expq2 (section 3.11)

Percentage Accurate: 37.2% → 99.1%

Time: 5.1s

Alternatives: 9

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: 99.1% 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 40.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Final simplification 100.0%

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

Alternative 2: 83.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;\left(-\frac{2}{x \cdot x}\right) - \frac{4}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left({x}^{3} \cdot -0.001388888888888889 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.6)
   (- (- (/ 2.0 (* x x))) (/ 4.0 (pow x 3.0)))
   (+
    0.5
    (+
     (* (pow x 3.0) -0.001388888888888889)
     (+ (* x 0.08333333333333333) (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -(2.0 / (x * x)) - (4.0 / pow(x, 3.0));
	} else {
		tmp = 0.5 + ((pow(x, 3.0) * -0.001388888888888889) + ((x * 0.08333333333333333) + (1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.6d0)) then
        tmp = -(2.0d0 / (x * x)) - (4.0d0 / (x ** 3.0d0))
    else
        tmp = 0.5d0 + (((x ** 3.0d0) * (-0.001388888888888889d0)) + ((x * 0.08333333333333333d0) + (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -(2.0 / (x * x)) - (4.0 / Math.pow(x, 3.0));
	} else {
		tmp = 0.5 + ((Math.pow(x, 3.0) * -0.001388888888888889) + ((x * 0.08333333333333333) + (1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.6:
		tmp = -(2.0 / (x * x)) - (4.0 / math.pow(x, 3.0))
	else:
		tmp = 0.5 + ((math.pow(x, 3.0) * -0.001388888888888889) + ((x * 0.08333333333333333) + (1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.6)
		tmp = Float64(Float64(-Float64(2.0 / Float64(x * x))) - Float64(4.0 / (x ^ 3.0)));
	else
		tmp = Float64(0.5 + Float64(Float64((x ^ 3.0) * -0.001388888888888889) + Float64(Float64(x * 0.08333333333333333) + Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.6)
		tmp = -(2.0 / (x * x)) - (4.0 / (x ^ 3.0));
	else
		tmp = 0.5 + (((x ^ 3.0) * -0.001388888888888889) + ((x * 0.08333333333333333) + (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.6], N[((-N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]) - N[(4.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.001388888888888889), $MachinePrecision] + N[(N[(x * 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009

   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 54.4%

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

       1. unpow2 54.4%

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

       2. associate-/r* 53.8%

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

       3. *-rgt-identity 53.8%

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

       4. associate-*r/ 53.8%

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

       5. metadata-eval 53.8%

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

       6. distribute-neg-frac 53.8%

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

       7. distribute-lft-neg-in 53.8%

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

       8. unpow-1 53.8%

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

       9. unpow-1 53.8%

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

       10. pow-sqr 53.8%

           \[\leadsto \frac{-\color{blue}{{x}^{\left(2 \cdot -1\right)}}}{0.5 - \frac{1}{x}} \]

       11. metadata-eval 53.8%

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

   11. Simplified 53.8%

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

   12. Taylor expanded in x around inf 54.4%

       \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{2}} + 4 \cdot \frac{1}{{x}^{3}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 54.4%

          \[\leadsto -\left(\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + 4 \cdot \frac{1}{{x}^{3}}\right) \]

       2. metadata-eval 54.4%

          \[\leadsto -\left(\frac{\color{blue}{2}}{{x}^{2}} + 4 \cdot \frac{1}{{x}^{3}}\right) \]

       3. unpow2 54.4%

          \[\leadsto -\left(\frac{2}{\color{blue}{x \cdot x}} + 4 \cdot \frac{1}{{x}^{3}}\right) \]

       4. associate-*r/ 54.4%

          \[\leadsto -\left(\frac{2}{x \cdot x} + \color{blue}{\frac{4 \cdot 1}{{x}^{3}}}\right) \]

