expq2 (section 3.11)

Percentage Accurate: 37.3% → 99.7%

Time: 5.4s

Alternatives: 4

Speedup: 68.3×

Specification

\[710 > x\]

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: 37.3% 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: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00155:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.00155)
   (/ 1.0 (- 1.0 (exp (- x))))
   (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.00155) {
		tmp = 1.0 / (1.0 - exp(-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 <= (-0.00155d0)) then
        tmp = 1.0d0 / (1.0d0 - exp(-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 <= -0.00155) {
		tmp = 1.0 / (1.0 - Math.exp(-x));
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.00155:
		tmp = 1.0 / (1.0 - math.exp(-x))
	else:
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.00155)
		tmp = Float64(1.0 / Float64(1.0 - exp(Float64(-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 <= -0.00155)
		tmp = 1.0 / (1.0 - exp(-x));
	else
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.00155], N[(1.0 / N[(1.0 - N[Exp[(-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 < -0.00154999999999999995

   1. Initial program 100.0%

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

      1. expm1-define 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. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 100.0%

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

      2. metadata-eval 100.0%

         \[\leadsto {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{\color{blue}{\left(0.5 + 0.5\right)}} \]

      3. pow-prod-up 97.7%

         \[\leadsto \color{blue}{{\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5}} \]

      4. clear-num 97.7%

         \[\leadsto {\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}\right)}}^{0.5} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      5. inv-pow 97.7%

         \[\leadsto {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{-1}\right)}}^{0.5} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      6. metadata-eval 97.7%

         \[\leadsto {\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{0.5} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      7. pow-pow 97.7%

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)}} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      8. expm1-undefine 97.7%

         \[\leadsto {\left(\frac{\color{blue}{e^{x} - 1}}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      9. div-sub 0.0%

         \[\leadsto {\color{blue}{\left(\frac{e^{x}}{e^{x}} - \frac{1}{e^{x}}\right)}}^{\left(\left(-1\right) \cdot 0.5\right)} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      10. pow1 0.0%

          \[\leadsto {\left(\frac{\color{blue}{{\left(e^{x}\right)}^{1}}}{e^{x}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      11. pow1 0.0%

          \[\leadsto {\left(\frac{{\left(e^{x}\right)}^{1}}{\color{blue}{{\left(e^{x}\right)}^{1}}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      12. pow-div 97.7%

          \[\leadsto {\left(\color{blue}{{\left(e^{x}\right)}^{\left(1 - 1\right)}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      13. metadata-eval 97.7%

          \[\leadsto {\left({\left(e^{x}\right)}^{\color{blue}{0}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      14. metadata-eval 97.7%

          \[\leadsto {\left(\color{blue}{1} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      15. rec-exp 97.7%

          \[\leadsto {\left(1 - \color{blue}{e^{-x}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      16. metadata-eval 97.7%

          \[\leadsto {\left(1 - e^{-x}\right)}^{\left(\color{blue}{-1} \cdot 0.5\right)} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      17. metadata-eval 97.7%

          \[\leadsto {\left(1 - e^{-x}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.5} \]

      18. clear-num 97.7%

          \[\leadsto {\left(1 - e^{-x}\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}\right)}}^{0.5} \]

      19. inv-pow 97.7%

          \[\leadsto {\left(1 - e^{-x}\right)}^{-0.5} \cdot {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{-1}\right)}}^{0.5} \]

      20. metadata-eval 97.7%

          \[\leadsto {\left(1 - e^{-x}\right)}^{-0.5} \cdot {\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{0.5} \]

   6. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{{\left(1 - e^{-x}\right)}^{-0.5} \cdot {\left(1 - e^{-x}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. pow-sqr 100.0%

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

      2. metadata-eval 100.0%

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

      3. unpow-1 100.0%

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

   8. Simplified 100.0%

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

   if -0.00154999999999999995 < x

   1. Initial program 5.3%

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

      1. expm1-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00155:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{x \cdot 0.5}}{\mathsf{expm1}\left(x\right)}}{{\left(e^{x}\right)}^{-0.5}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (exp (* x 0.5)) (expm1 x)) (pow (exp x) -0.5)))

C

double code(double x) {
	return (exp((x * 0.5)) / expm1(x)) / pow(exp(x), -0.5);
}

Java

public static double code(double x) {
	return (Math.exp((x * 0.5)) / Math.expm1(x)) / Math.pow(Math.exp(x), -0.5);
}

Python

def code(x):
	return (math.exp((x * 0.5)) / math.expm1(x)) / math.pow(math.exp(x), -0.5)

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * 0.5)) / expm1(x)) / (exp(x) ^ -0.5))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] / N[Power[N[Exp[x], $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.5%

