expq2 (section 3.11)

Percentage Accurate: 37.8% → 99.2%

Time: 5.2s

Alternatives: 8

Speedup: 40.5×

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: 37.8% 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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.5%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. expm1-log1p-u 63.2%

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

   2. expm1-udef 63.2%

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

   3. log1p-udef 63.2%

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

   4. add-exp-log 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 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 37.5%

   \[\frac{e^{x}}{e^{x} - 1} \]
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: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.9)
   0.0
   (+
    (+
     1.0
     (+
      0.5
      (+
       (* -0.001388888888888889 (pow x 3.0))
       (+ (* x 0.08333333333333333) (/ 1.0 x)))))
    -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -3.9) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + (0.5 + ((-0.001388888888888889 * pow(x, 3.0)) + ((x * 0.08333333333333333) + (1.0 / x))))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.9d0)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 + (0.5d0 + (((-0.001388888888888889d0) * (x ** 3.0d0)) + ((x * 0.08333333333333333d0) + (1.0d0 / x))))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.9) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + (0.5 + ((-0.001388888888888889 * Math.pow(x, 3.0)) + ((x * 0.08333333333333333) + (1.0 / x))))) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.9:
		tmp = 0.0
	else:
		tmp = (1.0 + (0.5 + ((-0.001388888888888889 * math.pow(x, 3.0)) + ((x * 0.08333333333333333) + (1.0 / x))))) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.9)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 + Float64(Float64(-0.001388888888888889 * (x ^ 3.0)) + Float64(Float64(x * 0.08333333333333333) + Float64(1.0 / x))))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.9)
		tmp = 0.0;
	else
		tmp = (1.0 + (0.5 + ((-0.001388888888888889 * (x ^ 3.0)) + ((x * 0.08333333333333333) + (1.0 / x))))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.9], 0.0, N[(N[(1.0 + 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]), $MachinePrecision] + -1.0), $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. 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. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 96.6%

      \[\leadsto \left(1 + \color{blue}{\frac{1}{x}}\right) - 1 \]
   7. Step-by-step derivation

      1. sub-neg 96.6%

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

      2. expm1-log1p-u 96.6%

         \[\leadsto \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}\right) + \left(-1\right) \]

      3. expm1-udef 96.6%

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

      4. log1p-udef 96.6%

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

      5. add-exp-log 96.6%

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

      6. add-exp-log 96.6%

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

      7. log1p-udef 96.6%

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

      8. expm1-udef 96.6%

         \[\leadsto \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}\right) + \left(-1\right) \]

      9. expm1-log1p-u 96.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-prod 96.6%

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

      12. frac-times 96.6%

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

      13. metadata-eval 96.6%

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

      14. metadata-eval 96.6%

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

      15. frac-times 96.6%

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

      16. sqrt-unprod 96.6%

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

      17. add-sqr-sqrt 96.6%

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

      18. metadata-eval 96.6%

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

   8. Applied egg-rr 96.6%

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

   9. Taylor expanded in x around inf 100.0%

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

   if -3.89999999999999991 < x

   1. Initial program 6.5%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. expm1-log1p-u 44.8%

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

      2. expm1-udef 44.8%

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

      3. log1p-udef 44.8%

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

      4. add-exp-log 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\right)\right) + -1\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.9)
   0.0
   (+
    0.5
    (+
     (* -0.001388888888888889 (pow x 3.0))
     (+ (* x 0.08333333333333333) (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -3.9) {
		tmp = 0.0;
	} 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 <= (-3.9d0)) then
        tmp = 0.0d0
    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 <= -3.9) {
		tmp = 0.0;
	} 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 <= -3.9:
		tmp = 0.0
	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 <= -3.9)
		tmp = 0.0;
	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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.9)
		tmp = 0.0;
	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, -3.9], 0.0, 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 < -3.89999999999999991

   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. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 96.6%

      \[\leadsto \left(1 + \color{blue}{\frac{1}{x}}\right) - 1 \]
   7. Step-by-step derivation

      1. sub-neg 96.6%

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

      2. expm1-log1p-u 96.6%

         \[\leadsto \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}\right) + \left(-1\right) \]

      3. expm1-udef 96.6%

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

      4. log1p-udef 96.6%

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

      5. add-exp-log 96.6%

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

      6. add-exp-log 96.6%

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

      7. log1p-udef 96.6%

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

      8. expm1-udef 96.6%

         \[\leadsto \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}\right) + \left(-1\right) \]

      9. expm1-log1p-u 96.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-prod 96.6%

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

      12. frac-times 96.6%

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

      13. metadata-eval 96.6%

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

      14. metadata-eval 96.6%

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

      15. frac-times 96.6%

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

      16. sqrt-unprod 96.6%

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

      17. add-sqr-sqrt 96.6%

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

      18. metadata-eval 96.6%

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

   8. Applied egg-rr 96.6%

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

   9. Taylor expanded in x around inf 100.0%

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

   if -3.89999999999999991 < x

   1. Initial program 6.5%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\\ \end{array} \]

Alternative 5: 98.5% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -350.0) 0.0 (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -350.0) {
		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 <= (-350.0d0)) 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 <= -350.0) {
		tmp = 0.0;
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -350.0:
		tmp = 0.0
	else:
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -350.0)
		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 <= -350.0)
		tmp = 0.0;
	else
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -350.0], 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 < -350

