expq2 (section 3.11)

Percentage Accurate: 37.2% → 99.5%

Time: 4.7s

Alternatives: 7

Speedup: 68.3×

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x}}{e^{x} - 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return Math.exp(x) / (Math.exp(x) - 1.0);
}

Python

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

Julia

function code(x)
	return Float64(exp(x) / Float64(exp(x) - 1.0))
end

MATLAB

function tmp = code(x)
	tmp = exp(x) / (exp(x) - 1.0);
end

Wolfram

code[x_] := N[(N[Exp[x], $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 37.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x}}{e^{x} - 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return Math.exp(x) / (Math.exp(x) - 1.0);
}

Python

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

Julia

function code(x)
	return Float64(exp(x) / Float64(exp(x) - 1.0))
end

MATLAB

function tmp = code(x)
	tmp = exp(x) / (exp(x) - 1.0);
end

Wolfram

code[x_] := N[(N[Exp[x], $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x}}{\mathsf{expm1}\left(x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp x) (expm1 x)))

C

double code(double x) {
	return exp(x) / expm1(x);
}

Java

public static double code(double x) {
	return Math.exp(x) / Math.expm1(x);
}

Python

def code(x):
	return math.exp(x) / math.expm1(x)

Julia

function code(x)
	return Float64(exp(x) / expm1(x))
end

Wolfram

code[x_] := N[(N[Exp[x], $MachinePrecision] / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.1%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Step-by-step derivation

   1. expm1-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \frac{e^{x}}{\mathsf{expm1}\left(x\right)} \]

Alternative 2: 83.7% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{\frac{-1}{x \cdot x}}{0.5 + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.0)
   (/ (/ -1.0 (* x x)) (+ 0.5 (/ -1.0 x)))
   (+ 0.5 (+ (/ 1.0 x) (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.0) {
		tmp = (-1.0 / (x * x)) / (0.5 + (-1.0 / x));
	} else {
		tmp = 0.5 + ((1.0 / x) + (x * 0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.0d0)) then
        tmp = ((-1.0d0) / (x * x)) / (0.5d0 + ((-1.0d0) / x))
    else
        tmp = 0.5d0 + ((1.0d0 / x) + (x * 0.08333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.0) {
		tmp = (-1.0 / (x * x)) / (0.5 + (-1.0 / x));
	} else {
		tmp = 0.5 + ((1.0 / x) + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.0:
		tmp = (-1.0 / (x * x)) / (0.5 + (-1.0 / x))
	else:
		tmp = 0.5 + ((1.0 / x) + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(Float64(-1.0 / Float64(x * x)) / Float64(0.5 + Float64(-1.0 / x)));
	else
		tmp = Float64(0.5 + Float64(Float64(1.0 / x) + Float64(x * 0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.0)
		tmp = (-1.0 / (x * x)) / (0.5 + (-1.0 / x));
	else
		tmp = 0.5 + ((1.0 / x) + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.0], N[(N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(1.0 / x), $MachinePrecision] + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 100.0%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Step-by-step derivation

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 3.1%

      \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]
   5. Step-by-step derivation

      1. +-commutative 3.1%

         \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]

   6. Simplified 3.1%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
   7. Step-by-step derivation

      1. +-commutative 3.1%

         \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]

      2. flip-+ 3.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot 0.5 - \frac{1}{x} \cdot \frac{1}{x}}{0.5 - \frac{1}{x}}} \]

      3. metadata-eval 3.1%

         \[\leadsto \frac{\color{blue}{0.25} - \frac{1}{x} \cdot \frac{1}{x}}{0.5 - \frac{1}{x}} \]

      4. inv-pow 3.1%

         \[\leadsto \frac{0.25 - \color{blue}{{x}^{-1}} \cdot \frac{1}{x}}{0.5 - \frac{1}{x}} \]

      5. inv-pow 3.1%

         \[\leadsto \frac{0.25 - {x}^{-1} \cdot \color{blue}{{x}^{-1}}}{0.5 - \frac{1}{x}} \]

      6. pow-prod-up 3.1%

         \[\leadsto \frac{0.25 - \color{blue}{{x}^{\left(-1 + -1\right)}}}{0.5 - \frac{1}{x}} \]

      7. metadata-eval 3.1%

         \[\leadsto \frac{0.25 - {x}^{\color{blue}{-2}}}{0.5 - \frac{1}{x}} \]

   8. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{0.25 - {x}^{-2}}{0.5 - \frac{1}{x}}} \]