       5. metadata-eval 54.4%

          \[\leadsto -\left(\frac{2}{x \cdot x} + \frac{\color{blue}{4}}{{x}^{3}}\right) \]

   14. Simplified 54.4%

       \[\leadsto \color{blue}{-\left(\frac{2}{x \cdot x} + \frac{4}{{x}^{3}}\right)} \]

   if -3.60000000000000009 < x

   1. Initial program 7.4%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 99.4%

      \[\leadsto \color{blue}{0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(0.08333333333333333 \cdot x + \frac{1}{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;\left(-\frac{2}{x \cdot x}\right) - \frac{4}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left({x}^{3} \cdot -0.001388888888888889 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\\ \end{array} \]

Alternative 3: 83.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;\left(-\frac{2}{x \cdot x}\right) - \frac{4}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.5)
   (- (- (/ 2.0 (* x x))) (/ 4.0 (pow x 3.0)))
   (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = -(2.0 / (x * x)) - (4.0 / pow(x, 3.0));
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.5d0)) then
        tmp = -(2.0d0 / (x * x)) - (4.0d0 / (x ** 3.0d0))
    else
        tmp = 0.5d0 + ((x * 0.08333333333333333d0) + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = -(2.0 / (x * x)) - (4.0 / Math.pow(x, 3.0));
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.5:
		tmp = -(2.0 / (x * x)) - (4.0 / math.pow(x, 3.0))
	else:
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.5)
		tmp = Float64(Float64(-Float64(2.0 / Float64(x * x))) - Float64(4.0 / (x ^ 3.0)));
	else
		tmp = Float64(0.5 + Float64(Float64(x * 0.08333333333333333) + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.5)
		tmp = -(2.0 / (x * x)) - (4.0 / (x ^ 3.0));
	else
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.5], N[((-N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]) - N[(4.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(x * 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5

   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 54.4%

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

       1. unpow2 54.4%

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

       2. associate-/r* 53.8%

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

       3. *-rgt-identity 53.8%

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

       4. associate-*r/ 53.8%

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

       5. metadata-eval 53.8%

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

       6. distribute-neg-frac 53.8%

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

       7. distribute-lft-neg-in 53.8%

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

       8. unpow-1 53.8%

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

       9. unpow-1 53.8%

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

       10. pow-sqr 53.8%

           \[\leadsto \frac{-\color{blue}{{x}^{\left(2 \cdot -1\right)}}}{0.5 - \frac{1}{x}} \]

       11. metadata-eval 53.8%

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

   11. Simplified 53.8%

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

   12. Taylor expanded in x around inf 54.4%

       \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{2}} + 4 \cdot \frac{1}{{x}^{3}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 54.4%

          \[\leadsto -\left(\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + 4 \cdot \frac{1}{{x}^{3}}\right) \]

       2. metadata-eval 54.4%

          \[\leadsto -\left(\frac{\color{blue}{2}}{{x}^{2}} + 4 \cdot \frac{1}{{x}^{3}}\right) \]

       3. unpow2 54.4%

          \[\leadsto -\left(\frac{2}{\color{blue}{x \cdot x}} + 4 \cdot \frac{1}{{x}^{3}}\right) \]

       4. associate-*r/ 54.4%

          \[\leadsto -\left(\frac{2}{x \cdot x} + \color{blue}{\frac{4 \cdot 1}{{x}^{3}}}\right) \]

       5. metadata-eval 54.4%

          \[\leadsto -\left(\frac{2}{x \cdot x} + \frac{\color{blue}{4}}{{x}^{3}}\right) \]

   14. Simplified 54.4%

       \[\leadsto \color{blue}{-\left(\frac{2}{x \cdot x} + \frac{4}{{x}^{3}}\right)} \]

   if -2.5 < x

   1. Initial program 7.4%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 99.2%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;\left(-\frac{2}{x \cdot x}\right) - \frac{4}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 4: 83.5% accurate, 15.7× speedup

