expfmod (used to be hard to sample)

Percentage Accurate: 6.8% → 98.7%

Time: 16.4s

Alternatives: 7

Speedup: 4.9×

• could not determine a ground truth

  1. x = -1.406232936304372e+157

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: 6.8% 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

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

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 (sqrt (cos x))) (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, sqrt(cos(x))) / 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, sqrt(cos(x))) / 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 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, 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 6.0%

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

      1. /-rgt-identity 6.0%

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

      2. associate-/r/ 6.0%

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

      3. exp-neg 6.0%

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

      4. remove-double-neg 6.0%

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

   3. Simplified 6.0%

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

      1. add-log-exp 6.0%

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

      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 -4.999999999999985e-310 < x

   1. Initial program 7.0%

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

      1. /-rgt-identity 7.0%

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

      2. associate-/r/ 7.0%

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

      3. exp-neg 7.1%

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

      4. remove-double-neg 7.1%

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

   3. Simplified 7.1%

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

   5. Taylor expanded in x around 0 34.6%

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

      1. +-commutative 34.6%

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

   7. Simplified 34.6%

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

   8. Taylor expanded in x around inf 97.1%

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

Alternative 2: 97.7% accurate, 0.5× speedup

Math

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

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 (sqrt (cos x))) (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, sqrt(cos(x))) / 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, sqrt(cos(x))) / 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 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, 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 6.0%

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

      1. /-rgt-identity 6.0%

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

      2. associate-/r/ 6.0%

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

      3. exp-neg 6.0%

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

      4. remove-double-neg 6.0%

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

   3. Simplified 6.0%

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

   5. Taylor expanded in x around 0 4.9%

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

   6. Taylor expanded in x around 0 3.2%

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

      1. add-log-exp 6.0%

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

      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%

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

   8. Applied egg-rr 99.1%

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

   if -4.999999999999985e-310 < x

   1. Initial program 7.0%

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

      1. /-rgt-identity 7.0%

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

      2. associate-/r/ 7.0%

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

      3. exp-neg 7.1%

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

      4. remove-double-neg 7.1%

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

   3. Simplified 7.1%

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

   5. Taylor expanded in x around 0 34.6%

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

      1. +-commutative 34.6%

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

   7. Simplified 34.6%

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

   8. Taylor expanded in x around inf 97.1%

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

Alternative 3: 66.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6e-309)
   (/ (fmod (* x (+ 1.0 (/ 1.0 x))) 1.0) (exp x))
   (/ (fmod x (sqrt (cos x))) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -6e-309) {
		tmp = fmod((x * (1.0 + (1.0 / x))), 1.0) / exp(x);
	} else {
		tmp = fmod(x, sqrt(cos(x))) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-6d-309)) then
        tmp = mod((x * (1.0d0 + (1.0d0 / x))), 1.0d0) / exp(x)
    else
        tmp = mod(x, sqrt(cos(x))) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -6e-309:
		tmp = math.fmod((x * (1.0 + (1.0 / x))), 1.0) / math.exp(x)
	else:
		tmp = math.fmod(x, math.sqrt(math.cos(x))) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -6e-309)
		tmp = Float64(rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), 1.0) / exp(x));
	else
		tmp = Float64(rem(x, sqrt(cos(x))) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -6e-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 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.000000000000001e-309

   1. Initial program 6.0%

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

      1. /-rgt-identity 6.0%

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

      2. associate-/r/ 6.0%

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

      3. exp-neg 6.0%

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

      4. remove-double-neg 6.0%

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

   3. Simplified 6.0%

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

   5. Taylor expanded in x around 0 5.3%

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

      1. +-commutative 5.3%

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

   7. Simplified 5.3%

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

      1. add-cube-cbrt 5.3%

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

      2. associate-*l* 5.3%

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

      3. pow1/3 5.3%

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

      4. pow1/2 5.3%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\color{blue}{\left({\cos x}^{0.5}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

      5. pow-pow 5.3%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left(\color{blue}{{\cos x}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

      6. metadata-eval 5.3%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

      7. cbrt-prod 5.3%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{\cos x} \cdot \sqrt{\cos x}}}\right)\right)}{e^{x}} \]

      8. add-sqr-sqrt 5.3%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\cos x}}\right)\right)}{e^{x}} \]

   9. Applied egg-rr 5.3%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\cos x}^{0.16666666666666666} \cdot \sqrt[3]{\cos x}\right)}\right)}{e^{x}} \]
   10. Step-by-step derivation

       1. unpow1/3 5.3%

          \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \color{blue}{{\cos x}^{0.3333333333333333}}\right)\right)}{e^{x}} \]

       2. metadata-eval 5.3%

          \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot {\cos x}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}}\right)\right)}{e^{x}} \]

       3. pow-sqr 5.3%

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

       4. cube-mult 5.3%

          \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\left({\cos x}^{0.16666666666666666}\right)}^{3}\right)}\right)}{e^{x}} \]

