expq2 (section 3.11)

Percentage Accurate: 37.9% → 100.0%

Time: 3.5s

Alternatives: 7

Speedup: 40.5×

Specification

\[715 > x\]

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: 37.9% 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 37.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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: 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(-1.0 / expm1(Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 37.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 inf 37.3%

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

   1. expm1-def 100.0%

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

   2. *-lft-identity 100.0%

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

   3. metadata-eval 100.0%

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

   4. times-frac 100.0%

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

   5. neg-mul-1 100.0%

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

   6. associate-/l* 100.0%

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

   7. neg-sub0 100.0%

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

   8. expm1-def 37.3%

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

   9. associate--r- 37.3%

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

   10. neg-sub0 37.3%

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

   11. +-commutative 37.3%

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

   12. sub-neg 37.3%

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

   13. div-sub 3.7%

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

   14. exp-neg 3.7%

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

   15. *-inverses 37.3%

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

   16. expm1-def 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 3: 99.2% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -360:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -360.0) 0.0 (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -360.0) {
		tmp = 0.0;
	} 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 <= (-360.0d0)) then
        tmp = 0.0d0
    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 <= -360.0) {
		tmp = 0.0;
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -360.0:
		tmp = 0.0
	else:
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -360.0)
		tmp = 0.0;
	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 <= -360.0)
		tmp = 0.0;
	else
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -360.0], 0.0, 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 < -360

   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. add-cbrt-cube 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right) \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}}} \]

      2. pow1/3 100.0%

         \[\leadsto \color{blue}{{\left(\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right) \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.3333333333333333}} \]

      3. pow3 100.0%

         \[\leadsto {\color{blue}{\left({\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. clear-num 100.0%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. inv-pow 100.0%

         \[\leadsto {\left({\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. metadata-eval 100.0%

         \[\leadsto {\left({\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{3}\right)}^{0.3333333333333333} \]

      7. pow-pow 100.0%

         \[\leadsto {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      8. expm1-udef 100.0%

         \[\leadsto {\left({\left(\frac{\color{blue}{e^{x} - 1}}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      9. div-sub 0.0%

         \[\leadsto {\left({\color{blue}{\left(\frac{e^{x}}{e^{x}} - \frac{1}{e^{x}}\right)}}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      10. pow1 0.0%

          \[\leadsto {\left({\left(\frac{\color{blue}{{\left(e^{x}\right)}^{1}}}{e^{x}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      11. pow1 0.0%

          \[\leadsto {\left({\left(\frac{{\left(e^{x}\right)}^{1}}{\color{blue}{{\left(e^{x}\right)}^{1}}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      12. pow-div 100.0%

          \[\leadsto {\left({\left(\color{blue}{{\left(e^{x}\right)}^{\left(1 - 1\right)}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      13. metadata-eval 100.0%

          \[\leadsto {\left({\left({\left(e^{x}\right)}^{\color{blue}{0}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      14. metadata-eval 100.0%

          \[\leadsto {\left({\left(\color{blue}{1} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      15. rec-exp 100.0%

          \[\leadsto {\left({\left(1 - \color{blue}{e^{-x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      16. metadata-eval 100.0%

          \[\leadsto {\left({\left(1 - e^{-x}\right)}^{\left(\color{blue}{-1} \cdot 3\right)}\right)}^{0.3333333333333333} \]

      17. metadata-eval 100.0%

          \[\leadsto {\left({\left(1 - e^{-x}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left({\left(1 - e^{-x}\right)}^{-3}\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(1 - e^{-x}\right)}^{-3}}} \]

      2. sub-neg 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(1 + \left(-e^{-x}\right)\right)}}^{-3}} \]

      3. +-commutative 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(-e^{-x}\right) + 1\right)}}^{-3}} \]

      4. neg-sub0 100.0%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(0 - e^{-x}\right)} + 1\right)}^{-3}} \]

      5. associate-+l- 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(0 - \left(e^{-x} - 1\right)\right)}}^{-3}} \]

      6. sub0-neg 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(-\left(e^{-x} - 1\right)\right)}}^{-3}} \]

      7. expm1-def 100.0%

         \[\leadsto \sqrt[3]{{\left(-\color{blue}{\mathsf{expm1}\left(-x\right)}\right)}^{-3}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(-\mathsf{expm1}\left(-x\right)\right)}^{-3}}} \]

