expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 62.4%

Time: 24.2s

Alternatives: 12

Speedup: 5.0×

• could not determine a ground truth

  1. x = -1.4605987518297014e+308

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 62.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0 \lor \neg \left(t\_2 \leq 2\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{t\_1}\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))
        (t_1 (fmod (exp x) (sqrt (cos x))))
        (t_2 (* t_0 t_1)))
   (if (or (<= t_2 0.0) (not (<= t_2 2.0))) t_0 (/ (log (exp t_1)) (exp x)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x)));
	double t_2 = t_0 * t_1;
	double tmp;
	if ((t_2 <= 0.0) || !(t_2 <= 2.0)) {
		tmp = t_0;
	} else {
		tmp = log(exp(t_1)) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = exp(-x)
    t_1 = mod(exp(x), sqrt(cos(x)))
    t_2 = t_0 * t_1
    if ((t_2 <= 0.0d0) .or. (.not. (t_2 <= 2.0d0))) then
        tmp = t_0
    else
        tmp = log(exp(t_1)) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	t_1 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	t_2 = t_0 * t_1
	tmp = 0
	if (t_2 <= 0.0) or not (t_2 <= 2.0):
		tmp = t_0
	else:
		tmp = math.log(math.exp(t_1)) / math.exp(x)
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = rem(exp(x), sqrt(cos(x)))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if ((t_2 <= 0.0) || !(t_2 <= 2.0))
		tmp = t_0;
	else
		tmp = Float64(log(exp(t_1)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, 2.0]], $MachinePrecision]], t$95$0, N[(N[Log[N[Exp[t$95$1], $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0 or 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 3.4%

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

      1. /-rgt-identity 3.4%

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

      2. associate-/r/ 3.4%

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

      3. exp-neg 3.4%

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

      4. remove-double-neg 3.4%

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

   3. Simplified 3.4%

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

   5. Taylor expanded in x around 0 3.4%

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

      1. add-exp-log 3.4%

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

      2. div-exp 3.4%

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

   7. Applied egg-rr 3.4%

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

   8. Taylor expanded in x around inf 62.1%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 62.1%

         \[\leadsto e^{\color{blue}{-x}} \]

   10. Simplified 62.1%

       \[\leadsto e^{\color{blue}{-x}} \]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 72.7%

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

      1. /-rgt-identity 72.7%

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

      2. associate-/r/ 72.4%

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

      3. exp-neg 72.9%

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

      4. remove-double-neg 72.9%

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

   3. Simplified 72.9%

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

      1. add-log-exp 73.0%

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

   6. Applied egg-rr 73.0%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0 \lor \neg \left(e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0 \lor \neg \left(t\_2 \leq 2\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))
        (t_1 (fmod (exp x) (sqrt (cos x))))
        (t_2 (* t_0 t_1)))
   (if (or (<= t_2 0.0) (not (<= t_2 2.0))) t_0 (/ t_1 (exp x)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x)));
	double t_2 = t_0 * t_1;
	double tmp;
	if ((t_2 <= 0.0) || !(t_2 <= 2.0)) {
		tmp = t_0;
	} else {
		tmp = t_1 / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = exp(-x)
    t_1 = mod(exp(x), sqrt(cos(x)))
    t_2 = t_0 * t_1
    if ((t_2 <= 0.0d0) .or. (.not. (t_2 <= 2.0d0))) then
        tmp = t_0
    else
        tmp = t_1 / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	t_1 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	t_2 = t_0 * t_1
	tmp = 0
	if (t_2 <= 0.0) or not (t_2 <= 2.0):
		tmp = t_0
	else:
		tmp = t_1 / math.exp(x)
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = rem(exp(x), sqrt(cos(x)))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if ((t_2 <= 0.0) || !(t_2 <= 2.0))
		tmp = t_0;
	else
		tmp = Float64(t_1 / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, 2.0]], $MachinePrecision]], t$95$0, N[(t$95$1 / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0 or 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 3.4%

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

      1. /-rgt-identity 3.4%

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

      2. associate-/r/ 3.4%

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

      3. exp-neg 3.4%

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

      4. remove-double-neg 3.4%

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

   3. Simplified 3.4%

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

   5. Taylor expanded in x around 0 3.4%

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

      1. add-exp-log 3.4%

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

      2. div-exp 3.4%

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

   7. Applied egg-rr 3.4%

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

   8. Taylor expanded in x around inf 62.1%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 62.1%

         \[\leadsto e^{\color{blue}{-x}} \]

