expq2 (section 3.11)

Percentage Accurate: 37.2% → 99.9%

Time: 4.4s

Alternatives: 3

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: 99.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{-1}{\mathsf{expm1}\left(x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (/ -1.0 (expm1 x))))

C

double code(double x) {
	return 1.0 - (-1.0 / expm1(x));
}

Java

public static double code(double x) {
	return 1.0 - (-1.0 / Math.expm1(x));
}

Python

def code(x):
	return 1.0 - (-1.0 / math.expm1(x))

Julia

function code(x)
	return Float64(1.0 - Float64(-1.0 / expm1(x)))
end

Wolfram

code[x_] := N[(1.0 - N[(-1.0 / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.6%

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

   1. expm1-log1p-u 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{x}\right)\right)}}{\mathsf{expm1}\left(x\right)} \]

   2. expm1-udef 99.2%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(e^{x}\right)} - 1}}{\mathsf{expm1}\left(x\right)} \]

   3. div-sub 99.2%

      \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(e^{x}\right)}}{\mathsf{expm1}\left(x\right)} - \frac{1}{\mathsf{expm1}\left(x\right)}} \]

5. Applied egg-rr 1.6%

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(x\right)} - \frac{1}{\mathsf{expm1}\left(x\right)}} \]
6. Step-by-step derivation

   1. *-inverses 2.4%

      \[\leadsto \color{blue}{1} - \frac{1}{\mathsf{expm1}\left(x\right)} \]

7. Simplified 2.4%

   \[\leadsto \color{blue}{1 - \frac{1}{\mathsf{expm1}\left(x\right)}} \]
8. Step-by-step derivation

   1. frac-2neg 2.4%

      \[\leadsto 1 - \color{blue}{\frac{-1}{-\mathsf{expm1}\left(x\right)}} \]

   2. metadata-eval 2.4%

      \[\leadsto 1 - \frac{\color{blue}{-1}}{-\mathsf{expm1}\left(x\right)} \]

   3. metadata-eval 2.4%

      \[\leadsto 1 - \frac{\color{blue}{1 \cdot -1}}{-\mathsf{expm1}\left(x\right)} \]

   4. neg-sub0 2.4%

      \[\leadsto 1 - \frac{1 \cdot -1}{\color{blue}{0 - \mathsf{expm1}\left(x\right)}} \]

   5. expm1-udef 3.8%

      \[\leadsto 1 - \frac{1 \cdot -1}{0 - \color{blue}{\left(e^{x} - 1\right)}} \]

   6. add-sqr-sqrt 2.7%

      \[\leadsto 1 - \frac{1 \cdot -1}{0 - \left(e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} - 1\right)} \]

   7. sqrt-unprod 4.1%

      \[\leadsto 1 - \frac{1 \cdot -1}{0 - \left(e^{\color{blue}{\sqrt{x \cdot x}}} - 1\right)} \]

   8. sqr-neg 4.1%

      \[\leadsto 1 - \frac{1 \cdot -1}{0 - \left(e^{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}} - 1\right)} \]

   9. sqrt-unprod 1.4%

      \[\leadsto 1 - \frac{1 \cdot -1}{0 - \left(e^{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} - 1\right)} \]

   10. add-sqr-sqrt 3.9%

       \[\leadsto 1 - \frac{1 \cdot -1}{0 - \left(e^{\color{blue}{-x}} - 1\right)} \]

   11. associate-+l- 3.9%

       \[\leadsto 1 - \frac{1 \cdot -1}{\color{blue}{\left(0 - e^{-x}\right) + 1}} \]

   12. neg-sub0 3.9%

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

   13. +-commutative 3.9%

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

   14. sub-neg 3.9%

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

   15. add-sqr-sqrt 2.6%

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

   16. sqrt-unprod 3.6%

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

9. Applied egg-rr 100.0%

   \[\leadsto 1 - \color{blue}{\frac{1}{\mathsf{expm1}\left(x\right)} \cdot -1} \]
10. Step-by-step derivation

    1. associate-*l/ 100.0%

       \[\leadsto 1 - \color{blue}{\frac{1 \cdot -1}{\mathsf{expm1}\left(x\right)}} \]

    2. metadata-eval 100.0%

       \[\leadsto 1 - \frac{\color{blue}{-1}}{\mathsf{expm1}\left(x\right)} \]

11. Simplified 100.0%

    \[\leadsto 1 - \color{blue}{\frac{-1}{\mathsf{expm1}\left(x\right)}} \]

12. Final simplification 100.0%

    \[\leadsto 1 - \frac{-1}{\mathsf{expm1}\left(x\right)} \]

Alternative 2: 98.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;1 + \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -350.0)
   (+ 1.0 (expm1 x))
   (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -350.0) {
		tmp = 1.0 + expm1(x);
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -350.0) {
		tmp = 1.0 + Math.expm1(x);
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -350.0:
		tmp = 1.0 + math.expm1(x)
	else:
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -350.0)
		tmp = Float64(1.0 + expm1(x));
	else
		tmp = Float64(0.5 + Float64(Float64(x * 0.08333333333333333) + Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -350.0], N[(1.0 + N[(Exp[x] - 1), $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 < -350

