expq2 (section 3.11)

Percentage Accurate: 36.8% → 100.0%

Time: 5.1s

Alternatives: 7

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: 36.8% 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: 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 34.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0.01:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.01)
   (/ 1.0 (- 1.0 (exp (- x))))
   (+
    0.5
    (+
     (* -0.001388888888888889 (pow x 3.0))
     (+ (* x 0.08333333333333333) (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.01) {
		tmp = 1.0 / (1.0 - exp(-x));
	} else {
		tmp = 0.5 + ((-0.001388888888888889 * pow(x, 3.0)) + ((x * 0.08333333333333333) + (1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (exp(x) <= 0.01d0) then
        tmp = 1.0d0 / (1.0d0 - exp(-x))
    else
        tmp = 0.5d0 + (((-0.001388888888888889d0) * (x ** 3.0d0)) + ((x * 0.08333333333333333d0) + (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.exp(x) <= 0.01) {
		tmp = 1.0 / (1.0 - Math.exp(-x));
	} else {
		tmp = 0.5 + ((-0.001388888888888889 * Math.pow(x, 3.0)) + ((x * 0.08333333333333333) + (1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.exp(x) <= 0.01:
		tmp = 1.0 / (1.0 - math.exp(-x))
	else:
		tmp = 0.5 + ((-0.001388888888888889 * math.pow(x, 3.0)) + ((x * 0.08333333333333333) + (1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.01)
		tmp = Float64(1.0 / Float64(1.0 - exp(Float64(-x))));
	else
		tmp = Float64(0.5 + Float64(Float64(-0.001388888888888889 * (x ^ 3.0)) + Float64(Float64(x * 0.08333333333333333) + Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (exp(x) <= 0.01)
		tmp = 1.0 / (1.0 - exp(-x));
	else
		tmp = 0.5 + ((-0.001388888888888889 * (x ^ 3.0)) + ((x * 0.08333333333333333) + (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.01], N[(1.0 / N[(1.0 - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(-0.001388888888888889 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0100000000000000002

   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. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 97.4%

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

      2. pow1/2 97.4%

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

      3. clear-num 97.4%

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

      4. inv-pow 97.4%

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

      5. metadata-eval 97.4%

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

      6. pow-pow 97.4%

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

      7. expm1-udef 97.4%

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

      8. div-sub 0.0%

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

      9. pow1 0.0%

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

      10. pow1 0.0%

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

      11. pow-div 97.4%

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

      12. metadata-eval 97.4%

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

      13. metadata-eval 97.4%

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

      14. rec-exp 97.4%

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

      15. metadata-eval 97.4%

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

      16. metadata-eval 97.4%

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

      17. pow1/2 97.4%

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

      18. clear-num 97.4%

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

      19. inv-pow 97.4%

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

      20. metadata-eval 97.4%

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

   6. Applied egg-rr 97.4%

      \[\leadsto \color{blue}{{\left(1 - e^{-x}\right)}^{-0.5} \cdot {\left(1 - e^{-x}\right)}^{-0.5}} \]
   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 0.0100000000000000002 < (exp.f64 x)

   1. Initial program 6.0%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. Add Preprocessing

   5. Taylor expanded in x around 0 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0.01:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0) (exp x) (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = 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 (exp(x) <= 0.0d0) then
        tmp = 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 (Math.exp(x) <= 0.0) {
		tmp = Math.exp(x);
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.exp(x) <= 0.0:
		tmp = math.exp(x)
	else:
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x))
	return tmp

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.0], N[Exp[x], $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 (exp.f64 x) < 0.0

   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. Add Preprocessing
   5. Step-by-step derivation

      1. add-exp-log 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. exp-prod 100.0%

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

      4. exp-1-e 100.0%

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

      5. log-div 0.0%

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

      6. add-log-exp 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. e-exp-1 100.0%

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

      4. pow-exp 100.0%

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

      5. *-un-lft-identity 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{x}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

   11. Simplified 100.0%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 7.0%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. Add Preprocessing

   5. Taylor expanded in x around 0 98.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Add Preprocessing

5. Taylor expanded in x around 0 97.7%

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

6. Final simplification 97.7%

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

Alternative 5: 67.7% 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 34.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Add Preprocessing

5. Taylor expanded in x around 0 70.1%

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

6. Final simplification 70.1%

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

Alternative 6: 3.2% 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 34.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Add Preprocessing

5. Taylor expanded in x around 0 69.7%

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

   1. +-commutative 69.7%

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

7. Simplified 69.7%

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

8. Taylor expanded in x around inf 3.3%

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

9. Final simplification 3.3%

   \[\leadsto 0.5 \]
10. Add Preprocessing

Alternative 7: 3.3% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 34.3%

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

   1. add-exp-log 62.1%

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

   2. *-un-lft-identity 62.1%

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

   3. exp-prod 62.1%

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

   4. exp-1-e 62.1%

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

   5. log-div 32.8%

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

   6. add-log-exp 32.8%

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

6. Applied egg-rr 32.8%

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

7. Taylor expanded in x around inf 31.7%

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

8. Taylor expanded in x around 0 3.4%

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

9. Final simplification 3.4%

   \[\leadsto 1 \]
10. 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 2024020 
(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)))