expq2 (section 3.11)

Percentage Accurate: 37.5% → 100.0%

Time: 7.3s

Alternatives: 14

Speedup: 68.3×

Specification

\[710 > x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.5% 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, 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 41.5%

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

   1. sub-neg 41.5%

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

   2. +-commutative 41.5%

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

   3. rgt-mult-inverse 6.0%

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

   4. exp-neg 6.0%

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

   5. distribute-rgt-neg-out 6.0%

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

   6. *-rgt-identity 6.0%

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

   7. distribute-lft-in 6.0%

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

   8. neg-sub0 6.0%

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

   9. associate-+l- 6.0%

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

   10. neg-sub0 5.7%

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

   11. associate-/r* 5.7%

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

   12. *-rgt-identity 5.7%

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

   13. associate-*r/ 5.7%

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

   14. rgt-mult-inverse 41.3%

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

   15. distribute-frac-neg2 41.3%

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

   16. distribute-neg-frac 41.3%

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

   17. metadata-eval 41.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -3.9) {
		tmp = exp(x) / x;
	} else {
		tmp = (1.0 + (x * (0.5 + (x * 0.08333333333333333)))) / 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 = (1.0d0 + (x * (0.5d0 + (x * 0.08333333333333333d0)))) / 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 = (1.0 + (x * (0.5 + (x * 0.08333333333333333)))) / x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -3.9], N[(N[Exp[x], $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.1%

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

   if -3.89999999999999991 < x

   1. Initial program 8.8%

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

      1. sub-neg 8.8%

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

      2. +-commutative 8.8%

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

      3. rgt-mult-inverse 8.7%

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

      4. exp-neg 8.7%

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

      5. distribute-rgt-neg-out 8.7%

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

      6. *-rgt-identity 8.7%

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

      7. distribute-lft-in 8.7%

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

      8. neg-sub0 8.7%

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

      9. associate-+l- 8.7%

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

      10. neg-sub0 8.3%

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

      11. associate-/r* 8.3%

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

      12. *-rgt-identity 8.3%

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

      13. associate-*r/ 8.3%

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

      14. rgt-mult-inverse 8.3%

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

      15. distribute-frac-neg2 8.3%

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

      16. distribute-neg-frac 8.3%

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

      17. metadata-eval 8.3%

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

      18. expm1-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.6%

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(0.5 + 0.08333333333333333 \cdot x\right)}{x}} \]
   6. Step-by-step derivation

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

Alternative 3: 91.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (*
   x
   (+
    -1.0
    (* x (+ 0.5 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))))

C

double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0))))))
end function

Java

public static double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666))))));
}

Python

def code(x):
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666))))))

Julia

function code(x)
	return Float64(-1.0 / Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / (x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666))))));
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.5%

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

   1. sub-neg 41.5%

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

   2. +-commutative 41.5%

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

   3. rgt-mult-inverse 6.0%

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

   4. exp-neg 6.0%

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

   5. distribute-rgt-neg-out 6.0%

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

   6. *-rgt-identity 6.0%

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

   7. distribute-lft-in 6.0%

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

   8. neg-sub0 6.0%

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

   9. associate-+l- 6.0%

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

   10. neg-sub0 5.7%

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

   11. associate-/r* 5.7%

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

   12. *-rgt-identity 5.7%

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

   13. associate-*r/ 5.7%

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

   14. rgt-mult-inverse 41.3%

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

   15. distribute-frac-neg2 41.3%

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

   16. distribute-neg-frac 41.3%

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

   17. metadata-eval 41.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 88.6%

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}} \]

6. Final simplification 88.6%

   \[\leadsto \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)} \]
7. Add Preprocessing

Alternative 4: 88.6% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot \left(0.5 + x \cdot 0.08333333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.7)
   (/ -1.0 (* x (+ -1.0 (* x (* x -0.16666666666666666)))))
   (/ (+ 1.0 (* x (+ 0.5 (* x 0.08333333333333333)))) x)))