   8. Taylor expanded in x around 0 63.0%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{{x}^{3}}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 63.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{1}{{x}^{3}}}\right)\right)} \]

      2. expm1-udef 96.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{1}{{x}^{3}}}\right)} - 1} \]

      3. log1p-udef 96.6%

         \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt[3]{\frac{1}{{x}^{3}}}\right)}} - 1 \]

      4. rem-exp-log 96.6%

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

      5. cbrt-div 96.6%

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

      6. metadata-eval 96.6%

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

      7. rem-cbrt-cube 96.6%

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

   10. Applied egg-rr 96.6%

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

   11. Taylor expanded in x around inf 100.0%

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

   if -360 < x

   1. Initial program 5.6%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -360:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 4: 98.7% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -2.0) 0.0 (+ 0.5 (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = 0.0;
	} 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 <= (-2.0d0)) then
        tmp = 0.0d0
    else
        tmp = 0.5d0 + (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = 0.0;
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = 0.0
	else:
		tmp = 0.5 + (1.0 / x)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -2.0], 0.0, N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2

   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. add-cbrt-cube 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right) \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}}} \]

      2. pow1/3 98.9%

         \[\leadsto \color{blue}{{\left(\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right) \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.3333333333333333}} \]

      3. pow3 98.9%

         \[\leadsto {\color{blue}{\left({\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. clear-num 98.9%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. inv-pow 98.9%

         \[\leadsto {\left({\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. metadata-eval 98.9%

         \[\leadsto {\left({\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{3}\right)}^{0.3333333333333333} \]

      7. pow-pow 98.9%

         \[\leadsto {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      8. expm1-udef 98.9%

         \[\leadsto {\left({\left(\frac{\color{blue}{e^{x} - 1}}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      9. div-sub 0.0%

         \[\leadsto {\left({\color{blue}{\left(\frac{e^{x}}{e^{x}} - \frac{1}{e^{x}}\right)}}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      10. pow1 0.0%

          \[\leadsto {\left({\left(\frac{\color{blue}{{\left(e^{x}\right)}^{1}}}{e^{x}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      11. pow1 0.0%

          \[\leadsto {\left({\left(\frac{{\left(e^{x}\right)}^{1}}{\color{blue}{{\left(e^{x}\right)}^{1}}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      12. pow-div 98.9%

          \[\leadsto {\left({\left(\color{blue}{{\left(e^{x}\right)}^{\left(1 - 1\right)}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      13. metadata-eval 98.9%

          \[\leadsto {\left({\left({\left(e^{x}\right)}^{\color{blue}{0}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      14. metadata-eval 98.9%

          \[\leadsto {\left({\left(\color{blue}{1} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      15. rec-exp 98.9%

          \[\leadsto {\left({\left(1 - \color{blue}{e^{-x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      16. metadata-eval 98.9%

          \[\leadsto {\left({\left(1 - e^{-x}\right)}^{\left(\color{blue}{-1} \cdot 3\right)}\right)}^{0.3333333333333333} \]

      17. metadata-eval 98.9%

          \[\leadsto {\left({\left(1 - e^{-x}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

   5. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left({\left(1 - e^{-x}\right)}^{-3}\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(1 - e^{-x}\right)}^{-3}}} \]

      2. sub-neg 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(1 + \left(-e^{-x}\right)\right)}}^{-3}} \]

      3. +-commutative 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(-e^{-x}\right) + 1\right)}}^{-3}} \]

      4. neg-sub0 100.0%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(0 - e^{-x}\right)} + 1\right)}^{-3}} \]

      5. associate-+l- 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(0 - \left(e^{-x} - 1\right)\right)}}^{-3}} \]

      6. sub0-neg 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(-\left(e^{-x} - 1\right)\right)}}^{-3}} \]

      7. expm1-def 100.0%

         \[\leadsto \sqrt[3]{{\left(-\color{blue}{\mathsf{expm1}\left(-x\right)}\right)}^{-3}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(-\mathsf{expm1}\left(-x\right)\right)}^{-3}}} \]

   8. Taylor expanded in x around 0 62.4%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{{x}^{3}}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 62.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{1}{{x}^{3}}}\right)\right)} \]

      2. expm1-udef 95.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{1}{{x}^{3}}}\right)} - 1} \]

      3. log1p-udef 95.7%

         \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt[3]{\frac{1}{{x}^{3}}}\right)}} - 1 \]