   11. Simplified 5.3%

       \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\left({\cos x}^{0.16666666666666666}\right)}^{3}\right)}\right)}{e^{x}} \]

   12. Taylor expanded in x around 0 5.3%

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

   13. Taylor expanded in x around inf 20.4%

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

   if -6.000000000000001e-309 < x

   1. Initial program 7.0%

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

      1. /-rgt-identity 7.0%

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

      2. associate-/r/ 7.0%

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

      3. exp-neg 7.1%

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

      4. remove-double-neg 7.1%

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

   3. Simplified 7.1%

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

   5. Taylor expanded in x around 0 34.6%

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

      1. +-commutative 34.6%

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

   7. Simplified 34.6%

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

   8. Taylor expanded in x around inf 97.1%

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

Alternative 4: 66.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6e-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 <= -6e-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 <= (-6d-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 <= -6e-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 <= -6e-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, -6e-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 < -6.000000000000001e-309

   1. Initial program 6.0%

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

      1. /-rgt-identity 6.0%

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

      2. associate-/r/ 6.0%

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

      3. exp-neg 6.0%

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

      4. remove-double-neg 6.0%

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

   3. Simplified 6.0%

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

   5. Taylor expanded in x around 0 5.3%

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

      1. +-commutative 5.3%

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

   7. Simplified 5.3%

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

      1. add-cube-cbrt 5.3%

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

      2. associate-*l* 5.3%

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

      3. pow1/3 5.3%

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

      4. pow1/2 5.3%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\color{blue}{\left({\cos x}^{0.5}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

      5. pow-pow 5.3%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left(\color{blue}{{\cos x}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

      6. metadata-eval 5.3%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

      7. cbrt-prod 5.3%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{\cos x} \cdot \sqrt{\cos x}}}\right)\right)}{e^{x}} \]

      8. add-sqr-sqrt 5.3%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\cos x}}\right)\right)}{e^{x}} \]

   9. Applied egg-rr 5.3%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\cos x}^{0.16666666666666666} \cdot \sqrt[3]{\cos x}\right)}\right)}{e^{x}} \]
   10. Step-by-step derivation

       1. unpow1/3 5.3%

          \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \color{blue}{{\cos x}^{0.3333333333333333}}\right)\right)}{e^{x}} \]

       2. metadata-eval 5.3%

          \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot {\cos x}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}}\right)\right)}{e^{x}} \]

       3. pow-sqr 5.3%

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

       4. cube-mult 5.3%

          \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\left({\cos x}^{0.16666666666666666}\right)}^{3}\right)}\right)}{e^{x}} \]

   11. Simplified 5.3%

       \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\left({\cos x}^{0.16666666666666666}\right)}^{3}\right)}\right)}{e^{x}} \]

   12. Taylor expanded in x around 0 5.3%

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

   13. Taylor expanded in x around inf 20.4%

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

   if -6.000000000000001e-309 < x

   1. Initial program 7.0%

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

      1. /-rgt-identity 7.0%

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

      2. associate-/r/ 7.0%

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

      3. exp-neg 7.1%

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

      4. remove-double-neg 7.1%

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

   3. Simplified 7.1%

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

   5. Taylor expanded in x around 0 34.6%

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

      1. +-commutative 34.6%

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

   7. Simplified 34.6%

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

      1. add-cube-cbrt 34.6%

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

      2. associate-*l* 34.6%

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

      3. pow1/3 34.6%

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

      4. pow1/2 34.6%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\color{blue}{\left({\cos x}^{0.5}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

      5. pow-pow 34.6%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left(\color{blue}{{\cos x}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

      6. metadata-eval 34.6%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

      7. cbrt-prod 34.6%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{\cos x} \cdot \sqrt{\cos x}}}\right)\right)}{e^{x}} \]

      8. add-sqr-sqrt 34.6%

         \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\cos x}}\right)\right)}{e^{x}} \]

   9. Applied egg-rr 34.6%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\cos x}^{0.16666666666666666} \cdot \sqrt[3]{\cos x}\right)}\right)}{e^{x}} \]
   10. Step-by-step derivation

       1. unpow1/3 34.6%

          \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \color{blue}{{\cos x}^{0.3333333333333333}}\right)\right)}{e^{x}} \]

       2. metadata-eval 34.6%

          \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot {\cos x}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}}\right)\right)}{e^{x}} \]

       3. pow-sqr 34.6%

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

       4. cube-mult 34.6%

          \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\left({\cos x}^{0.16666666666666666}\right)}^{3}\right)}\right)}{e^{x}} \]

   11. Simplified 34.6%

       \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\left({\cos x}^{0.16666666666666666}\right)}^{3}\right)}\right)}{e^{x}} \]

   12. Taylor expanded in x around 0 34.6%

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

   13. Taylor expanded in x around inf 97.1%

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

Alternative 5: 59.4% 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 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 23.1%

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

   1. +-commutative 23.1%

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

7. Simplified 23.1%

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

   1. add-cube-cbrt 23.1%

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

   2. associate-*l* 23.1%

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

   3. pow1/3 23.1%

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

   4. pow1/2 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\color{blue}{\left({\cos x}^{0.5}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

