expfmod (used to be hard to sample)

Percentage Accurate: 7.1% → 63.3%

Time: 17.9s

Alternatives: 7

Speedup: 505.0×

• could not determine a ground truth

  1. x = -3.249536792344826e+83

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: 7.1% 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: 63.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   1.0
   (if (<= x 10.0)
     (pow (exp -1.0) (- x (log (log (exp (fmod (exp x) (sqrt (cos x))))))))
     (pow (exp -1.0) x))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = 1.0;
	} else if (x <= 10.0) {
		tmp = pow(exp(-1.0), (x - log(log(exp(fmod(exp(x), sqrt(cos(x))))))));
	} else {
		tmp = pow(exp(-1.0), x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = 1.0d0
    else if (x <= 10.0d0) then
        tmp = exp((-1.0d0)) ** (x - log(log(exp(mod(exp(x), sqrt(cos(x)))))))
    else
        tmp = exp((-1.0d0)) ** x
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = 1.0
	elif x <= 10.0:
		tmp = math.pow(math.exp(-1.0), (x - math.log(math.log(math.exp(math.fmod(math.exp(x), math.sqrt(math.cos(x))))))))
	else:
		tmp = math.pow(math.exp(-1.0), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = 1.0;
	elseif (x <= 10.0)
		tmp = exp(-1.0) ^ Float64(x - log(log(exp(rem(exp(x), sqrt(cos(x)))))));
	else
		tmp = exp(-1.0) ^ x;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], 1.0, If[LessEqual[x, 10.0], N[Power[N[Exp[-1.0], $MachinePrecision], N[(x - N[Log[N[Log[N[Exp[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Exp[-1.0], $MachinePrecision], x], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 9.0%

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

      1. exp-neg 9.0%

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

      2. associate-*r/ 9.0%

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

      3. *-rgt-identity 9.0%

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

   3. Simplified 9.0%

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

      1. pow1/2 9.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\cos x}^{0.5}\right)}\right)}{e^{x}} \]

      2. metadata-eval 9.0%

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

      3. metadata-eval 9.0%

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

      4. pow-pow 9.0%

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

      5. metadata-eval 9.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left({\cos x}^{\color{blue}{0.25}}\right)}^{2}\right)\right)}{e^{x}} \]

   5. Applied egg-rr 9.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\left({\cos x}^{0.25}\right)}^{2}\right)}\right)}{e^{x}} \]

   7. Applied egg-rr 97.7%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{1} \]

   if -1.999999999999994e-310 < x < 10

   1. Initial program 11.3%

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

      1. exp-neg 11.3%

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

      2. associate-*r/ 11.3%

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

      3. *-rgt-identity 11.3%

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

   3. Simplified 11.3%

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

      1. clear-num 11.3%

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

      2. inv-pow 11.3%

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

      3. pow-to-exp 11.3%

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

      4. *-commutative 11.3%

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

      5. exp-prod 11.3%

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

      6. log-div 11.3%

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

      7. rem-log-exp 11.3%

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

   5. Applied egg-rr 11.3%

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

      1. add-log-exp_binary64 11.3%

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

   7. Applied rewrite-once 11.3%

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

   if 10 < 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%

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

      1. clear-num 0.0%

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

      2. inv-pow 0.0%

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

      3. pow-to-exp 0.0%

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

      4. *-commutative 0.0%

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

      5. exp-prod 0.0%

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

      6. log-div 0.0%

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

      7. rem-log-exp 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 62.9%

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

Alternative 2: 63.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   1.0
   (if (<= x 10.0)
     (/ (log (exp (fmod (exp x) (sqrt (cos x))))) (exp x))
     (pow (exp -1.0) x))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = 1.0;
	} else if (x <= 10.0) {
		tmp = log(exp(fmod(exp(x), sqrt(cos(x))))) / exp(x);
	} else {
		tmp = pow(exp(-1.0), x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = 1.0d0
    else if (x <= 10.0d0) then
        tmp = log(exp(mod(exp(x), sqrt(cos(x))))) / exp(x)
    else
        tmp = exp((-1.0d0)) ** x
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = 1.0
	elif x <= 10.0:
		tmp = math.log(math.exp(math.fmod(math.exp(x), math.sqrt(math.cos(x))))) / math.exp(x)
	else:
		tmp = math.pow(math.exp(-1.0), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = 1.0;
	elseif (x <= 10.0)
		tmp = Float64(log(exp(rem(exp(x), sqrt(cos(x))))) / exp(x));
	else
		tmp = exp(-1.0) ^ x;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], 1.0, If[LessEqual[x, 10.0], N[(N[Log[N[Exp[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[Power[N[Exp[-1.0], $MachinePrecision], x], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 9.0%

