expfmod (used to be hard to sample)

Percentage Accurate: 6.5% → 98.8%

Time: 18.0s

Alternatives: 6

Speedup: 5.0×

• could not determine a ground truth

  1. x = -5.386225707464393e+296

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: 6.5% 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: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right) + \log \left(\sqrt[3]{e}\right)}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (/
    (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 <= -5e-310) {
		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 <= -5e-310)
		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, -5e-310], 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 < -4.999999999999985e-310

   1. Initial program 5.1%

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

      1. /-rgt-identity 5.1%

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

      2. associate-/r/ 5.1%

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

      3. exp-neg 5.1%

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

      4. remove-double-neg 5.1%

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

   3. Simplified 5.1%

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

      1. add-log-exp 5.1%

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

      2. add-cube-cbrt 97.9%

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

      3. log-prod 97.9%

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

      4. pow2 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in x around 0 97.9%

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

      1. exp-1-e 97.9%

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

   9. Simplified 97.9%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.2%

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

      1. /-rgt-identity 5.2%

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

      2. associate-/r/ 5.2%

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

      3. exp-neg 5.2%

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

      4. remove-double-neg 5.2%

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

   3. Simplified 5.2%

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

   5. Taylor expanded in x around 0 36.4%

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

      1. +-commutative 36.4%

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

   7. Simplified 36.4%

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

   8. Taylor expanded in x around inf 98.5%

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

   9. Taylor expanded in x around 0 98.5%

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

   10. Taylor expanded in x around 0 98.6%

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

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{\cos x}}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\log \left({t\_0}^{2}\right) + \log t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (exp (cos x)))))
   (if (<= x -5e-310)
     (fmod 1.0 (sqrt (+ (log (pow t_0 2.0)) (log t_0))))
     (/ (fmod x 1.0) (exp x)))))

C

double code(double x) {
	double t_0 = cbrt(exp(cos(x)));
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(1.0, sqrt((log(pow(t_0, 2.0)) + log(t_0))));
	} else {
		tmp = fmod(x, 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(exp(cos(x)))
	tmp = 0.0
	if (x <= -5e-310)
		tmp = rem(1.0, sqrt(Float64(log((t_0 ^ 2.0)) + log(t_0))));
	else
		tmp = Float64(rem(x, 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[N[Cos[x], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, -5e-310], N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[(N[Log[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision] + N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, 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 < -4.999999999999985e-310

   1. Initial program 5.1%

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

      1. /-rgt-identity 5.1%

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

      2. associate-/r/ 5.1%

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

      3. exp-neg 5.1%

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

      4. remove-double-neg 5.1%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in x around 0 4.1%

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

   6. Taylor expanded in x around 0 3.5%

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

      1. add-log-exp 5.1%

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

      2. add-cube-cbrt 97.9%

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

      3. log-prod 97.9%

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

      4. pow2 97.9%

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

   8. Applied egg-rr 97.3%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.2%

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

      1. /-rgt-identity 5.2%

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

      2. associate-/r/ 5.2%

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

      3. exp-neg 5.2%

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

      4. remove-double-neg 5.2%

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

   3. Simplified 5.2%

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

   5. Taylor expanded in x around 0 36.4%

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

      1. +-commutative 36.4%

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

   7. Simplified 36.4%

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

   8. Taylor expanded in x around inf 98.5%

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

   9. Taylor expanded in x around 0 98.5%

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

   10. Taylor expanded in x around 0 98.6%

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

Alternative 3: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(\frac{e^{x}}{\sqrt{e}} \cdot \frac{e}{\sqrt{e}}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (/ (fmod (* (/ (exp x) (sqrt E)) (/ E (sqrt E))) (sqrt (cos x))) (exp x))
   (/ (fmod x 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(((exp(x) / sqrt(((double) M_E))) * (((double) M_E) / sqrt(((double) M_E)))), sqrt(cos(x))) / exp(x);
	} else {
		tmp = fmod(x, 1.0) / exp(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = math.fmod(((math.exp(x) / math.sqrt(math.e)) * (math.e / math.sqrt(math.e))), math.sqrt(math.cos(x))) / 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-310)
		tmp = Float64(rem(Float64(Float64(exp(x) / sqrt(exp(1))) * Float64(exp(1) / sqrt(exp(1)))), 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-310], N[(N[With[{TMP1 = N[(N[(N[Exp[x], $MachinePrecision] / N[Sqrt[E], $MachinePrecision]), $MachinePrecision] * N[(E / N[Sqrt[E], $MachinePrecision]), $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 < -4.999999999999985e-310

