expq2 (section 3.11)

Percentage Accurate: 38.1% → 100.0%

Time: 3.8s

Alternatives: 8

Speedup: 68.3×

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: 38.1% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. clear-num 99.9%

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

   2. inv-pow 99.9%

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

   3. div-inv 100.0%

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

   4. metadata-eval 100.0%

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

   5. unpow-prod-down 100.0%

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

   6. metadata-eval 100.0%

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

   7. inv-pow 100.0%

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

   8. rec-exp 100.0%

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

   9. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 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 32.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 3: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;\frac{e^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.7)
   (/ (exp x) x)
   (+
    0.5
    (+
     (* -0.001388888888888889 (pow x 3.0))
     (+ (* x 0.08333333333333333) (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -3.7) {
		tmp = exp(x) / 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 (x <= (-3.7d0)) then
        tmp = exp(x) / 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 (x <= -3.7) {
		tmp = Math.exp(x) / 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 x <= -3.7:
		tmp = math.exp(x) / 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 (x <= -3.7)
		tmp = Float64(exp(x) / 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 (x <= -3.7)
		tmp = exp(x) / 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[x, -3.7], N[(N[Exp[x], $MachinePrecision] / x), $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 x < -3.7000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if -3.7000000000000002 < x

   1. Initial program 7.6%

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

      1. expm1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;\frac{e^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0021:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0021)
   (/ 1.0 (- 1.0 (exp (- x))))
   (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0021) {
		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 <= (-0.0021d0)) 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 <= -0.0021) {
		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 <= -0.0021:
		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 <= -0.0021)
		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 <= -0.0021)
		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, -0.0021], 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 < -0.00209999999999999987

   1. Initial program 99.8%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 98.6%

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

      2. pow1/2 98.6%

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

      3. clear-num 98.6%

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

      4. inv-pow 98.6%

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

      5. metadata-eval 98.6%

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

      6. pow-pow 98.6%

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

      7. expm1-udef 98.6%

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

      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 98.6%

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

      12. metadata-eval 98.6%

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

      13. metadata-eval 98.6%

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

      14. rec-exp 98.6%

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

      15. metadata-eval 98.6%

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

      16. metadata-eval 98.6%

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

      17. pow1/2 98.6%

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

      18. clear-num 98.6%

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

      19. inv-pow 98.6%

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

      20. metadata-eval 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. pow-sqr 99.8%

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

      2. metadata-eval 99.8%

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

      3. unpow-1 99.8%

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

   7. Simplified 99.8%

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

   if -0.00209999999999999987 < x

   1. Initial program 7.2%

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

      1. expm1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0021:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 5: 99.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9:\\ \;\;\;\;\frac{e^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.9)
   (/ (exp x) x)
   (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.9) {
		tmp = exp(x) / x;
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.9d0)) then
        tmp = exp(x) / x
    else
        tmp = 0.5d0 + ((x * 0.08333333333333333d0) + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.9) {
		tmp = Math.exp(x) / x;
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.9:
		tmp = math.exp(x) / x
	else:
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.9)
		tmp = Float64(exp(x) / x);
	else
		tmp = Float64(0.5 + Float64(Float64(x * 0.08333333333333333) + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.9)
		tmp = exp(x) / x;
	else
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.9], N[(N[Exp[x], $MachinePrecision] / 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 x < -3.89999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if -3.89999999999999991 < x

   1. Initial program 7.6%

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

      1. expm1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9:\\ \;\;\;\;\frac{e^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 6: 66.3% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ 0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x))))

C

double code(double x) {
	return 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 + ((x * 0.08333333333333333d0) + (1.0d0 / x))
end function

Java

public static double code(double x) {
	return 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
}

Python

def code(x):
	return 0.5 + ((x * 0.08333333333333333) + (1.0 / x))

Julia

function code(x)
	return Float64(0.5 + Float64(Float64(x * 0.08333333333333333) + Float64(1.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
end

Wolfram

code[x_] := N[(0.5 + N[(N[(x * 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 72.6%

   \[\leadsto \color{blue}{0.5 + \left(0.08333333333333333 \cdot x + \frac{1}{x}\right)} \]

5. Final simplification 72.6%

   \[\leadsto 0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) \]

Alternative 7: 66.2% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + \frac{1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 0.5 (/ 1.0 x)))

C

double code(double x) {
	return 0.5 + (1.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 + (1.0d0 / x)
end function

Java

public static double code(double x) {
	return 0.5 + (1.0 / x);
}

Python

def code(x):
	return 0.5 + (1.0 / x)

Julia

function code(x)
	return Float64(0.5 + Float64(1.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 + (1.0 / x);
end

Wolfram

code[x_] := N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 71.9%

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

   1. +-commutative 71.9%

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

6. Simplified 71.9%

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

7. Final simplification 71.9%

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

Alternative 8: 66.3% 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 32.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 71.6%

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

5. Final simplification 71.6%

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

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 2023337 
(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)))