   10. Simplified 62.1%

       \[\leadsto e^{\color{blue}{-x}} \]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 72.7%

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

      1. /-rgt-identity 72.7%

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

      2. associate-/r/ 72.4%

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

      3. exp-neg 72.9%

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

      4. remove-double-neg 72.9%

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

   3. Simplified 72.9%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0 \lor \neg \left(e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ \mathbf{if}\;x \leq 500:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(t\_0 \cdot 0.6666666666666666 + \log \left(\sqrt[3]{e^{t\_0}}\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))))
   (if (<= x 500.0)
     (/
      (fmod (exp x) (+ (* t_0 0.6666666666666666) (log (cbrt (exp t_0)))))
      (exp x))
     (exp (- x)))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double tmp;
	if (x <= 500.0) {
		tmp = fmod(exp(x), ((t_0 * 0.6666666666666666) + log(cbrt(exp(t_0))))) / exp(x);
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	tmp = 0.0
	if (x <= 500.0)
		tmp = Float64(rem(exp(x), Float64(Float64(t_0 * 0.6666666666666666) + log(cbrt(exp(t_0))))) / exp(x));
	else
		tmp = exp(Float64(-x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 500.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(t$95$0 * 0.6666666666666666), $MachinePrecision] + N[Log[N[Power[N[Exp[t$95$0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[Exp[(-x)], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 10.4%

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

      1. /-rgt-identity 10.4%

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

      2. associate-/r/ 10.4%

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

      3. exp-neg 10.4%

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

      4. remove-double-neg 10.4%

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

   3. Simplified 10.4%

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

      1. add-log-exp 10.4%

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

      2. add-cube-cbrt 51.7%

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

      3. log-prod 51.7%

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

      4. pow2 51.7%

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

   6. Applied egg-rr 51.7%

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

      1. unpow2 51.7%

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

      2. log-prod 51.7%

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

      3. pow1/3 51.7%

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

      4. log-pow 51.7%

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

      5. add-log-exp 51.7%

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

      6. pow1/3 51.8%

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

      7. log-pow 51.8%

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

      8. add-log-exp 51.8%

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

   8. Applied egg-rr 51.8%

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

      1. distribute-rgt-out 51.8%

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

      2. metadata-eval 51.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot \color{blue}{0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   10. Simplified 51.8%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\sqrt{\cos x} \cdot 0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   if 500 < x

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in x around 0 0.0%

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

      1. add-exp-log 0.0%

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

      2. div-exp 0.0%

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

   7. Applied egg-rr 0.0%

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

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto e^{\color{blue}{-x}} \]

   10. Simplified 100.0%

       \[\leadsto e^{\color{blue}{-x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 62.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot 0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 500.0)
   (/
    (fmod (exp x) (+ (* (sqrt (cos x)) 0.6666666666666666) (log (cbrt E))))
    (exp x))
   (exp (- x))))

C

double code(double x) {
	double tmp;
	if (x <= 500.0) {
		tmp = fmod(exp(x), ((sqrt(cos(x)) * 0.6666666666666666) + log(cbrt(((double) M_E))))) / exp(x);
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 500.0)
		tmp = Float64(rem(exp(x), Float64(Float64(sqrt(cos(x)) * 0.6666666666666666) + log(cbrt(exp(1))))) / exp(x));
	else
		tmp = exp(Float64(-x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 500.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision] * 0.6666666666666666), $MachinePrecision] + N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[Exp[(-x)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 10.4%

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

      1. /-rgt-identity 10.4%

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

      2. associate-/r/ 10.4%

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

      3. exp-neg 10.4%

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

      4. remove-double-neg 10.4%

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

   3. Simplified 10.4%

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

      1. add-log-exp 10.4%

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

      2. add-cube-cbrt 51.7%

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

      3. log-prod 51.7%

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

      4. pow2 51.7%

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

   6. Applied egg-rr 51.7%

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

      1. unpow2 51.7%

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

      2. log-prod 51.7%

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

      3. pow1/3 51.7%

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

      4. log-pow 51.7%

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

      5. add-log-exp 51.7%

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

      6. pow1/3 51.8%

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

      7. log-pow 51.8%

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

      8. add-log-exp 51.8%

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

   8. Applied egg-rr 51.8%

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

      1. distribute-rgt-out 51.8%

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

      2. metadata-eval 51.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot \color{blue}{0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   10. Simplified 51.8%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\sqrt{\cos x} \cdot 0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   11. Taylor expanded in x around 0 51.3%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot 0.6666666666666666 + \color{blue}{\log \left(\sqrt[3]{e^{1}}\right)}\right)\right)}{e^{x}} \]
   12. Step-by-step derivation