   5. pow-pow 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left(\color{blue}{{\cos x}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

   6. metadata-eval 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

   7. cbrt-prod 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{\cos x} \cdot \sqrt{\cos x}}}\right)\right)}{e^{x}} \]

   8. add-sqr-sqrt 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\cos x}}\right)\right)}{e^{x}} \]

9. Applied egg-rr 23.1%

   \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\cos x}^{0.16666666666666666} \cdot \sqrt[3]{\cos x}\right)}\right)}{e^{x}} \]
10. Step-by-step derivation

    1. unpow1/3 23.1%

       \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \color{blue}{{\cos x}^{0.3333333333333333}}\right)\right)}{e^{x}} \]

    2. metadata-eval 23.1%

       \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot {\cos x}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}}\right)\right)}{e^{x}} \]

    3. pow-sqr 23.1%

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

    4. cube-mult 23.1%

       \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\left({\cos x}^{0.16666666666666666}\right)}^{3}\right)}\right)}{e^{x}} \]

11. Simplified 23.1%

    \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\left({\cos x}^{0.16666666666666666}\right)}^{3}\right)}\right)}{e^{x}} \]

12. Taylor expanded in x around 0 23.0%

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

13. Taylor expanded in x around inf 59.7%

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

Alternative 6: 24.7% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $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 23.1%

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

   1. +-commutative 23.1%

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

7. Simplified 23.1%

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

   1. add-cube-cbrt 23.1%

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

   2. associate-*l* 23.1%

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

   3. pow1/3 23.1%

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

   4. pow1/2 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\color{blue}{\left({\cos x}^{0.5}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

   5. pow-pow 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left(\color{blue}{{\cos x}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

   6. metadata-eval 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

   7. cbrt-prod 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{\cos x} \cdot \sqrt{\cos x}}}\right)\right)}{e^{x}} \]

   8. add-sqr-sqrt 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\cos x}}\right)\right)}{e^{x}} \]

9. Applied egg-rr 23.1%

   \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\cos x}^{0.16666666666666666} \cdot \sqrt[3]{\cos x}\right)}\right)}{e^{x}} \]
10. Step-by-step derivation

    1. unpow1/3 23.1%

       \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \color{blue}{{\cos x}^{0.3333333333333333}}\right)\right)}{e^{x}} \]

    2. metadata-eval 23.1%

       \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot {\cos x}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}}\right)\right)}{e^{x}} \]

    3. pow-sqr 23.1%

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

    4. cube-mult 23.1%

       \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\left({\cos x}^{0.16666666666666666}\right)}^{3}\right)}\right)}{e^{x}} \]

11. Simplified 23.1%

    \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\left({\cos x}^{0.16666666666666666}\right)}^{3}\right)}\right)}{e^{x}} \]

12. Taylor expanded in x around 0 23.0%

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

13. Taylor expanded in x around 0 22.9%

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

    1. +-commutative 23.1%

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

15. Simplified 22.9%

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

Alternative 7: 23.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], 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 23.1%

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

   1. +-commutative 23.1%

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

7. Simplified 23.1%

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

   1. add-cube-cbrt 23.1%

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

   2. associate-*l* 23.1%

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

   3. pow1/3 23.1%

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

   4. pow1/2 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\color{blue}{\left({\cos x}^{0.5}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

   5. pow-pow 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left(\color{blue}{{\cos x}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

   6. metadata-eval 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)\right)}{e^{x}} \]

   7. cbrt-prod 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{\cos x} \cdot \sqrt{\cos x}}}\right)\right)}{e^{x}} \]

   8. add-sqr-sqrt 23.1%

      \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\cos x}}\right)\right)}{e^{x}} \]

9. Applied egg-rr 23.1%

   \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\cos x}^{0.16666666666666666} \cdot \sqrt[3]{\cos x}\right)}\right)}{e^{x}} \]
10. Step-by-step derivation

    1. unpow1/3 23.1%

       \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot \color{blue}{{\cos x}^{0.3333333333333333}}\right)\right)}{e^{x}} \]

    2. metadata-eval 23.1%

       \[\leadsto \frac{\left(\left(x + 1\right) \bmod \left({\cos x}^{0.16666666666666666} \cdot {\cos x}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}}\right)\right)}{e^{x}} \]

    3. pow-sqr 23.1%

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

    4. cube-mult 23.1%

       \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\left({\cos x}^{0.16666666666666666}\right)}^{3}\right)}\right)}{e^{x}} \]

11. Simplified 23.1%

    \[\leadsto \frac{\left(\left(x + 1\right) \bmod \color{blue}{\left({\left({\cos x}^{0.16666666666666666}\right)}^{3}\right)}\right)}{e^{x}} \]

12. Taylor expanded in x around 0 23.0%

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

13. Taylor expanded in x around 0 22.5%

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

14. Final simplification 22.5%

    \[\leadsto \left(\left(x + 1\right) \bmod 1\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))