expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 63.0%

Time: 16.2s

Alternatives: 9

Speedup: 5.0×

• could not determine a ground truth

  1. x = -5.1001131269516345e+233

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 63.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (/ (fmod (exp x) (* 3.0 (log (cbrt E)))) (exp x))
   (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = fmod(exp(x), (3.0 * log(cbrt(((double) M_E))))) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(rem(exp(x), Float64(3.0 * log(cbrt(exp(1))))) / exp(x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(3.0 * N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.4%

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

      1. exp-neg 8.5%

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

      2. associate-*r/ 8.5%

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

      3. *-rgt-identity 8.5%

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

   3. Simplified 8.5%

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

      1. add-log-exp 8.5%

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

      2. add-cube-cbrt 56.6%

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

      3. log-prod 56.7%

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

   5. Applied egg-rr 56.1%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]
   6. Step-by-step derivation

      1. log-pow 56.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      2. distribute-lft1-in 56.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]

      3. metadata-eval 56.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      4. cos-0 56.1%

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

      5. metadata-eval 56.1%

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

      6. exp-1-e 56.1%

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

   7. Simplified 56.1%

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

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

   1. Initial program 0.0%

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

      1. exp-neg 0.0%

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

      2. associate-*r/ 0.0%

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

      3. *-rgt-identity 0.0%

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

   3. Simplified 0.0%

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

   5. Applied egg-rr 98.1%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   6. Step-by-step derivation

      1. +-lft-identity 98.1%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 98.1%

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

      3. metadata-eval 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in x around 0 98.2%

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

4. Final simplification 65.0%

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

Alternative 2: 62.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (* (fmod (exp x) (* 3.0 (log (cbrt E)))) (- 1.0 x))
   (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = fmod(exp(x), (3.0 * log(cbrt(((double) M_E))))) * (1.0 - x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(rem(exp(x), Float64(3.0 * log(cbrt(exp(1))))) * Float64(1.0 - x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(3.0 * N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 8.4%

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

   2. Taylor expanded in x around 0 7.3%

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

      1. associate-*r* 7.3%

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

      2. neg-mul-1 7.3%

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

      3. distribute-lft1-in 7.3%

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

   4. Simplified 7.3%

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

      1. add-log-exp 8.4%

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

      2. add-cube-cbrt 56.3%

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

      3. log-prod 56.4%

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

   6. Applied egg-rr 55.1%

      \[\leadsto \left(\left(-x\right) + 1\right) \cdot \left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. log-pow 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      2. distribute-lft1-in 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]

      3. metadata-eval 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      4. cos-0 55.8%

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

      5. metadata-eval 55.8%

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

      6. exp-1-e 55.8%

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

   8. Simplified 55.1%

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

   if 1 < x

   1. Initial program 0.0%

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

      1. exp-neg 0.0%

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

      2. associate-*r/ 0.0%

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

      3. *-rgt-identity 0.0%

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

   3. Simplified 0.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   6. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 100.0%

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

      3. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 64.4%

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

Alternative 3: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 200:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 200.0) (fmod (exp x) (* 3.0 (log (cbrt E)))) (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 200.0) {
		tmp = fmod(exp(x), (3.0 * log(cbrt(((double) M_E)))));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 200.0)
		tmp = rem(exp(x), Float64(3.0 * log(cbrt(exp(1)))));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 200.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(3.0 * N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 200

   1. Initial program 8.4%

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

   2. Taylor expanded in x around 0 7.3%

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

      1. associate-*r* 7.3%

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

      2. neg-mul-1 7.3%

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

      3. distribute-lft1-in 7.3%

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

   4. Simplified 7.3%

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

      1. add-log-exp 8.4%

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

      2. add-cube-cbrt 56.3%

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

      3. log-prod 56.4%

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

   6. Applied egg-rr 55.1%

      \[\leadsto \left(\left(-x\right) + 1\right) \cdot \left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. log-pow 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      2. distribute-lft1-in 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]

      3. metadata-eval 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      4. cos-0 55.8%

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

      5. metadata-eval 55.8%

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

      6. exp-1-e 55.8%

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

   8. Simplified 55.1%

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

   9. Taylor expanded in x around 0 6.6%

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. unpow1/3 54.6%

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

   11. Simplified 54.6%

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

   if 200 < x

   1. Initial program 0.0%

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

      1. exp-neg 0.0%

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

      2. associate-*r/ 0.0%

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

      3. *-rgt-identity 0.0%

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

   3. Simplified 0.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   6. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 100.0%

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

      3. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 200:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]