C

double code(double x) {
	double tmp;
	if (x <= -3.7) {
		tmp = -1.0 / (x * (-1.0 + (x * (x * -0.16666666666666666))));
	} else {
		tmp = (1.0 + (x * (0.5 + (x * 0.08333333333333333)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.7d0)) then
        tmp = (-1.0d0) / (x * ((-1.0d0) + (x * (x * (-0.16666666666666666d0)))))
    else
        tmp = (1.0d0 + (x * (0.5d0 + (x * 0.08333333333333333d0)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.7) {
		tmp = -1.0 / (x * (-1.0 + (x * (x * -0.16666666666666666))));
	} else {
		tmp = (1.0 + (x * (0.5 + (x * 0.08333333333333333)))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.7:
		tmp = -1.0 / (x * (-1.0 + (x * (x * -0.16666666666666666))))
	else:
		tmp = (1.0 + (x * (0.5 + (x * 0.08333333333333333)))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.7)
		tmp = Float64(-1.0 / Float64(x * Float64(-1.0 + Float64(x * Float64(x * -0.16666666666666666)))));
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.08333333333333333)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.7)
		tmp = -1.0 / (x * (-1.0 + (x * (x * -0.16666666666666666))));
	else
		tmp = (1.0 + (x * (0.5 + (x * 0.08333333333333333)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.7], N[(-1.0 / N[(x * N[(-1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. rgt-mult-inverse 1.1%

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

      4. exp-neg 1.1%

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

      5. distribute-rgt-neg-out 1.1%

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

      6. *-rgt-identity 1.1%

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

      7. distribute-lft-in 1.1%

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

      8. neg-sub0 1.1%

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

      9. associate-+l- 1.1%

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

      10. neg-sub0 1.1%

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

      11. associate-/r* 1.1%

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

      12. *-rgt-identity 1.1%

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

      13. associate-*r/ 1.1%

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

      14. rgt-mult-inverse 100.0%

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

      15. distribute-frac-neg2 100.0%

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

      16. distribute-neg-frac 100.0%

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

      17. metadata-eval 100.0%

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

      18. expm1-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 67.0%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)}} \]

   6. Taylor expanded in x around inf 67.0%

      \[\leadsto \frac{-1}{x \cdot \left(x \cdot \color{blue}{\left(-0.16666666666666666 \cdot x\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto \frac{-1}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)} - 1\right)} \]

   8. Simplified 67.0%

      \[\leadsto \frac{-1}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)} - 1\right)} \]

   if -3.7000000000000002 < x

   1. Initial program 8.8%

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

      1. sub-neg 8.8%

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

      2. +-commutative 8.8%

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

      3. rgt-mult-inverse 8.7%

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

      4. exp-neg 8.7%

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

      5. distribute-rgt-neg-out 8.7%

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

      6. *-rgt-identity 8.7%

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

      7. distribute-lft-in 8.7%

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

      8. neg-sub0 8.7%

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

      9. associate-+l- 8.7%

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

      10. neg-sub0 8.3%

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

      11. associate-/r* 8.3%

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

      12. *-rgt-identity 8.3%

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

      13. associate-*r/ 8.3%

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

      14. rgt-mult-inverse 8.3%

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

      15. distribute-frac-neg2 8.3%

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

      16. distribute-neg-frac 8.3%

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

      17. metadata-eval 8.3%

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

      18. expm1-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.6%

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(0.5 + 0.08333333333333333 \cdot x\right)}{x}} \]
   6. Step-by-step derivation

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot \left(0.5 + x \cdot 0.08333333333333333\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ -1.0 (* x (+ -1.0 (* x (+ 0.5 (* x (* x 0.041666666666666664))))))))

C

double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * (x * 0.041666666666666664))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (x * ((-1.0d0) + (x * (0.5d0 + (x * (x * 0.041666666666666664d0))))))
end function

Java

public static double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * (x * 0.041666666666666664))))));
}

Python

def code(x):
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * (x * 0.041666666666666664))))))

Julia

function code(x)
	return Float64(-1.0 / Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664)))))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / (x * (-1.0 + (x * (0.5 + (x * (x * 0.041666666666666664))))));
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.5%

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

   1. sub-neg 41.5%

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

   2. +-commutative 41.5%

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

   3. rgt-mult-inverse 6.0%

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

   4. exp-neg 6.0%

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

   5. distribute-rgt-neg-out 6.0%

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

   6. *-rgt-identity 6.0%

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

   7. distribute-lft-in 6.0%

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

   8. neg-sub0 6.0%

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

   9. associate-+l- 6.0%

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

   10. neg-sub0 5.7%

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

   11. associate-/r* 5.7%

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

   12. *-rgt-identity 5.7%

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

   13. associate-*r/ 5.7%

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

   14. rgt-mult-inverse 41.3%

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

   15. distribute-frac-neg2 41.3%

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

   16. distribute-neg-frac 41.3%

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

   17. metadata-eval 41.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 88.6%

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}} \]

6. Taylor expanded in x around inf 88.0%

   \[\leadsto \frac{-1}{x \cdot \left(x \cdot \left(0.5 + x \cdot \color{blue}{\left(0.041666666666666664 \cdot x\right)}\right) - 1\right)} \]
7. Step-by-step derivation