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{x}\right)\right)}}{\mathsf{expm1}\left(x\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(e^{x}\right)} - 1}}{\mathsf{expm1}\left(x\right)} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(e^{x}\right)}}{\mathsf{expm1}\left(x\right)} - \frac{1}{\mathsf{expm1}\left(x\right)}} \]

   5. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(x\right)} - \frac{1}{\mathsf{expm1}\left(x\right)}} \]
   6. Step-by-step derivation

      1. *-inverses 3.1%

         \[\leadsto \color{blue}{1} - \frac{1}{\mathsf{expm1}\left(x\right)} \]

   7. Simplified 3.1%

      \[\leadsto \color{blue}{1 - \frac{1}{\mathsf{expm1}\left(x\right)}} \]
   8. Step-by-step derivation

      1. frac-2neg 3.1%

         \[\leadsto 1 - \color{blue}{\frac{-1}{-\mathsf{expm1}\left(x\right)}} \]

      2. metadata-eval 3.1%

         \[\leadsto 1 - \frac{\color{blue}{-1}}{-\mathsf{expm1}\left(x\right)} \]

      3. metadata-eval 3.1%

         \[\leadsto 1 - \frac{\color{blue}{1 \cdot -1}}{-\mathsf{expm1}\left(x\right)} \]

      4. neg-sub0 3.1%

         \[\leadsto 1 - \frac{1 \cdot -1}{\color{blue}{0 - \mathsf{expm1}\left(x\right)}} \]

      5. expm1-udef 3.1%

         \[\leadsto 1 - \frac{1 \cdot -1}{0 - \color{blue}{\left(e^{x} - 1\right)}} \]

      6. add-sqr-sqrt 0.0%

         \[\leadsto 1 - \frac{1 \cdot -1}{0 - \left(e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} - 1\right)} \]

      7. sqrt-unprod 3.1%

         \[\leadsto 1 - \frac{1 \cdot -1}{0 - \left(e^{\color{blue}{\sqrt{x \cdot x}}} - 1\right)} \]

      8. sqr-neg 3.1%

         \[\leadsto 1 - \frac{1 \cdot -1}{0 - \left(e^{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}} - 1\right)} \]

      9. sqrt-unprod 3.1%

         \[\leadsto 1 - \frac{1 \cdot -1}{0 - \left(e^{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} - 1\right)} \]

      10. add-sqr-sqrt 3.1%

          \[\leadsto 1 - \frac{1 \cdot -1}{0 - \left(e^{\color{blue}{-x}} - 1\right)} \]

      11. associate-+l- 3.1%

          \[\leadsto 1 - \frac{1 \cdot -1}{\color{blue}{\left(0 - e^{-x}\right) + 1}} \]

      12. neg-sub0 3.1%

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

      13. +-commutative 3.1%

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

      14. sub-neg 3.1%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 3.1%

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

   9. Applied egg-rr 100.0%

      \[\leadsto 1 - \color{blue}{\frac{1}{\mathsf{expm1}\left(x\right)} \cdot -1} \]
   10. Step-by-step derivation

       1. associate-*l/ 100.0%

          \[\leadsto 1 - \color{blue}{\frac{1 \cdot -1}{\mathsf{expm1}\left(x\right)}} \]

       2. metadata-eval 100.0%

          \[\leadsto 1 - \frac{\color{blue}{-1}}{\mathsf{expm1}\left(x\right)} \]

   11. Simplified 100.0%

       \[\leadsto 1 - \color{blue}{\frac{-1}{\mathsf{expm1}\left(x\right)}} \]
   12. Step-by-step derivation

       1. add-exp-log 100.0%

          \[\leadsto 1 - \color{blue}{e^{\log \left(\frac{-1}{\mathsf{expm1}\left(x\right)}\right)}} \]

       2. add-sqr-sqrt 100.0%

          \[\leadsto 1 - e^{\color{blue}{\sqrt{\log \left(\frac{-1}{\mathsf{expm1}\left(x\right)}\right)} \cdot \sqrt{\log \left(\frac{-1}{\mathsf{expm1}\left(x\right)}\right)}}} \]

       3. sqrt-unprod 100.0%

          \[\leadsto 1 - e^{\color{blue}{\sqrt{\log \left(\frac{-1}{\mathsf{expm1}\left(x\right)}\right) \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(x\right)}\right)}}} \]

       4. frac-2neg 100.0%

          \[\leadsto 1 - e^{\sqrt{\log \color{blue}{\left(\frac{--1}{-\mathsf{expm1}\left(x\right)}\right)} \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(x\right)}\right)}} \]

       5. metadata-eval 100.0%

          \[\leadsto 1 - e^{\sqrt{\log \left(\frac{\color{blue}{1}}{-\mathsf{expm1}\left(x\right)}\right) \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(x\right)}\right)}} \]