   1. Initial program 5.1%

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

      1. /-rgt-identity 5.1%

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

      2. associate-/r/ 5.1%

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

      3. exp-neg 5.1%

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

      4. remove-double-neg 5.1%

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

   3. Simplified 5.1%

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

      1. add-cube-cbrt 5.1%

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

      2. pow3 5.1%

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

   6. Applied egg-rr 5.1%

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

      1. rem-cube-cbrt 5.1%

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

      2. *-un-lft-identity 5.1%

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

      3. pow-exp 5.1%

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

      4. e-exp-1 5.1%

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

      5. expm1-log1p-u 5.1%

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

      6. log1p-define 5.1%

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

      7. +-commutative 5.1%

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

      8. expm1-undefine 5.1%

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

      9. add-exp-log 5.1%

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

      10. pow-div 5.1%

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

      11. pow1 5.1%

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

      12. e-exp-1 5.1%

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

      13. pow-exp 5.1%

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

      14. *-un-lft-identity 5.1%

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

      15. exp-sum 5.1%

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

      16. e-exp-1 5.1%

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

      17. add-sqr-sqrt 96.3%

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

      18. times-frac 96.3%

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

   8. Applied egg-rr 96.3%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.2%

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

      1. /-rgt-identity 5.2%

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

      2. associate-/r/ 5.2%

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

      3. exp-neg 5.2%

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

      4. remove-double-neg 5.2%

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

   3. Simplified 5.2%

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

   5. Taylor expanded in x around 0 36.4%

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

      1. +-commutative 36.4%

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

   7. Simplified 36.4%

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

   8. Taylor expanded in x around inf 98.5%

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

   9. Taylor expanded in x around 0 98.5%

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

   10. Taylor expanded in x around 0 98.6%

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

Alternative 4: 66.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-309)
   (/ (fmod (* x (+ 1.0 (/ 1.0 x))) (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 * (1.0 + (1.0 / x))), sqrt(cos(x))) / 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))), sqrt(cos(x))) / 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))), math.sqrt(math.cos(x))) / 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))), 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[(1.0 + N[(1.0 / x), $MachinePrecision]), $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 5.1%

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

      1. /-rgt-identity 5.1%

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

      2. associate-/r/ 5.1%

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

      3. exp-neg 5.1%

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

      4. remove-double-neg 5.1%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in x around 0 4.7%

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

      1. +-commutative 4.7%

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

   7. Simplified 4.7%

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

   8. Taylor expanded in x around inf 22.9%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 5.2%

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

      1. /-rgt-identity 5.2%

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

      2. associate-/r/ 5.2%

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

      3. exp-neg 5.2%

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

      4. remove-double-neg 5.2%

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

   3. Simplified 5.2%

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

   5. Taylor expanded in x around 0 36.4%

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

      1. +-commutative 36.4%

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

   7. Simplified 36.4%

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

   8. Taylor expanded in x around inf 98.5%

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

   9. Taylor expanded in x around 0 98.5%

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

   10. Taylor expanded in x around 0 98.6%

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

Alternative 5: 59.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.2%

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

   1. /-rgt-identity 5.2%

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

   2. associate-/r/ 5.2%

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

   3. exp-neg 5.2%

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

   4. remove-double-neg 5.2%

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

3. Simplified 5.2%

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

5. Taylor expanded in x around 0 24.7%

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

   1. +-commutative 24.7%

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

7. Simplified 24.7%

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

8. Taylor expanded in x around inf 63.2%

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

9. Taylor expanded in x around 0 63.2%

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

10. Taylor expanded in x around 0 63.2%

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

Alternative 6: 23.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.2%

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

   1. /-rgt-identity 5.2%

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

   2. associate-/r/ 5.2%

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

   3. exp-neg 5.2%

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

   4. remove-double-neg 5.2%

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

3. Simplified 5.2%

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

5. Taylor expanded in x around 0 4.5%

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

6. Taylor expanded in x around 0 4.3%

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

7. Taylor expanded in x around 0 4.9%

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

8. Taylor expanded in x around 0 23.4%

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

Reproduce

herbie shell --seed 2024179 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))