       1. exp-1-e 51.3%

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot 0.6666666666666666 + \log \left(\sqrt[3]{\color{blue}{e}}\right)\right)\right)}{e^{x}} \]

   13. Simplified 51.3%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot 0.6666666666666666 + \color{blue}{\log \left(\sqrt[3]{e}\right)}\right)\right)}{e^{x}} \]

   if 500 < x

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in x around 0 0.0%

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

      1. add-exp-log 0.0%

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

      2. div-exp 0.0%

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

   7. Applied egg-rr 0.0%

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

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto e^{\color{blue}{-x}} \]

   10. Simplified 100.0%

       \[\leadsto e^{\color{blue}{-x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 61.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-310) (not (<= x 0.05)))
   (exp (- x))
   (/ (fmod (exp x) (+ 1.0 (* -0.25 (pow x 2.0)))) (exp x))))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-310) || !(x <= 0.05)) {
		tmp = exp(-x);
	} else {
		tmp = fmod(exp(x), (1.0 + (-0.25 * pow(x, 2.0)))) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-310)) .or. (.not. (x <= 0.05d0))) then
        tmp = exp(-x)
    else
        tmp = mod(exp(x), (1.0d0 + ((-0.25d0) * (x ** 2.0d0)))) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if (x <= -5e-310) or not (x <= 0.05):
		tmp = math.exp(-x)
	else:
		tmp = math.fmod(math.exp(x), (1.0 + (-0.25 * math.pow(x, 2.0)))) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-310) || !(x <= 0.05))
		tmp = exp(Float64(-x));
	else
		tmp = Float64(rem(exp(x), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-310], N[Not[LessEqual[x, 0.05]], $MachinePrecision]], N[Exp[(-x)], $MachinePrecision], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $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 or 0.050000000000000003 < x

   1. Initial program 6.4%

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

      1. /-rgt-identity 6.4%

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

      2. associate-/r/ 6.4%

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

      3. exp-neg 6.4%

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

      4. remove-double-neg 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in x around 0 6.4%

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

      1. add-exp-log 6.4%

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

      2. div-exp 6.4%

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

   7. Applied egg-rr 6.4%

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

   8. Taylor expanded in x around inf 97.9%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 97.9%

         \[\leadsto e^{\color{blue}{-x}} \]

   10. Simplified 97.9%

       \[\leadsto e^{\color{blue}{-x}} \]

   if -4.999999999999985e-310 < x < 0.050000000000000003

   1. Initial program 10.4%

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

      1. /-rgt-identity 10.4%

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

      2. associate-/r/ 10.4%

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

      3. exp-neg 10.4%

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

      4. remove-double-neg 10.4%

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

   3. Simplified 10.4%

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

   5. Taylor expanded in x around 0 10.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-310) (not (<= x 0.05)))
   (exp (- x))
   (* (fmod (exp x) (sqrt (cos x))) (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-310) || !(x <= 0.05)) {
		tmp = exp(-x);
	} else {
		tmp = fmod(exp(x), sqrt(cos(x))) * (1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-310)) .or. (.not. (x <= 0.05d0))) then
        tmp = exp(-x)
    else
        tmp = mod(exp(x), sqrt(cos(x))) * (1.0d0 - x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if (x <= -5e-310) or not (x <= 0.05):
		tmp = math.exp(-x)
	else:
		tmp = math.fmod(math.exp(x), math.sqrt(math.cos(x))) * (1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-310) || !(x <= 0.05))
		tmp = exp(Float64(-x));
	else
		tmp = Float64(rem(exp(x), sqrt(cos(x))) * Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-310], N[Not[LessEqual[x, 0.05]], $MachinePrecision]], N[Exp[(-x)], $MachinePrecision], 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[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310 or 0.050000000000000003 < x