   9. Taylor expanded in x around 0 55.0%

      \[\leadsto \frac{\color{blue}{\frac{-1}{{x}^{2}}}}{0.5 - \frac{1}{x}} \]
   10. Step-by-step derivation

       1. unpow2 55.0%

          \[\leadsto \frac{\frac{-1}{\color{blue}{x \cdot x}}}{0.5 - \frac{1}{x}} \]

   11. Simplified 55.0%

       \[\leadsto \frac{\color{blue}{\frac{-1}{x \cdot x}}}{0.5 - \frac{1}{x}} \]

   if -4 < x

   1. Initial program 6.4%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Step-by-step derivation

      1. expm1-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{0.5 + \left(0.08333333333333333 \cdot x + \frac{1}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{\frac{-1}{x \cdot x}}{0.5 + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\\ \end{array} \]

Alternative 3: 83.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.5)
   (/ -2.0 (* x x))
   (+ 0.5 (+ (/ 1.0 x) (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.5) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + ((1.0 / x) + (x * 0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.5d0)) then
        tmp = (-2.0d0) / (x * x)
    else
        tmp = 0.5d0 + ((1.0d0 / x) + (x * 0.08333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.5) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + ((1.0 / x) + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.5:
		tmp = -2.0 / (x * x)
	else:
		tmp = 0.5 + ((1.0 / x) + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.5)
		tmp = Float64(-2.0 / Float64(x * x));
	else
		tmp = Float64(0.5 + Float64(Float64(1.0 / x) + Float64(x * 0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.5)
		tmp = -2.0 / (x * x);
	else
		tmp = 0.5 + ((1.0 / x) + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.5], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(1.0 / x), $MachinePrecision] + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5

   1. Initial program 100.0%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Step-by-step derivation

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 3.1%

      \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]
   5. Step-by-step derivation

      1. +-commutative 3.1%

         \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]

   6. Simplified 3.1%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
   7. Step-by-step derivation

      1. +-commutative 3.1%

         \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]

      2. flip-+ 3.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot 0.5 - \frac{1}{x} \cdot \frac{1}{x}}{0.5 - \frac{1}{x}}} \]

      3. metadata-eval 3.1%

         \[\leadsto \frac{\color{blue}{0.25} - \frac{1}{x} \cdot \frac{1}{x}}{0.5 - \frac{1}{x}} \]

      4. inv-pow 3.1%

         \[\leadsto \frac{0.25 - \color{blue}{{x}^{-1}} \cdot \frac{1}{x}}{0.5 - \frac{1}{x}} \]

      5. inv-pow 3.1%

         \[\leadsto \frac{0.25 - {x}^{-1} \cdot \color{blue}{{x}^{-1}}}{0.5 - \frac{1}{x}} \]

      6. pow-prod-up 3.1%

         \[\leadsto \frac{0.25 - \color{blue}{{x}^{\left(-1 + -1\right)}}}{0.5 - \frac{1}{x}} \]

      7. metadata-eval 3.1%

         \[\leadsto \frac{0.25 - {x}^{\color{blue}{-2}}}{0.5 - \frac{1}{x}} \]

   8. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{0.25 - {x}^{-2}}{0.5 - \frac{1}{x}}} \]

   9. Taylor expanded in x around 0 55.0%

      \[\leadsto \frac{\color{blue}{\frac{-1}{{x}^{2}}}}{0.5 - \frac{1}{x}} \]
   10. Step-by-step derivation

       1. unpow2 55.0%

          \[\leadsto \frac{\frac{-1}{\color{blue}{x \cdot x}}}{0.5 - \frac{1}{x}} \]

   11. Simplified 55.0%

       \[\leadsto \frac{\color{blue}{\frac{-1}{x \cdot x}}}{0.5 - \frac{1}{x}} \]

   12. Taylor expanded in x around inf 55.0%

       \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 55.0%

          \[\leadsto \frac{-2}{\color{blue}{x \cdot x}} \]

   14. Simplified 55.0%

       \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]

   if -4.5 < x

   1. Initial program 6.4%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Step-by-step derivation

      1. expm1-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{0.5 + \left(0.08333333333333333 \cdot x + \frac{1}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\\ \end{array} \]

Alternative 4: 83.4% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.75) (/ -2.0 (* x x)) (+ 0.5 (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.75d0)) then
        tmp = (-2.0d0) / (x * x)
    else
        tmp = 0.5d0 + (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.75:
		tmp = -2.0 / (x * x)
	else:
		tmp = 0.5 + (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.75)
		tmp = Float64(-2.0 / Float64(x * x));
	else
		tmp = Float64(0.5 + Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.75)
		tmp = -2.0 / (x * x);
	else
		tmp = 0.5 + (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.75], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75

