taylor expanded in x around 0 6.0%

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

      1. +-commutative 7.3%

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

   9. Simplified 6.0%

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

   10. Taylor expanded in x around inf 20.4%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 5.5%

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

      1. /-rgt-identity 5.5%

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

      2. associate-/r/ 5.5%

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

      3. exp-neg 5.5%

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

      4. remove-double-neg 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in x around 0 39.3%

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

      1. +-commutative 39.3%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in x around inf 97.8%

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

   9. Taylor expanded in x around 0 97.8%

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

Alternative 11: 66.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-309) (fmod (* x (+ 1.0 (/ 1.0 x))) 1.0) (fmod x 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -5e-309) {
		tmp = fmod((x * (1.0 + (1.0 / x))), 1.0);
	} else {
		tmp = fmod(x, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-309)) then
        tmp = mod((x * (1.0d0 + (1.0d0 / x))), 1.0d0)
    else
        tmp = mod(x, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-309:
		tmp = math.fmod((x * (1.0 + (1.0 / x))), 1.0)
	else:
		tmp = math.fmod(x, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-309)
		tmp = rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), 1.0);
	else
		tmp = rem(x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-309], N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999995e-309

   1. Initial program 8.3%

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

      1. /-rgt-identity 8.3%

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

      2. associate-/r/ 8.3%

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

      3. exp-neg 8.3%

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

      4. remove-double-neg 8.3%

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

   3. Simplified 8.3%

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

   5. Taylor expanded in x around 0 8.3%

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

   6. Taylor expanded in x around 0 6.0%

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

   7. Taylor expanded in x around 0 6.0%

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

      1. +-commutative 7.3%

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

   9. Simplified 6.0%

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

   10. Taylor expanded in x around inf 20.4%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 5.5%

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

      1. /-rgt-identity 5.5%

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

      2. associate-/r/ 5.5%

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

      3. exp-neg 5.5%

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

      4. remove-double-neg 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in x around 0 4.8%

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

   6. Taylor expanded in x around 0 4.7%

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

   7. Taylor expanded in x around 0 37.9%

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

      1. +-commutative 39.3%

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

   9. Simplified 37.9%

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

   10. Taylor expanded in x around inf 96.5%

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

Alternative 12: 59.5% 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 6.6%

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

   1. /-rgt-identity 6.6%

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

   2. associate-/r/ 6.6%

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

   3. exp-neg 6.6%

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

   4. remove-double-neg 6.6%

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

3. Simplified 6.6%

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

5. Taylor expanded in x around 0 6.1%

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

6. Taylor expanded in x around 0 5.2%

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

7. Taylor expanded in x around 0 26.3%

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

   1. +-commutative 27.7%

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

9. Simplified 26.3%

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

10. Taylor expanded in x around inf 62.3%

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

Reproduce

herbie shell --seed 2024118 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))