expq2 (section 3.11)

Percentage Accurate: 37.2% → 99.3%

Time: 4.3s

Alternatives: 3

Speedup: 68.3×

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x}}{e^{x} - 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = exp(x) / (exp(x) - 1.0);
end

Wolfram

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

Initial Program: 37.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x}}{e^{x} - 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = exp(x) / (exp(x) - 1.0);
end

Wolfram

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

Alternative 1: 99.3% 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 32.8%

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

   1. expm1-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\sqrt[3]{{x}^{-3}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.8)
   (cbrt (pow x -3.0))
   (+
    0.5
    (+
     (* -0.001388888888888889 (pow x 3.0))
     (+ (* x 0.08333333333333333) (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -3.8) {
		tmp = cbrt(pow(x, -3.0));
	} else {
		tmp = 0.5 + ((-0.001388888888888889 * pow(x, 3.0)) + ((x * 0.08333333333333333) + (1.0 / x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.8) {
		tmp = Math.cbrt(Math.pow(x, -3.0));
	} 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 <= -3.8)
		tmp = cbrt((x ^ -3.0));
	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

Wolfram

code[x_] := If[LessEqual[x, -3.8], N[Power[N[Power[x, -3.0], $MachinePrecision], 1/3], $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 < -3.7999999999999998

   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. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 3.1%

      \[\leadsto \sqrt[3]{\color{blue}{0.375 + \left(1.5 \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{x} + \frac{1}{{x}^{3}}\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 3.1%

         \[\leadsto \sqrt[3]{\color{blue}{\left(1.5 \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{x} + \frac{1}{{x}^{3}}\right)\right) + 0.375}} \]

      2. associate-+l+ 3.1%

         \[\leadsto \sqrt[3]{\color{blue}{1.5 \cdot \frac{1}{{x}^{2}} + \left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) + 0.375\right)}} \]

      3. unpow2 3.1%

         \[\leadsto \sqrt[3]{1.5 \cdot \frac{1}{\color{blue}{x \cdot x}} + \left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) + 0.375\right)} \]

      4. associate-*r/ 3.1%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1.5 \cdot 1}{x \cdot x}} + \left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) + 0.375\right)} \]

      5. metadata-eval 3.1%

         \[\leadsto \sqrt[3]{\frac{\color{blue}{1.5}}{x \cdot x} + \left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) + 0.375\right)} \]

   8. Simplified 3.1%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1.5}{x \cdot x} + \left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) + 0.375\right)}} \]

   9. Taylor expanded in x around 0 70.0%

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

       1. exp-to-pow 0.0%

          \[\leadsto \sqrt[3]{\frac{1}{\color{blue}{e^{\log x \cdot 3}}}} \]

       2. *-commutative 0.0%

          \[\leadsto \sqrt[3]{\frac{1}{e^{\color{blue}{3 \cdot \log x}}}} \]

       3. exp-neg 0.0%

          \[\leadsto \sqrt[3]{\color{blue}{e^{-3 \cdot \log x}}} \]

       4. distribute-lft-neg-in 0.0%

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

       5. metadata-eval 0.0%

          \[\leadsto \sqrt[3]{e^{\color{blue}{-3} \cdot \log x}} \]

       6. *-commutative 0.0%

          \[\leadsto \sqrt[3]{e^{\color{blue}{\log x \cdot -3}}} \]

       7. exp-to-pow 67.4%

          \[\leadsto \sqrt[3]{\color{blue}{{x}^{-3}}} \]

   11. Simplified 67.4%

       \[\leadsto \sqrt[3]{\color{blue}{{x}^{-3}}} \]

   if -3.7999999999999998 < x

   1. Initial program 5.5%

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

      1. expm1-def 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in x around 0 98.4%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\sqrt[3]{{x}^{-3}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\\ \end{array} \]

Alternative 3: 66.7% 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 32.8%

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

   1. expm1-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 70.6%

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

5. Final simplification 70.6%

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

Developer target: 37.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 - e^{-x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 - exp(-x));
end

Wolfram

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

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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