   1. *-commutative 88.0%

      \[\leadsto \frac{-1}{x \cdot \left(x \cdot \left(0.5 + x \cdot \color{blue}{\left(x \cdot 0.041666666666666664\right)}\right) - 1\right)} \]

8. Simplified 88.0%

   \[\leadsto \frac{-1}{x \cdot \left(x \cdot \left(0.5 + x \cdot \color{blue}{\left(x \cdot 0.041666666666666664\right)}\right) - 1\right)} \]

9. Final simplification 88.0%

   \[\leadsto \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \]
10. Add Preprocessing

Alternative 6: 88.1% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9:\\ \;\;\;\;\frac{\frac{\frac{4}{x}}{-x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 - \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.9) (/ (/ (/ 4.0 x) (- x)) x) (- 0.5 (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -2.9) {
		tmp = ((4.0 / x) / -x) / x;
	} 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 <= (-2.9d0)) then
        tmp = ((4.0d0 / x) / -x) / x
    else
        tmp = 0.5d0 - ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.9) {
		tmp = ((4.0 / x) / -x) / x;
	} else {
		tmp = 0.5 - (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.9:
		tmp = ((4.0 / x) / -x) / x
	else:
		tmp = 0.5 - (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.9)
		tmp = Float64(Float64(Float64(4.0 / x) / Float64(-x)) / x);
	else
		tmp = Float64(0.5 - Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.9)
		tmp = ((4.0 / x) / -x) / x;
	else
		tmp = 0.5 - (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.9], N[(N[(N[(4.0 / x), $MachinePrecision] / (-x)), $MachinePrecision] / x), $MachinePrecision], N[(0.5 - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.89999999999999991

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. rgt-mult-inverse 1.1%

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

      4. exp-neg 1.1%

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

      5. distribute-rgt-neg-out 1.1%

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

      6. *-rgt-identity 1.1%

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

      7. distribute-lft-in 1.1%

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

      8. neg-sub0 1.1%

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

      9. associate-+l- 1.1%

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

      10. neg-sub0 1.1%

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

      11. associate-/r* 1.1%

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

      12. *-rgt-identity 1.1%

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

      13. associate-*r/ 1.1%

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

      14. rgt-mult-inverse 100.0%

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

      15. distribute-frac-neg2 100.0%

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

      16. distribute-neg-frac 100.0%

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

      17. metadata-eval 100.0%

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

      18. expm1-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 44.5%

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

      1. *-un-lft-identity 44.5%

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

      2. associate-/r* 43.2%

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

      3. *-commutative 43.2%

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

      4. fma-neg 43.2%

         \[\leadsto 1 \cdot \frac{\frac{-1}{x}}{\color{blue}{\mathsf{fma}\left(x, 0.5, -1\right)}} \]

      5. metadata-eval 43.2%

         \[\leadsto 1 \cdot \frac{\frac{-1}{x}}{\mathsf{fma}\left(x, 0.5, \color{blue}{-1}\right)} \]

   7. Applied egg-rr 43.2%

      \[\leadsto \color{blue}{1 \cdot \frac{\frac{-1}{x}}{\mathsf{fma}\left(x, 0.5, -1\right)}} \]
   8. Step-by-step derivation

      1. *-lft-identity 43.2%

         \[\leadsto \color{blue}{\frac{\frac{-1}{x}}{\mathsf{fma}\left(x, 0.5, -1\right)}} \]

      2. associate-/l/ 44.5%

         \[\leadsto \color{blue}{\frac{-1}{\mathsf{fma}\left(x, 0.5, -1\right) \cdot x}} \]

      3. associate-/r* 43.2%

         \[\leadsto \color{blue}{\frac{\frac{-1}{\mathsf{fma}\left(x, 0.5, -1\right)}}{x}} \]

   9. Simplified 43.2%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\mathsf{fma}\left(x, 0.5, -1\right)}}{x}} \]

   10. Taylor expanded in x around inf 43.2%

       \[\leadsto \frac{\color{blue}{-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x}}}{x} \]
   11. Step-by-step derivation

       1. mul-1-neg 43.2%

          \[\leadsto \frac{\color{blue}{-\frac{2 + 4 \cdot \frac{1}{x}}{x}}}{x} \]

       2. distribute-neg-frac2 43.2%

          \[\leadsto \frac{\color{blue}{\frac{2 + 4 \cdot \frac{1}{x}}{-x}}}{x} \]