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

      1. exp-neg 9.0%

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

      2. associate-*r/ 9.0%

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

      3. *-rgt-identity 9.0%

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

   3. Simplified 9.0%

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

      1. pow1/2 9.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\cos x}^{0.5}\right)}\right)}{e^{x}} \]

      2. metadata-eval 9.0%

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

      3. metadata-eval 9.0%

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

      4. pow-pow 9.0%

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

      5. metadata-eval 9.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left({\cos x}^{\color{blue}{0.25}}\right)}^{2}\right)\right)}{e^{x}} \]

   5. Applied egg-rr 9.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\left({\cos x}^{0.25}\right)}^{2}\right)}\right)}{e^{x}} \]

   7. Applied egg-rr 97.7%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{1} \]

   if -1.999999999999994e-310 < x < 10

   1. Initial program 11.3%

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

      1. exp-neg 11.3%

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

      2. associate-*r/ 11.3%

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

      3. *-rgt-identity 11.3%

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

   3. Simplified 11.3%

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

      1. add-log-exp_binary64 11.3%

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

   5. Applied rewrite-once 11.3%

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

   if 10 < 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%

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

      1. clear-num 0.0%

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

      2. inv-pow 0.0%

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

      3. pow-to-exp 0.0%

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

      4. *-commutative 0.0%

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

      5. exp-prod 0.0%

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

      6. log-div 0.0%

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

      7. rem-log-exp 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 62.9%

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

Alternative 3: 63.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   1.0
   (if (<= x 10.0)
     (exp (- (log (fmod (exp x) (sqrt (cos x)))) x))
     (pow (exp -1.0) x))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = 1.0;
	} else if (x <= 10.0) {
		tmp = exp((log(fmod(exp(x), sqrt(cos(x)))) - x));
	} else {
		tmp = pow(exp(-1.0), x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = 1.0d0
    else if (x <= 10.0d0) then
        tmp = exp((log(mod(exp(x), sqrt(cos(x)))) - x))
    else
        tmp = exp((-1.0d0)) ** x
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = 1.0
	elif x <= 10.0:
		tmp = math.exp((math.log(math.fmod(math.exp(x), math.sqrt(math.cos(x)))) - x))
	else:
		tmp = math.pow(math.exp(-1.0), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = 1.0;
	elseif (x <= 10.0)
		tmp = exp(Float64(log(rem(exp(x), sqrt(cos(x)))) - x));
	else
		tmp = exp(-1.0) ^ x;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], 1.0, If[LessEqual[x, 10.0], N[Exp[N[(N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision], N[Power[N[Exp[-1.0], $MachinePrecision], x], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 9.0%

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

      1. exp-neg 9.0%

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

      2. associate-*r/ 9.0%

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

      3. *-rgt-identity 9.0%

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

   3. Simplified 9.0%

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

      1. pow1/2 9.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\cos x}^{0.5}\right)}\right)}{e^{x}} \]

      2. metadata-eval 9.0%

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

      3. metadata-eval 9.0%

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

      4. pow-pow 9.0%

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

      5. metadata-eval 9.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left({\cos x}^{\color{blue}{0.25}}\right)}^{2}\right)\right)}{e^{x}} \]

   5. Applied egg-rr 9.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\left({\cos x}^{0.25}\right)}^{2}\right)}\right)}{e^{x}} \]