Alternative 4: 26.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(1 + {x}^{2} \cdot -0.25\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 500.0)
   (/ (fmod (exp x) (+ 1.0 (* (pow x 2.0) -0.25))) (exp x))
   (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 500.0) {
		tmp = fmod(exp(x), (1.0 + (pow(x, 2.0) * -0.25))) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 500.0d0) then
        tmp = mod(exp(x), (1.0d0 + ((x ** 2.0d0) * (-0.25d0)))) / exp(x)
    else
        tmp = mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 500.0:
		tmp = math.fmod(math.exp(x), (1.0 + (math.pow(x, 2.0) * -0.25))) / math.exp(x)
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 500.0)
		tmp = Float64(rem(exp(x), Float64(1.0 + Float64((x ^ 2.0) * -0.25))) / exp(x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 500.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 500

   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. exp-neg 8.4%

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

      2. associate-*r/ 8.4%

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

      3. *-rgt-identity 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in x around 0 8.2%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]
   5. Step-by-step derivation

      1. *-commutative 8.2%

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

   6. Simplified 8.2%

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

   if 500 < x

   1. Initial program 0.0%

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

      1. exp-neg 0.0%

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

      2. associate-*r/ 0.0%

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

      3. *-rgt-identity 0.0%

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

   3. Simplified 0.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   6. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 100.0%

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

      3. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(1 + {x}^{2} \cdot -0.25\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]

Alternative 5: 25.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 20:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 20.0) (/ 1.0 (/ (exp x) (fmod (exp x) 1.0))) (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 20.0) {
		tmp = 1.0 / (exp(x) / fmod(exp(x), 1.0));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 20.0d0) then
        tmp = 1.0d0 / (exp(x) / mod(exp(x), 1.0d0))
    else
        tmp = mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 20.0:
		tmp = 1.0 / (math.exp(x) / math.fmod(math.exp(x), 1.0))
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 20.0)
		tmp = Float64(1.0 / Float64(exp(x) / rem(exp(x), 1.0)));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 20.0], N[(1.0 / N[(N[Exp[x], $MachinePrecision] / N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 20

   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. exp-neg 8.4%

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

      2. associate-*r/ 8.4%

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

      3. *-rgt-identity 8.4%

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

   3. Simplified 8.4%

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

      1. add-log-exp 8.4%

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

      2. add-cube-cbrt 56.3%

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

      3. log-prod 56.4%

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

   5. Applied egg-rr 55.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]
   6. Step-by-step derivation

      1. log-pow 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      2. distribute-lft1-in 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]

      3. metadata-eval 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      4. cos-0 55.8%

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

      5. metadata-eval 55.8%

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

      6. exp-1-e 55.8%

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

   7. Simplified 55.8%

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

      1. *-commutative 55.8%

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

      2. add-log-exp 55.8%

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

      3. exp-to-pow 55.8%

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

      4. pow3 55.8%

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

      5. add-cube-cbrt 7.9%

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

      6. log-E 7.9%

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

      7. frac-2neg 7.9%

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

      8. div-inv 7.9%

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

   9. Applied egg-rr 7.9%

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

       1. un-div-inv 7.9%

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

       2. clear-num 7.9%

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

       3. add-sqr-sqrt 4.0%

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

       4. sqrt-unprod 4.1%

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

       5. sqr-neg 4.1%

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

       6. sqrt-unprod 4.1%

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

       7. add-sqr-sqrt 4.1%

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

       8. add-sqr-sqrt 0.0%

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

       9. sqrt-unprod 7.9%

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

       10. sqr-neg 7.9%

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

       11. sqrt-unprod 7.9%

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

       12. add-sqr-sqrt 7.9%

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

   11. Applied egg-rr 7.9%

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

   if 20 < x

   1. Initial program 0.0%

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

      1. exp-neg 0.0%

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

      2. associate-*r/ 0.0%

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

      3. *-rgt-identity 0.0%

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

   3. Simplified 0.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   6. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 100.0%

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

      3. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 20:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]

Alternative 6: 25.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 20:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 20.0) (/ (fmod (exp x) 1.0) (exp x)) (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 20.0) {
		tmp = fmod(exp(x), 1.0) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 20.0d0) then
        tmp = mod(exp(x), 1.0d0) / exp(x)
    else
        tmp = mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 20.0:
		tmp = math.fmod(math.exp(x), 1.0) / math.exp(x)
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 20.0)
		tmp = Float64(rem(exp(x), 1.0) / exp(x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 20.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 20

   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. exp-neg 8.4%

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

      2. associate-*r/ 8.4%

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

      3. *-rgt-identity 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in x around 0 7.9%

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

   if 20 < x

   1. Initial program 0.0%

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

      1. exp-neg 0.0%

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

      2. associate-*r/ 0.0%

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

      3. *-rgt-identity 0.0%

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

   3. Simplified 0.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   6. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 100.0%