   1. Initial program 6.4%

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

      1. /-rgt-identity 6.4%

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

      2. associate-/r/ 6.4%

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

      3. exp-neg 6.4%

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

      4. remove-double-neg 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in x around 0 6.4%

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

      1. add-exp-log 6.4%

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

      2. div-exp 6.4%

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

   7. Applied egg-rr 6.4%

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

   8. Taylor expanded in x around inf 97.9%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 97.9%

         \[\leadsto e^{\color{blue}{-x}} \]

   10. Simplified 97.9%

       \[\leadsto e^{\color{blue}{-x}} \]

   if -4.999999999999985e-310 < x < 0.050000000000000003

   1. Initial program 10.4%

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

      1. /-rgt-identity 10.4%

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

      2. associate-/r/ 10.4%

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

      3. exp-neg 10.4%

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

      4. remove-double-neg 10.4%

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

   3. Simplified 10.4%

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

   5. Taylor expanded in x around 0 9.7%

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

      1. +-commutative 9.7%

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

      2. mul-1-neg 9.7%

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

      3. unsub-neg 9.7%

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

      4. *-lft-identity 9.7%

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

      5. distribute-rgt-out-- 9.7%

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

   7. Simplified 9.7%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right) \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-310) (not (<= x 0.05)))
   (exp (- x))
   (* (fmod (exp x) (+ 1.0 (* -0.25 (pow x 2.0)))) (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-310) || !(x <= 0.05)) {
		tmp = exp(-x);
	} else {
		tmp = fmod(exp(x), (1.0 + (-0.25 * pow(x, 2.0)))) * (1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-310)) .or. (.not. (x <= 0.05d0))) then
        tmp = exp(-x)
    else
        tmp = mod(exp(x), (1.0d0 + ((-0.25d0) * (x ** 2.0d0)))) * (1.0d0 - x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if (x <= -5e-310) or not (x <= 0.05):
		tmp = math.exp(-x)
	else:
		tmp = math.fmod(math.exp(x), (1.0 + (-0.25 * math.pow(x, 2.0)))) * (1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-310) || !(x <= 0.05))
		tmp = exp(Float64(-x));
	else
		tmp = Float64(rem(exp(x), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))) * Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-310], N[Not[LessEqual[x, 0.05]], $MachinePrecision]], N[Exp[(-x)], $MachinePrecision], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310 or 0.050000000000000003 < x

   1. Initial program 6.4%

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

      1. /-rgt-identity 6.4%

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

      2. associate-/r/ 6.4%

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

      3. exp-neg 6.4%

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

      4. remove-double-neg 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in x around 0 6.4%

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

      1. add-exp-log 6.4%

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

      2. div-exp 6.4%

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

   7. Applied egg-rr 6.4%

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

   8. Taylor expanded in x around inf 97.9%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 97.9%

         \[\leadsto e^{\color{blue}{-x}} \]

   10. Simplified 97.9%

       \[\leadsto e^{\color{blue}{-x}} \]

   if -4.999999999999985e-310 < x < 0.050000000000000003

   1. Initial program 10.4%

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

      1. /-rgt-identity 10.4%

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

      2. associate-/r/ 10.4%

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

      3. exp-neg 10.4%

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

      4. remove-double-neg 10.4%

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

   3. Simplified 10.4%

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

   5. Taylor expanded in x around 0 9.7%

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

      1. +-commutative 9.7%

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

      2. mul-1-neg 9.7%

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

      3. unsub-neg 9.7%

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

      4. *-lft-identity 9.7%

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

      5. distribute-rgt-out-- 9.7%

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

   7. Simplified 9.7%

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

   8. Taylor expanded in x around 0 9.7%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot \left(1 - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right) \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-310) (not (<= x 0.05)))
   (exp (- x))
   (/ 1.0 (/ (exp x) (fmod (exp x) 1.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-310) || !(x <= 0.05)) {
		tmp = exp(-x);
	} else {
		tmp = 1.0 / (exp(x) / fmod(exp(x), 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-310)) .or. (.not. (x <= 0.05d0))) then
        tmp = exp(-x)
    else
        tmp = 1.0d0 / (exp(x) / mod(exp(x), 1.0d0))
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if (x <= -5e-310) or not (x <= 0.05):
		tmp = math.exp(-x)
	else:
		tmp = 1.0 / (math.exp(x) / math.fmod(math.exp(x), 1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-310) || !(x <= 0.05))
		tmp = exp(Float64(-x));
	else
		tmp = Float64(1.0 / Float64(exp(x) / rem(exp(x), 1.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-310], N[Not[LessEqual[x, 0.05]], $MachinePrecision]], N[Exp[(-x)], $MachinePrecision], N[(1.0 / N[(N[Exp[x], $MachinePrecision] / N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310 or 0.050000000000000003 < x