   7. Applied egg-rr 97.7%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{1} \]

   if -1.999999999999994e-310 < x < 10

   1. Initial program 11.3%

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

      1. exp-neg 11.3%

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

      2. associate-*r/ 11.3%

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

      3. *-rgt-identity 11.3%

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

   3. Simplified 11.3%

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

      1. clear-num 11.3%

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

      2. inv-pow 11.3%

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

      3. pow-to-exp 11.3%

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

      4. *-commutative 11.3%

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

      5. log-pow 11.3%

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

      6. inv-pow 11.3%

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

      7. clear-num 11.3%

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

      8. log-div 11.3%

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

      9. rem-log-exp 11.3%

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

   5. Applied egg-rr 11.3%

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

   if 10 < 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%

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

      1. clear-num 0.0%

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

      2. inv-pow 0.0%

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

      3. pow-to-exp 0.0%

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

      4. *-commutative 0.0%

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

      5. exp-prod 0.0%

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

      6. log-div 0.0%

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

      7. rem-log-exp 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 62.9%

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

Alternative 4: 63.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   1.0
   (if (<= x 10.0)
     (/ (fmod (exp x) (sqrt (cos x))) (exp x))
     (pow (exp -1.0) x))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = 1.0;
	} else if (x <= 10.0) {
		tmp = fmod(exp(x), sqrt(cos(x))) / exp(x);
	} else {
		tmp = pow(exp(-1.0), x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = 1.0d0
    else if (x <= 10.0d0) then
        tmp = mod(exp(x), sqrt(cos(x))) / exp(x)
    else
        tmp = exp((-1.0d0)) ** x
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = 1.0
	elif x <= 10.0:
		tmp = math.fmod(math.exp(x), math.sqrt(math.cos(x))) / math.exp(x)
	else:
		tmp = math.pow(math.exp(-1.0), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = 1.0;
	elseif (x <= 10.0)
		tmp = Float64(rem(exp(x), sqrt(cos(x))) / exp(x));
	else
		tmp = exp(-1.0) ^ x;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], 1.0, If[LessEqual[x, 10.0], 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], N[Power[N[Exp[-1.0], $MachinePrecision], x], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 9.0%

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

      1. exp-neg 9.0%

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

      2. associate-*r/ 9.0%

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

      3. *-rgt-identity 9.0%

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

   3. Simplified 9.0%

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

      1. pow1/2 9.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\cos x}^{0.5}\right)}\right)}{e^{x}} \]

      2. metadata-eval 9.0%

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

      3. metadata-eval 9.0%

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

      4. pow-pow 9.0%

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

      5. metadata-eval 9.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left({\cos x}^{\color{blue}{0.25}}\right)}^{2}\right)\right)}{e^{x}} \]

   5. Applied egg-rr 9.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\left({\cos x}^{0.25}\right)}^{2}\right)}\right)}{e^{x}} \]

   7. Applied egg-rr 97.7%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{1} \]

   if -1.999999999999994e-310 < x < 10

   1. Initial program 11.3%

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

      1. exp-neg 11.3%

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

      2. associate-*r/ 11.3%

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

      3. *-rgt-identity 11.3%

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

   3. Simplified 11.3%

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

   if 10 < 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%

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

      1. clear-num 0.0%

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

      2. inv-pow 0.0%

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

      3. pow-to-exp 0.0%

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

      4. *-commutative 0.0%

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

      5. exp-prod 0.0%

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

      6. log-div 0.0%

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

      7. rem-log-exp 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 62.9%

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

Alternative 5: 62.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-300}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-1}\right)}^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 5e-300) 1.0 (pow (exp -1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= 5e-300) {
		tmp = 1.0;
	} else {
		tmp = pow(exp(-1.0), x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5d-300) then
        tmp = 1.0d0
    else
        tmp = exp((-1.0d0)) ** x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e-300) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(Math.exp(-1.0), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-300:
		tmp = 1.0
	else:
		tmp = math.pow(math.exp(-1.0), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-300)
		tmp = 1.0;
	else
		tmp = exp(-1.0) ^ x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-300)
		tmp = 1.0;
	else
		tmp = exp(-1.0) ^ x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-300], 1.0, N[Power[N[Exp[-1.0], $MachinePrecision], x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999996e-300