   1. Initial program 100.0%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Step-by-step derivation

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 3.1%

      \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]
   5. Step-by-step derivation

      1. +-commutative 3.1%

         \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]

   6. Simplified 3.1%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
   7. Step-by-step derivation

      1. +-commutative 3.1%

         \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]

      2. flip-+ 3.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot 0.5 - \frac{1}{x} \cdot \frac{1}{x}}{0.5 - \frac{1}{x}}} \]

      3. metadata-eval 3.1%

         \[\leadsto \frac{\color{blue}{0.25} - \frac{1}{x} \cdot \frac{1}{x}}{0.5 - \frac{1}{x}} \]

      4. inv-pow 3.1%

         \[\leadsto \frac{0.25 - \color{blue}{{x}^{-1}} \cdot \frac{1}{x}}{0.5 - \frac{1}{x}} \]

      5. inv-pow 3.1%

         \[\leadsto \frac{0.25 - {x}^{-1} \cdot \color{blue}{{x}^{-1}}}{0.5 - \frac{1}{x}} \]

      6. pow-prod-up 3.1%

         \[\leadsto \frac{0.25 - \color{blue}{{x}^{\left(-1 + -1\right)}}}{0.5 - \frac{1}{x}} \]

      7. metadata-eval 3.1%

         \[\leadsto \frac{0.25 - {x}^{\color{blue}{-2}}}{0.5 - \frac{1}{x}} \]

   8. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{0.25 - {x}^{-2}}{0.5 - \frac{1}{x}}} \]

   9. Taylor expanded in x around 0 55.0%

      \[\leadsto \frac{\color{blue}{\frac{-1}{{x}^{2}}}}{0.5 - \frac{1}{x}} \]
   10. Step-by-step derivation

       1. unpow2 55.0%

          \[\leadsto \frac{\frac{-1}{\color{blue}{x \cdot x}}}{0.5 - \frac{1}{x}} \]

   11. Simplified 55.0%

       \[\leadsto \frac{\color{blue}{\frac{-1}{x \cdot x}}}{0.5 - \frac{1}{x}} \]

   12. Taylor expanded in x around inf 55.0%

       \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 55.0%

          \[\leadsto \frac{-2}{\color{blue}{x \cdot x}} \]

   14. Simplified 55.0%

       \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]

   if -1.75 < x

   1. Initial program 6.4%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Step-by-step derivation

      1. expm1-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]
   5. Step-by-step derivation

      1. +-commutative 98.2%

         \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]

   6. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \]

Alternative 5: 66.8% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + \frac{1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 0.5 (/ 1.0 x)))

C

double code(double x) {
	return 0.5 + (1.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 + (1.0d0 / x)
end function

Java

public static double code(double x) {
	return 0.5 + (1.0 / x);
}

Python

def code(x):
	return 0.5 + (1.0 / x)

Julia

function code(x)
	return Float64(0.5 + Float64(1.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 + (1.0 / x);
end

Wolfram

code[x_] := N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.1%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Step-by-step derivation

   1. expm1-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 67.0%

   \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]
5. Step-by-step derivation

   1. +-commutative 67.0%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]

6. Simplified 67.0%

   \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]

7. Final simplification 67.0%

   \[\leadsto 0.5 + \frac{1}{x} \]

Alternative 6: 66.9% accurate, 68.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	return 1.0 / x;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

function code(x)
	return Float64(1.0 / x)
end

MATLAB

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

Wolfram

code[x_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 37.1%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Step-by-step derivation

   1. expm1-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 66.9%

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

5. Final simplification 66.9%

   \[\leadsto \frac{1}{x} \]

Alternative 7: 3.4% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 37.1%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Step-by-step derivation

   1. expm1-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 67.0%

   \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]
5. Step-by-step derivation

   1. +-commutative 67.0%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]

6. Simplified 67.0%

   \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]

7. Taylor expanded in x around inf 3.4%

   \[\leadsto \color{blue}{0.5} \]

8. Final simplification 3.4%

   \[\leadsto 0.5 \]

Developer target: 37.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 - e^{-x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (1.0 - Math.exp(-x));
}

Python

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

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 - exp(Float64(-x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 - exp(-x));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023230 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

  (/ (exp x) (- (exp x) 1.0)))