   1. Initial program 6.4%

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

      1. /-rgt-identity 6.4%

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

      2. associate-/r/ 6.4%

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

      3. exp-neg 6.4%

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

      4. remove-double-neg 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in x around 0 6.4%

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

      1. add-exp-log 6.4%

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

      2. div-exp 6.4%

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

   7. Applied egg-rr 6.4%

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

   8. Taylor expanded in x around inf 97.9%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 97.9%

         \[\leadsto e^{\color{blue}{-x}} \]

   10. Simplified 97.9%

       \[\leadsto e^{\color{blue}{-x}} \]

   if -4.999999999999985e-310 < x < 0.050000000000000003

   1. Initial program 10.4%

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

      1. /-rgt-identity 10.4%

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

      2. associate-/r/ 10.4%

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

      3. exp-neg 10.4%

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

      4. remove-double-neg 10.4%

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

   3. Simplified 10.4%

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

   5. Taylor expanded in x around 0 9.4%

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

      1. add-exp-log 9.4%

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

      2. div-exp 9.4%

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

   7. Applied egg-rr 9.4%

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

      1. exp-diff 9.4%

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

      2. add-exp-log 9.4%

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

      3. clear-num 9.4%

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

   9. Applied egg-rr 9.4%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-310) (not (<= x 0.05)))
   (exp (- x))
   (/ (fmod (exp x) 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-310) || !(x <= 0.05)) {
		tmp = exp(-x);
	} else {
		tmp = fmod(exp(x), 1.0) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-310)) .or. (.not. (x <= 0.05d0))) then
        tmp = exp(-x)
    else
        tmp = mod(exp(x), 1.0d0) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if (x <= -5e-310) or not (x <= 0.05):
		tmp = math.exp(-x)
	else:
		tmp = math.fmod(math.exp(x), 1.0) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-310) || !(x <= 0.05))
		tmp = exp(Float64(-x));
	else
		tmp = Float64(rem(exp(x), 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-310], N[Not[LessEqual[x, 0.05]], $MachinePrecision]], N[Exp[(-x)], $MachinePrecision], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310 or 0.050000000000000003 < x

   1. Initial program 6.4%

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

      1. /-rgt-identity 6.4%

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

      2. associate-/r/ 6.4%

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

      3. exp-neg 6.4%

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

      4. remove-double-neg 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in x around 0 6.4%

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

      1. add-exp-log 6.4%

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

      2. div-exp 6.4%

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

   7. Applied egg-rr 6.4%

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

   8. Taylor expanded in x around inf 97.9%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 97.9%

         \[\leadsto e^{\color{blue}{-x}} \]

   10. Simplified 97.9%

       \[\leadsto e^{\color{blue}{-x}} \]

   if -4.999999999999985e-310 < x < 0.050000000000000003

   1. Initial program 10.4%

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

      1. /-rgt-identity 10.4%

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

      2. associate-/r/ 10.4%

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

      3. exp-neg 10.4%

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

      4. remove-double-neg 10.4%

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

   3. Simplified 10.4%

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

   5. Taylor expanded in x around 0 9.4%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(\left(e^{x}\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-310) (not (<= x 0.05)))
   (exp (- x))
   (* (- 1.0 x) (fmod (exp x) 1.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-310) || !(x <= 0.05)) {
		tmp = exp(-x);
	} else {
		tmp = (1.0 - x) * fmod(exp(x), 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-310)) .or. (.not. (x <= 0.05d0))) then
        tmp = exp(-x)
    else
        tmp = (1.0d0 - x) * mod(exp(x), 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if (x <= -5e-310) or not (x <= 0.05):
		tmp = math.exp(-x)
	else:
		tmp = (1.0 - x) * math.fmod(math.exp(x), 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-310) || !(x <= 0.05))
		tmp = exp(Float64(-x));
	else
		tmp = Float64(Float64(1.0 - x) * rem(exp(x), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-310], N[Not[LessEqual[x, 0.05]], $MachinePrecision]], N[Exp[(-x)], $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310 or 0.050000000000000003 < x