   1. Initial program 9.1%

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

      1. exp-neg 9.1%

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

      2. associate-*r/ 9.1%

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

      3. *-rgt-identity 9.1%

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

   3. Simplified 9.1%

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

      1. pow1/2 9.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\cos x}^{0.5}\right)}\right)}{e^{x}} \]

      2. metadata-eval 9.1%

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

      3. metadata-eval 9.1%

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

      4. pow-pow 9.1%

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

      5. metadata-eval 9.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left({\cos x}^{\color{blue}{0.25}}\right)}^{2}\right)\right)}{e^{x}} \]

   5. Applied egg-rr 9.1%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\left({\cos x}^{0.25}\right)}^{2}\right)}\right)}{e^{x}} \]

   7. Applied egg-rr 94.1%

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

   8. Taylor expanded in x around 0 96.2%

      \[\leadsto \color{blue}{1} \]

   if 4.99999999999999996e-300 < x

   1. Initial program 7.6%

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

      1. exp-neg 7.6%

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

      2. associate-*r/ 7.6%

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

      3. *-rgt-identity 7.6%

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

   3. Simplified 7.6%

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

      1. clear-num 7.6%

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

      2. inv-pow 7.6%

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

      3. pow-to-exp 7.6%

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

      4. *-commutative 7.6%

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

      5. exp-prod 7.6%

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

      6. log-div 7.6%

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

      7. rem-log-exp 7.6%

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

   5. Applied egg-rr 7.6%

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

   6. Taylor expanded in x around inf 36.7%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-300}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-1}\right)}^{x}\\ \end{array} \]

Alternative 6: 52.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ {\left(1 + \left(x \cdot x\right) \cdot -0.25\right)}^{-1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (+ 1.0 (* (* x x) -0.25)) -1.0))

C

double code(double x) {
	return pow((1.0 + ((x * x) * -0.25)), -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 + ((x * x) * (-0.25d0))) ** (-1.0d0)
end function

Java

public static double code(double x) {
	return Math.pow((1.0 + ((x * x) * -0.25)), -1.0);
}

Python

def code(x):
	return math.pow((1.0 + ((x * x) * -0.25)), -1.0)

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(x * x) * -0.25)) ^ -1.0
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((x * x) * -0.25)) ^ -1.0;
end

Wolfram

code[x_] := N[Power[N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 8.2%

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

   1. exp-neg 8.2%

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

   2. associate-*r/ 8.2%

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

   3. *-rgt-identity 8.2%

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

3. Simplified 8.2%

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

   1. pow1/2 8.2%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\cos x}^{0.5}\right)}\right)}{e^{x}} \]

   2. metadata-eval 8.2%

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

   3. metadata-eval 8.2%

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

   4. pow-pow 8.2%

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

   5. metadata-eval 8.2%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left({\cos x}^{\color{blue}{0.25}}\right)}^{2}\right)\right)}{e^{x}} \]

5. Applied egg-rr 8.2%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\left({\cos x}^{0.25}\right)}^{2}\right)}\right)}{e^{x}} \]

7. Applied egg-rr 40.8%

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

8. Taylor expanded in x around 0 50.4%

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

   1. *-commutative 50.4%

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

   2. unpow2 50.4%

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

10. Simplified 50.4%

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

11. Final simplification 50.4%

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

Alternative 7: 43.6% accurate, 505.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 8.2%

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

   1. exp-neg 8.2%

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

   2. associate-*r/ 8.2%

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

   3. *-rgt-identity 8.2%

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

3. Simplified 8.2%

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

   1. pow1/2 8.2%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\cos x}^{0.5}\right)}\right)}{e^{x}} \]

   2. metadata-eval 8.2%

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

   3. metadata-eval 8.2%

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

   4. pow-pow 8.2%

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

   5. metadata-eval 8.2%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left({\cos x}^{\color{blue}{0.25}}\right)}^{2}\right)\right)}{e^{x}} \]

5. Applied egg-rr 8.2%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\left({\cos x}^{0.25}\right)}^{2}\right)}\right)}{e^{x}} \]

7. Applied egg-rr 40.8%

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

8. Taylor expanded in x around 0 41.7%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 41.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))