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

      3. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 20:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]

Alternative 7: 25.4% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (* (- 1.0 x) (fmod (exp x) 1.0)) (fmod 1.0 1.0)))

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   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. exp-neg 8.4%

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

      2. associate-*r/ 8.4%

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

      3. *-rgt-identity 8.4%

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

   3. Simplified 8.4%

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

      1. add-log-exp 8.4%

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

      2. add-cube-cbrt 56.3%

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

      3. log-prod 56.4%

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

   5. Applied egg-rr 55.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]
   6. Step-by-step derivation

      1. log-pow 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      2. distribute-lft1-in 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]

      3. metadata-eval 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      4. cos-0 55.8%

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

      5. metadata-eval 55.8%

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

      6. exp-1-e 55.8%

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

   7. Simplified 55.8%

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

   8. Taylor expanded in x around 0 7.2%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right)\right) + \left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 7.2%

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right)} + \left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right) \]

      2. distribute-lft1-in 7.1%

         \[\leadsto \color{blue}{\left(-1 \cdot x + 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right)} \]

      3. log-pow 7.1%

         \[\leadsto \left(-1 \cdot x + 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log e\right)}\right)\right) \]

      4. log-E 7.1%

         \[\leadsto \left(-1 \cdot x + 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \left(0.3333333333333333 \cdot \color{blue}{1}\right)\right)\right) \]

      5. metadata-eval 7.1%

         \[\leadsto \left(-1 \cdot x + 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \color{blue}{0.3333333333333333}\right)\right) \]

      6. metadata-eval 7.1%

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

      7. distribute-rgt1-in 7.2%

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

      8. *-lft-identity 7.2%

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

      9. distribute-rgt-out 7.1%

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

      10. mul-1-neg 7.1%

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

      11. unsub-neg 7.1%

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

   10. Simplified 7.1%

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

   if 1 < x

   1. Initial program 0.0%

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

      1. exp-neg 0.0%

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

      2. associate-*r/ 0.0%

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

      3. *-rgt-identity 0.0%

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

   3. Simplified 0.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   6. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 100.0%

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

      3. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 26.4%

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

Alternative 8: 24.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 20:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 20.0) (fmod (exp x) 1.0) (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 20.0) {
		tmp = fmod(exp(x), 1.0);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 20.0d0) then
        tmp = mod(exp(x), 1.0d0)
    else
        tmp = mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 20.0:
		tmp = math.fmod(math.exp(x), 1.0)
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 20.0)
		tmp = rem(exp(x), 1.0);
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 20.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 20

   1. Initial program 8.4%

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

   2. Taylor expanded in x around 0 7.3%

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

      1. associate-*r* 7.3%

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

      2. neg-mul-1 7.3%

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

      3. distribute-lft1-in 7.3%

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

   4. Simplified 7.3%

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

      1. add-log-exp 8.4%

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

      2. add-cube-cbrt 56.3%

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

      3. log-prod 56.4%

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

   6. Applied egg-rr 55.1%

      \[\leadsto \left(\left(-x\right) + 1\right) \cdot \left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. log-pow 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      2. distribute-lft1-in 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]

      3. metadata-eval 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      4. cos-0 55.8%

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

      5. metadata-eval 55.8%

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

      6. exp-1-e 55.8%

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

   8. Simplified 55.1%

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

   9. Taylor expanded in x around 0 6.6%

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. log-pow 6.6%

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log e\right)}\right)\right) \]

       2. log-E 6.6%

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(3 \cdot \left(0.3333333333333333 \cdot \color{blue}{1}\right)\right)\right) \]

       3. metadata-eval 6.6%

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(3 \cdot \color{blue}{0.3333333333333333}\right)\right) \]

       4. metadata-eval 6.6%

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

   11. Simplified 6.6%

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

   if 20 < x

   1. Initial program 0.0%

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

      1. exp-neg 0.0%

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

      2. associate-*r/ 0.0%

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

      3. *-rgt-identity 0.0%

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

   3. Simplified 0.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   6. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 100.0%

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

      3. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 20:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]

Alternative 9: 22.9% 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 6.7%

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

   1. exp-neg 6.7%

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

   2. associate-*r/ 6.7%

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

   3. *-rgt-identity 6.7%

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

3. Simplified 6.7%

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

5. Applied egg-rr 24.0%

   \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
6. Step-by-step derivation

   1. +-lft-identity 24.0%

      \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

   2. cos-0 24.0%

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

   3. metadata-eval 24.0%

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

7. Simplified 24.0%

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

8. Taylor expanded in x around 0 24.0%

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

9. Final simplification 24.0%

   \[\leadsto \left(1 \bmod 1\right) \]

Reproduce

herbie shell --seed 2023310 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))