   1. Initial program 6.4%

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

      1. /-rgt-identity 6.4%

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

      2. associate-/r/ 6.4%

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

      3. exp-neg 6.4%

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

      4. remove-double-neg 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in x around 0 6.4%

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

      1. add-exp-log 6.4%

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

      2. div-exp 6.4%

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

   7. Applied egg-rr 6.4%

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

   8. Taylor expanded in x around inf 97.9%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 97.9%

         \[\leadsto e^{\color{blue}{-x}} \]

   10. Simplified 97.9%

       \[\leadsto e^{\color{blue}{-x}} \]

   if -4.999999999999985e-310 < x < 0.050000000000000003

   1. Initial program 10.4%

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

      1. /-rgt-identity 10.4%

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

      2. associate-/r/ 10.4%

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

      3. exp-neg 10.4%

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

      4. remove-double-neg 10.4%

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

   3. Simplified 10.4%

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

   5. Taylor expanded in x around 0 9.7%

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

      1. +-commutative 9.7%

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

      2. mul-1-neg 9.7%

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

      3. unsub-neg 9.7%

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

      4. *-lft-identity 9.7%

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

      5. distribute-rgt-out-- 9.7%

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

   7. Simplified 9.7%

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

   8. Taylor expanded in x around 0 9.4%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(\left(e^{x}\right) \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-310) (not (<= x 0.05))) (exp (- x)) (fmod (exp x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-310) || !(x <= 0.05)) {
		tmp = exp(-x);
	} else {
		tmp = fmod(exp(x), 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-310)) .or. (.not. (x <= 0.05d0))) then
        tmp = exp(-x)
    else
        tmp = mod(exp(x), 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if (x <= -5e-310) or not (x <= 0.05):
		tmp = math.exp(-x)
	else:
		tmp = math.fmod(math.exp(x), 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-310) || !(x <= 0.05))
		tmp = exp(Float64(-x));
	else
		tmp = rem(exp(x), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-310], N[Not[LessEqual[x, 0.05]], $MachinePrecision]], N[Exp[(-x)], $MachinePrecision], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310 or 0.050000000000000003 < x

   1. Initial program 6.4%

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

      1. /-rgt-identity 6.4%

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

      2. associate-/r/ 6.4%

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

      3. exp-neg 6.4%

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

      4. remove-double-neg 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in x around 0 6.4%

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

      1. add-exp-log 6.4%

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

      2. div-exp 6.4%

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

   7. Applied egg-rr 6.4%

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

   8. Taylor expanded in x around inf 97.9%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 97.9%

         \[\leadsto e^{\color{blue}{-x}} \]

   10. Simplified 97.9%

       \[\leadsto e^{\color{blue}{-x}} \]

   if -4.999999999999985e-310 < x < 0.050000000000000003

   1. Initial program 10.4%

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

      1. /-rgt-identity 10.4%

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

      2. associate-/r/ 10.4%

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

      3. exp-neg 10.4%

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

      4. remove-double-neg 10.4%

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

   3. Simplified 10.4%

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

   5. Taylor expanded in x around 0 9.4%

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

   6. Taylor expanded in x around 0 9.4%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- x)))

C

double code(double x) {
	return exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(-x)
end function

Java

public static double code(double x) {
	return Math.exp(-x);
}

Python

def code(x):
	return math.exp(-x)

Julia

function code(x)
	return exp(Float64(-x))
end

MATLAB

function tmp = code(x)
	tmp = exp(-x);
end

Wolfram

code[x_] := N[Exp[(-x)], $MachinePrecision]

Derivation

1. Initial program 8.0%

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

   1. /-rgt-identity 8.0%

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

   2. associate-/r/ 8.0%

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

   3. exp-neg 8.1%

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

   4. remove-double-neg 8.1%

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

3. Simplified 8.1%

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

5. Taylor expanded in x around 0 7.7%

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

   1. add-exp-log 7.7%

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

   2. div-exp 7.7%

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

7. Applied egg-rr 7.7%

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

8. Taylor expanded in x around inf 60.0%

   \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
9. Step-by-step derivation

   1. neg-mul-1 60.0%

      \[\leadsto e^{\color{blue}{-x}} \]

10. Simplified 60.0%

    \[\leadsto e^{\color{blue}{-x}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024106 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))