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

   1. expm1-define 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. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. times-frac 100.0%

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

   6. metadata-eval 100.0%

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

   7. unpow-prod-down 100.0%

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

   8. pow1/2 100.0%

      \[\leadsto {\left(\frac{1}{\color{blue}{{\left(e^{x}\right)}^{0.5}}}\right)}^{\left(-1\right)} \cdot {\left(\frac{\mathsf{expm1}\left(x\right)}{\sqrt{e^{x}}}\right)}^{\left(-1\right)} \]

   9. pow-flip 100.0%

      \[\leadsto {\color{blue}{\left({\left(e^{x}\right)}^{\left(-0.5\right)}\right)}}^{\left(-1\right)} \cdot {\left(\frac{\mathsf{expm1}\left(x\right)}{\sqrt{e^{x}}}\right)}^{\left(-1\right)} \]

   10. metadata-eval 100.0%

       \[\leadsto {\left({\left(e^{x}\right)}^{\color{blue}{-0.5}}\right)}^{\left(-1\right)} \cdot {\left(\frac{\mathsf{expm1}\left(x\right)}{\sqrt{e^{x}}}\right)}^{\left(-1\right)} \]

   11. metadata-eval 100.0%

       \[\leadsto {\left({\left(e^{x}\right)}^{-0.5}\right)}^{\color{blue}{-1}} \cdot {\left(\frac{\mathsf{expm1}\left(x\right)}{\sqrt{e^{x}}}\right)}^{\left(-1\right)} \]

   12. metadata-eval 100.0%

       \[\leadsto {\left({\left(e^{x}\right)}^{-0.5}\right)}^{-1} \cdot {\left(\frac{\mathsf{expm1}\left(x\right)}{\sqrt{e^{x}}}\right)}^{\color{blue}{-1}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left({\left(e^{x}\right)}^{-0.5}\right)}^{-1} \cdot {\left(\frac{\mathsf{expm1}\left(x\right)}{\sqrt{e^{x}}}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 100.0%

      \[\leadsto \color{blue}{\frac{1}{{\left(e^{x}\right)}^{-0.5}}} \cdot {\left(\frac{\mathsf{expm1}\left(x\right)}{\sqrt{e^{x}}}\right)}^{-1} \]

   2. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{1 \cdot {\left(\frac{\mathsf{expm1}\left(x\right)}{\sqrt{e^{x}}}\right)}^{-1}}{{\left(e^{x}\right)}^{-0.5}}} \]

   3. *-lft-identity 100.0%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\mathsf{expm1}\left(x\right)}{\sqrt{e^{x}}}\right)}^{-1}}}{{\left(e^{x}\right)}^{-0.5}} \]

   4. unpow-1 100.0%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\sqrt{e^{x}}}}}}{{\left(e^{x}\right)}^{-0.5}} \]

   5. associate-/l* 100.0%

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot \sqrt{e^{x}}}{\mathsf{expm1}\left(x\right)}}}{{\left(e^{x}\right)}^{-0.5}} \]

   6. *-lft-identity 100.0%

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{e^{x}}}}{\mathsf{expm1}\left(x\right)}}{{\left(e^{x}\right)}^{-0.5}} \]

8. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{e^{x}}}{\mathsf{expm1}\left(x\right)}}{{\left(e^{x}\right)}^{-0.5}}} \]
9. Step-by-step derivation

   1. pow1/2 100.0%

      \[\leadsto \frac{\frac{\color{blue}{{\left(e^{x}\right)}^{0.5}}}{\mathsf{expm1}\left(x\right)}}{{\left(e^{x}\right)}^{-0.5}} \]

   2. pow-exp 100.0%

      \[\leadsto \frac{\frac{\color{blue}{e^{x \cdot 0.5}}}{\mathsf{expm1}\left(x\right)}}{{\left(e^{x}\right)}^{-0.5}} \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{\frac{\color{blue}{e^{x \cdot 0.5}}}{\mathsf{expm1}\left(x\right)}}{{\left(e^{x}\right)}^{-0.5}} \]

11. Final simplification 100.0%

    \[\leadsto \frac{\frac{e^{x \cdot 0.5}}{\mathsf{expm1}\left(x\right)}}{{\left(e^{x}\right)}^{-0.5}} \]
12. Add Preprocessing

Alternative 3: 100.0% 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 37.5%

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

   1. expm1-define 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. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 4: 67.2% 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 37.5%

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

   1. expm1-define 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. Add Preprocessing

5. Taylor expanded in x around 0 67.1%

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

6. Final simplification 67.1%

   \[\leadsto \frac{1}{x} \]
7. Add Preprocessing

Developer target: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024039 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

  :herbie-target
  (/ (- 1.0) (expm1 (- x)))

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