       6. log-rec 100.0%

          \[\leadsto 1 - e^{\sqrt{\color{blue}{\left(-\log \left(-\mathsf{expm1}\left(x\right)\right)\right)} \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(x\right)}\right)}} \]

       7. frac-2neg 100.0%

          \[\leadsto 1 - e^{\sqrt{\left(-\log \left(-\mathsf{expm1}\left(x\right)\right)\right) \cdot \log \color{blue}{\left(\frac{--1}{-\mathsf{expm1}\left(x\right)}\right)}}} \]

       8. metadata-eval 100.0%

          \[\leadsto 1 - e^{\sqrt{\left(-\log \left(-\mathsf{expm1}\left(x\right)\right)\right) \cdot \log \left(\frac{\color{blue}{1}}{-\mathsf{expm1}\left(x\right)}\right)}} \]

       9. log-rec 100.0%

          \[\leadsto 1 - e^{\sqrt{\left(-\log \left(-\mathsf{expm1}\left(x\right)\right)\right) \cdot \color{blue}{\left(-\log \left(-\mathsf{expm1}\left(x\right)\right)\right)}}} \]

       10. sqr-neg 100.0%

           \[\leadsto 1 - e^{\sqrt{\color{blue}{\log \left(-\mathsf{expm1}\left(x\right)\right) \cdot \log \left(-\mathsf{expm1}\left(x\right)\right)}}} \]

       11. add-sqr-sqrt 100.0%

           \[\leadsto 1 - e^{\sqrt{\log \color{blue}{\left(\sqrt{-\mathsf{expm1}\left(x\right)} \cdot \sqrt{-\mathsf{expm1}\left(x\right)}\right)} \cdot \log \left(-\mathsf{expm1}\left(x\right)\right)}} \]

       12. sqrt-unprod 100.0%

           \[\leadsto 1 - e^{\sqrt{\log \color{blue}{\left(\sqrt{\left(-\mathsf{expm1}\left(x\right)\right) \cdot \left(-\mathsf{expm1}\left(x\right)\right)}\right)} \cdot \log \left(-\mathsf{expm1}\left(x\right)\right)}} \]

       13. sqr-neg 100.0%

           \[\leadsto 1 - e^{\sqrt{\log \left(\sqrt{\color{blue}{\mathsf{expm1}\left(x\right) \cdot \mathsf{expm1}\left(x\right)}}\right) \cdot \log \left(-\mathsf{expm1}\left(x\right)\right)}} \]

       14. sqrt-unprod 0.0%

           \[\leadsto 1 - e^{\sqrt{\log \color{blue}{\left(\sqrt{\mathsf{expm1}\left(x\right)} \cdot \sqrt{\mathsf{expm1}\left(x\right)}\right)} \cdot \log \left(-\mathsf{expm1}\left(x\right)\right)}} \]

       15. add-sqr-sqrt 0.0%

           \[\leadsto 1 - e^{\sqrt{\log \color{blue}{\left(\mathsf{expm1}\left(x\right)\right)} \cdot \log \left(-\mathsf{expm1}\left(x\right)\right)}} \]

       16. add-sqr-sqrt 0.0%

           \[\leadsto 1 - e^{\sqrt{\log \left(\mathsf{expm1}\left(x\right)\right) \cdot \log \color{blue}{\left(\sqrt{-\mathsf{expm1}\left(x\right)} \cdot \sqrt{-\mathsf{expm1}\left(x\right)}\right)}}} \]

       17. sqrt-unprod 0.0%

           \[\leadsto 1 - e^{\sqrt{\log \left(\mathsf{expm1}\left(x\right)\right) \cdot \log \color{blue}{\left(\sqrt{\left(-\mathsf{expm1}\left(x\right)\right) \cdot \left(-\mathsf{expm1}\left(x\right)\right)}\right)}}} \]

   13. Applied egg-rr 100.0%

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

       1. *-commutative 100.0%

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

       2. mul-1-neg 100.0%

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

       3. sub0-neg 100.0%

          \[\leadsto 1 - \left(\color{blue}{\left(0 - e^{x}\right)} + 1\right) \]

       4. associate-+l- 100.0%

          \[\leadsto 1 - \color{blue}{\left(0 - \left(e^{x} - 1\right)\right)} \]

       5. expm1-def 100.0%

          \[\leadsto 1 - \left(0 - \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

       6. sub0-neg 100.0%

          \[\leadsto 1 - \color{blue}{\left(-\mathsf{expm1}\left(x\right)\right)} \]

   15. Simplified 100.0%

       \[\leadsto 1 - \color{blue}{\left(-\mathsf{expm1}\left(x\right)\right)} \]

   if -350 < x

   1. Initial program 5.3%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;1 + \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 3: 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 35.6%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 67.9%

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

5. Final simplification 67.9%

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

Developer target: 37.6% 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 2023301 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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