      4. rem-exp-log 95.7%

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

      5. cbrt-div 95.7%

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

      6. metadata-eval 95.7%

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

      7. rem-cbrt-cube 95.7%

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

   10. Applied egg-rr 95.7%

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

   11. Taylor expanded in x around inf 98.9%

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

   if -2 < x

   1. Initial program 5.1%

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

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

      1. +-commutative 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \]

Alternative 5: 98.0% accurate, 40.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -360:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -360.0) 0.0 (/ 1.0 x)))

C

double code(double x) {
	double tmp;
	if (x <= -360.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-360.0d0)) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -360.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -360.0:
		tmp = 0.0
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -360.0)
		tmp = 0.0;
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -360.0)
		tmp = 0.0;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -360.0], 0.0, N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -360

   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. add-cbrt-cube 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right) \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}}} \]

      2. pow1/3 100.0%

         \[\leadsto \color{blue}{{\left(\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right) \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.3333333333333333}} \]

      3. pow3 100.0%

         \[\leadsto {\color{blue}{\left({\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. clear-num 100.0%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. inv-pow 100.0%

         \[\leadsto {\left({\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. metadata-eval 100.0%

         \[\leadsto {\left({\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{3}\right)}^{0.3333333333333333} \]

      7. pow-pow 100.0%

         \[\leadsto {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      8. expm1-udef 100.0%

         \[\leadsto {\left({\left(\frac{\color{blue}{e^{x} - 1}}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      9. div-sub 0.0%

         \[\leadsto {\left({\color{blue}{\left(\frac{e^{x}}{e^{x}} - \frac{1}{e^{x}}\right)}}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      10. pow1 0.0%

          \[\leadsto {\left({\left(\frac{\color{blue}{{\left(e^{x}\right)}^{1}}}{e^{x}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      11. pow1 0.0%

          \[\leadsto {\left({\left(\frac{{\left(e^{x}\right)}^{1}}{\color{blue}{{\left(e^{x}\right)}^{1}}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      12. pow-div 100.0%

          \[\leadsto {\left({\left(\color{blue}{{\left(e^{x}\right)}^{\left(1 - 1\right)}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      13. metadata-eval 100.0%

          \[\leadsto {\left({\left({\left(e^{x}\right)}^{\color{blue}{0}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      14. metadata-eval 100.0%

          \[\leadsto {\left({\left(\color{blue}{1} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      15. rec-exp 100.0%

          \[\leadsto {\left({\left(1 - \color{blue}{e^{-x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

      16. metadata-eval 100.0%

          \[\leadsto {\left({\left(1 - e^{-x}\right)}^{\left(\color{blue}{-1} \cdot 3\right)}\right)}^{0.3333333333333333} \]

      17. metadata-eval 100.0%

          \[\leadsto {\left({\left(1 - e^{-x}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left({\left(1 - e^{-x}\right)}^{-3}\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(1 - e^{-x}\right)}^{-3}}} \]

      2. sub-neg 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(1 + \left(-e^{-x}\right)\right)}}^{-3}} \]

      3. +-commutative 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(-e^{-x}\right) + 1\right)}}^{-3}} \]

      4. neg-sub0 100.0%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(0 - e^{-x}\right)} + 1\right)}^{-3}} \]

      5. associate-+l- 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(0 - \left(e^{-x} - 1\right)\right)}}^{-3}} \]

      6. sub0-neg 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(-\left(e^{-x} - 1\right)\right)}}^{-3}} \]

      7. expm1-def 100.0%

         \[\leadsto \sqrt[3]{{\left(-\color{blue}{\mathsf{expm1}\left(-x\right)}\right)}^{-3}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(-\mathsf{expm1}\left(-x\right)\right)}^{-3}}} \]

   8. Taylor expanded in x around 0 63.0%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{{x}^{3}}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 63.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{1}{{x}^{3}}}\right)\right)} \]

      2. expm1-udef 96.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{1}{{x}^{3}}}\right)} - 1} \]

      3. log1p-udef 96.6%

         \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt[3]{\frac{1}{{x}^{3}}}\right)}} - 1 \]

      4. rem-exp-log 96.6%

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

      5. cbrt-div 96.6%

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

      6. metadata-eval 96.6%

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

      7. rem-cbrt-cube 96.6%

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

   10. Applied egg-rr 96.6%

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

   11. Taylor expanded in x around inf 100.0%

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

   if -360 < x

   1. Initial program 5.6%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 97.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -360:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

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 37.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 66.5%