Math

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

FPCore

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

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 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	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 + ((x * 0.08333333333333333d0) + (1.0d0 / x))
    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 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	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 + ((x * 0.08333333333333333) + (1.0 / x))
	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(x * 0.08333333333333333) + Float64(1.0 / x)));
	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 + ((x * 0.08333333333333333) + (1.0 / x));
	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[(x * 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $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 54.4%

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

       1. unpow2 54.4%

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

   11. Simplified 54.4%

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

   if -4 < x

   1. Initial program 7.4%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 99.2%

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

4. Final simplification 83.3%

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

Alternative 5: 83.5% accurate, 18.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -4.5) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	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 + ((x * 0.08333333333333333d0) + (1.0d0 / x))
    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 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.5:
		tmp = -2.0 / (x * x)
	else:
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x))
	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(x * 0.08333333333333333) + Float64(1.0 / x)));
	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 + ((x * 0.08333333333333333) + (1.0 / x));
	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[(x * 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $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 54.4%

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

       1. unpow2 54.4%

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

       2. associate-/r* 53.8%

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

       3. *-rgt-identity 53.8%

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

       4. associate-*r/ 53.8%

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

       5. metadata-eval 53.8%

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

       6. distribute-neg-frac 53.8%

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

       7. distribute-lft-neg-in 53.8%

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

       8. unpow-1 53.8%

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

       9. unpow-1 53.8%

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

       10. pow-sqr 53.8%

           \[\leadsto \frac{-\color{blue}{{x}^{\left(2 \cdot -1\right)}}}{0.5 - \frac{1}{x}} \]

       11. metadata-eval 53.8%

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

   11. Simplified 53.8%

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

   12. Taylor expanded in x around inf 54.4%

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

       1. unpow2 54.4%

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

   14. Simplified 54.4%

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

   if -4.5 < x

   1. Initial program 7.4%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 99.2%

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

4. Final simplification 83.3%

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

Alternative 6: 83.2% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.76:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.76) (/ -2.0 (* x x)) (+ 0.5 (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.76) {
		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 <= (-1.76d0)) 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 <= -1.76) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.76:
		tmp = -2.0 / (x * x)
	else:
		tmp = 0.5 + (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.76)
		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 <= -1.76)
		tmp = -2.0 / (x * x);
	else
		tmp = 0.5 + (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.76], 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 < -1.76000000000000001

   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 54.0%

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

       1. unpow2 54.0%

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

       2. associate-/r* 53.4%

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

       3. *-rgt-identity 53.4%

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

       4. associate-*r/ 53.4%

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

       5. metadata-eval 53.4%

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

       6. distribute-neg-frac 53.4%

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

       7. distribute-lft-neg-in 53.4%

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

       8. unpow-1 53.4%

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

       9. unpow-1 53.4%

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

       10. pow-sqr 53.4%

           \[\leadsto \frac{-\color{blue}{{x}^{\left(2 \cdot -1\right)}}}{0.5 - \frac{1}{x}} \]

       11. metadata-eval 53.4%

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

   11. Simplified 53.4%

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

   12. Taylor expanded in x around inf 54.0%

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

       1. unpow2 54.0%

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

   14. Simplified 54.0%

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

   if -1.76000000000000001 < x

   1. Initial program 6.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 99.4%

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

      1. +-commutative 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.76:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \]

Alternative 7: 66.6% 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 40.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 64.8%

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

   1. +-commutative 64.8%

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

6. Simplified 64.8%

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

7. Final simplification 64.8%

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

Alternative 8: 66.6% 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 40.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 64.7%

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

5. Final simplification 64.7%

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

Alternative 9: 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 40.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 64.8%

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

   1. +-commutative 64.8%

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

6. Simplified 64.8%

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

7. Taylor expanded in x around inf 3.6%

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

8. Final simplification 3.6%

   \[\leadsto 0.5 \]

Developer target: 38.0% 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 2023240 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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