   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. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 96.6%

      \[\leadsto \left(1 + \color{blue}{\frac{1}{x}}\right) - 1 \]
   7. Step-by-step derivation

      1. sub-neg 96.6%

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

      2. expm1-log1p-u 96.6%

         \[\leadsto \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}\right) + \left(-1\right) \]

      3. expm1-udef 96.6%

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

      4. log1p-udef 96.6%

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

      5. add-exp-log 96.6%

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

      6. add-exp-log 96.6%

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

      7. log1p-udef 96.6%

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

      8. expm1-udef 96.6%

         \[\leadsto \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}\right) + \left(-1\right) \]

      9. expm1-log1p-u 96.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-prod 96.6%

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

      12. frac-times 96.6%

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

      13. metadata-eval 96.6%

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

      14. metadata-eval 96.6%

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

      15. frac-times 96.6%

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

      16. sqrt-unprod 96.6%

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

      17. add-sqr-sqrt 96.6%

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

      18. metadata-eval 96.6%

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

   8. Applied egg-rr 96.6%

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

   9. Taylor expanded in x around inf 100.0%

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

   if -350 < x

   1. Initial program 6.5%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 6: 98.1% 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. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 96.6%

      \[\leadsto \left(1 + \color{blue}{\frac{1}{x}}\right) - 1 \]
   7. Step-by-step derivation

      1. sub-neg 96.6%

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

      2. expm1-log1p-u 96.6%

         \[\leadsto \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}\right) + \left(-1\right) \]

      3. expm1-udef 96.6%

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

      4. log1p-udef 96.6%

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

      5. add-exp-log 96.6%

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

      6. add-exp-log 96.6%

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

      7. log1p-udef 96.6%

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

      8. expm1-udef 96.6%

         \[\leadsto \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}\right) + \left(-1\right) \]

      9. expm1-log1p-u 96.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-prod 96.6%

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

      12. frac-times 96.6%

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

      13. metadata-eval 96.6%

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

      14. metadata-eval 96.6%

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

      15. frac-times 96.6%

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

      16. sqrt-unprod 96.6%

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

      17. add-sqr-sqrt 96.6%

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

      18. metadata-eval 96.6%

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

   8. Applied egg-rr 96.6%

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

   9. Taylor expanded in x around inf 100.0%

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

   if -2 < x

   1. Initial program 6.5%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 98.7%

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

      1. +-commutative 98.7%

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

   6. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
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 7: 97.4% accurate, 40.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -350.0) 0.0 (/ 1.0 x)))

C

double code(double x) {
	double tmp;
	if (x <= -350.0) {
		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 <= (-350.0d0)) 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 <= -350.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -350.0:
		tmp = 0.0
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -350.0)
		tmp = 0.0;
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -350.0)
		tmp = 0.0;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -350.0], 0.0, N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -350

   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. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 96.6%

      \[\leadsto \left(1 + \color{blue}{\frac{1}{x}}\right) - 1 \]
   7. Step-by-step derivation

      1. sub-neg 96.6%

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

      2. expm1-log1p-u 96.6%

         \[\leadsto \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}\right) + \left(-1\right) \]

      3. expm1-udef 96.6%

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

      4. log1p-udef 96.6%

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

      5. add-exp-log 96.6%

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

      6. add-exp-log 96.6%

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

      7. log1p-udef 96.6%

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

      8. expm1-udef 96.6%

         \[\leadsto \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}\right) + \left(-1\right) \]

      9. expm1-log1p-u 96.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-prod 96.6%

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

      12. frac-times 96.6%

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

      13. metadata-eval 96.6%

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

      14. metadata-eval 96.6%

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

      15. frac-times 96.6%

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

      16. sqrt-unprod 96.6%

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

      17. add-sqr-sqrt 96.6%

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

      18. metadata-eval 96.6%

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

   8. Applied egg-rr 96.6%

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

   9. Taylor expanded in x around inf 100.0%

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

   if -350 < x

   1. Initial program 6.5%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 97.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 8: 34.6% 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 37.5%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. expm1-log1p-u 63.2%

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

   2. expm1-udef 63.2%

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

   3. log1p-udef 63.2%

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

   4. add-exp-log 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 97.2%

   \[\leadsto \left(1 + \color{blue}{\frac{1}{x}}\right) - 1 \]
7. Step-by-step derivation

   1. sub-neg 97.2%

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

   2. expm1-log1p-u 61.5%

      \[\leadsto \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}\right) + \left(-1\right) \]

   3. expm1-udef 61.5%

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

   4. log1p-udef 61.5%

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

   5. add-exp-log 97.2%

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

   6. add-exp-log 61.5%

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

   7. log1p-udef 61.5%

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

   8. expm1-udef 61.5%

      \[\leadsto \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}\right) + \left(-1\right) \]

   9. expm1-log1p-u 97.2%

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

   10. add-sqr-sqrt 32.0%

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

   11. sqrt-prod 50.4%

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

   12. frac-times 50.4%

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

   13. metadata-eval 50.4%

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

   14. metadata-eval 50.4%

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

   15. frac-times 50.4%

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

   16. sqrt-unprod 32.3%

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

   17. add-sqr-sqrt 32.5%

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

   18. metadata-eval 32.5%

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

8. Applied egg-rr 32.5%

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

9. Taylor expanded in x around inf 34.7%

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

10. Final simplification 34.7%

    \[\leadsto 0 \]

Developer target: 38.5% 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 2023264 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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