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

   1. +-commutative 66.5%

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

6. Simplified 66.5%

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

7. Taylor expanded in x around inf 3.2%

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

8. Final simplification 3.2%

   \[\leadsto 0.5 \]

Alternative 7: 34.7% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 37.3%

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

   1. add-cbrt-cube 60.8%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right) \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}}} \]

   2. pow1/3 44.6%

      \[\leadsto \color{blue}{{\left(\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right) \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{0.3333333333333333}} \]

   3. pow3 44.6%

      \[\leadsto {\color{blue}{\left({\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

   4. clear-num 44.6%

      \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

   5. inv-pow 44.6%

      \[\leadsto {\left({\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. metadata-eval 44.6%

      \[\leadsto {\left({\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{3}\right)}^{0.3333333333333333} \]

   7. pow-pow 44.6%

      \[\leadsto {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}}^{0.3333333333333333} \]

   8. expm1-udef 35.9%

      \[\leadsto {\left({\left(\frac{\color{blue}{e^{x} - 1}}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

   9. div-sub 2.3%

      \[\leadsto {\left({\color{blue}{\left(\frac{e^{x}}{e^{x}} - \frac{1}{e^{x}}\right)}}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

   10. pow1 2.3%

       \[\leadsto {\left({\left(\frac{\color{blue}{{\left(e^{x}\right)}^{1}}}{e^{x}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

   11. pow1 2.3%

       \[\leadsto {\left({\left(\frac{{\left(e^{x}\right)}^{1}}{\color{blue}{{\left(e^{x}\right)}^{1}}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

   12. pow-div 35.9%

       \[\leadsto {\left({\left(\color{blue}{{\left(e^{x}\right)}^{\left(1 - 1\right)}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

   13. metadata-eval 35.9%

       \[\leadsto {\left({\left({\left(e^{x}\right)}^{\color{blue}{0}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

   14. metadata-eval 35.9%

       \[\leadsto {\left({\left(\color{blue}{1} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

   15. rec-exp 35.9%

       \[\leadsto {\left({\left(1 - \color{blue}{e^{-x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} \]

   16. metadata-eval 35.9%

       \[\leadsto {\left({\left(1 - e^{-x}\right)}^{\left(\color{blue}{-1} \cdot 3\right)}\right)}^{0.3333333333333333} \]

   17. metadata-eval 35.9%

       \[\leadsto {\left({\left(1 - e^{-x}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

5. Applied egg-rr 35.9%

   \[\leadsto \color{blue}{{\left({\left(1 - e^{-x}\right)}^{-3}\right)}^{0.3333333333333333}} \]
6. Step-by-step derivation

   1. unpow1/3 37.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(1 - e^{-x}\right)}^{-3}}} \]

   2. sub-neg 37.3%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(1 + \left(-e^{-x}\right)\right)}}^{-3}} \]

   3. +-commutative 37.3%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(-e^{-x}\right) + 1\right)}}^{-3}} \]

   4. neg-sub0 37.3%

      \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(0 - e^{-x}\right)} + 1\right)}^{-3}} \]

   5. associate-+l- 37.3%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(0 - \left(e^{-x} - 1\right)\right)}}^{-3}} \]

   6. sub0-neg 37.3%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(-\left(e^{-x} - 1\right)\right)}}^{-3}} \]

   7. expm1-def 60.8%

      \[\leadsto \sqrt[3]{{\left(-\color{blue}{\mathsf{expm1}\left(-x\right)}\right)}^{-3}} \]

7. Simplified 60.8%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(-\mathsf{expm1}\left(-x\right)\right)}^{-3}}} \]

8. Taylor expanded in x around 0 47.0%

   \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{{x}^{3}}}} \]
9. Step-by-step derivation

   1. expm1-log1p-u 32.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{1}{{x}^{3}}}\right)\right)} \]

   2. expm1-udef 43.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{1}{{x}^{3}}}\right)} - 1} \]

   3. log1p-udef 43.3%

      \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt[3]{\frac{1}{{x}^{3}}}\right)}} - 1 \]

   4. rem-exp-log 58.3%

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

   5. cbrt-div 59.5%

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

   6. metadata-eval 59.5%

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

   7. rem-cbrt-cube 97.5%

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

10. Applied egg-rr 97.5%

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

11. Taylor expanded in x around inf 35.1%

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

12. Final simplification 35.1%

    \[\leadsto 0 \]

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 2023330 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 715.0 x)

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

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