       3. associate-*r/ 43.2%

          \[\leadsto \frac{\frac{2 + \color{blue}{\frac{4 \cdot 1}{x}}}{-x}}{x} \]

       4. metadata-eval 43.2%

          \[\leadsto \frac{\frac{2 + \frac{\color{blue}{4}}{x}}{-x}}{x} \]

   12. Simplified 43.2%

       \[\leadsto \frac{\color{blue}{\frac{2 + \frac{4}{x}}{-x}}}{x} \]

   13. Taylor expanded in x around 0 66.0%

       \[\leadsto \frac{\frac{\color{blue}{\frac{4}{x}}}{-x}}{x} \]

   if -2.89999999999999991 < x

   1. Initial program 8.8%

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

      1. sub-neg 8.8%

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

      2. +-commutative 8.8%

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

      3. rgt-mult-inverse 8.7%

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

      4. exp-neg 8.7%

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

      5. distribute-rgt-neg-out 8.7%

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

      6. *-rgt-identity 8.7%

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

      7. distribute-lft-in 8.7%

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

      8. neg-sub0 8.7%

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

      9. associate-+l- 8.7%

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

      10. neg-sub0 8.3%

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

      11. associate-/r* 8.3%

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

      12. *-rgt-identity 8.3%

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

      13. associate-*r/ 8.3%

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

      14. rgt-mult-inverse 8.3%

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

      15. distribute-frac-neg2 8.3%

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

      16. distribute-neg-frac 8.3%

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

      17. metadata-eval 8.3%

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

      18. expm1-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}} \]

   6. Taylor expanded in x around 0 97.5%

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

      1. +-commutative 97.5%

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

      2. *-commutative 97.5%

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

      3. metadata-eval 97.5%

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

      4. sub-neg 97.5%

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

      5. div-sub 97.5%

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

      6. associate-*r/ 97.5%

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

      7. metadata-eval 97.5%

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

      8. associate-*r/ 97.5%

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

      9. *-commutative 97.5%

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

      10. associate-*r* 97.5%

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

      11. rgt-mult-inverse 97.5%

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

      12. metadata-eval 97.5%

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

   8. Simplified 97.5%

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

Alternative 7: 88.5% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ -1.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666)))))))

C

double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))
end function

Java

public static double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
}

Python

def code(x):
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))

Julia

function code(x)
	return Float64(-1.0 / Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.5%

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

   1. sub-neg 41.5%

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

   2. +-commutative 41.5%

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

   3. rgt-mult-inverse 6.0%

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

   4. exp-neg 6.0%

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

   5. distribute-rgt-neg-out 6.0%

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

   6. *-rgt-identity 6.0%

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

   7. distribute-lft-in 6.0%

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

   8. neg-sub0 6.0%

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

   9. associate-+l- 6.0%

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

   10. neg-sub0 5.7%

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

   11. associate-/r* 5.7%

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

   12. *-rgt-identity 5.7%

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

   13. associate-*r/ 5.7%

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

   14. rgt-mult-inverse 41.3%

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

   15. distribute-frac-neg2 41.3%

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

   16. distribute-neg-frac 41.3%

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

   17. metadata-eval 41.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 86.8%

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)}} \]

6. Final simplification 86.8%

   \[\leadsto \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)} \]
7. Add Preprocessing

Alternative 8: 82.9% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\frac{-1}{x \cdot \left(x \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 - \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.75) (/ -1.0 (* x (* x 0.5))) (- 0.5 (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = -1.0 / (x * (x * 0.5));
	} 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 <= (-1.75d0)) then
        tmp = (-1.0d0) / (x * (x * 0.5d0))
    else
        tmp = 0.5d0 - ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = -1.0 / (x * (x * 0.5));
	} else {
		tmp = 0.5 - (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.75:
		tmp = -1.0 / (x * (x * 0.5))
	else:
		tmp = 0.5 - (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.75)
		tmp = Float64(-1.0 / Float64(x * Float64(x * 0.5)));
	else
		tmp = Float64(0.5 - Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.75)
		tmp = -1.0 / (x * (x * 0.5));
	else
		tmp = 0.5 - (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.75], N[(-1.0 / N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. rgt-mult-inverse 1.1%

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

      4. exp-neg 1.1%

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

      5. distribute-rgt-neg-out 1.1%

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

      6. *-rgt-identity 1.1%

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

      7. distribute-lft-in 1.1%

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

      8. neg-sub0 1.1%

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

      9. associate-+l- 1.1%

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

      10. neg-sub0 1.1%

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

      11. associate-/r* 1.1%

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

      12. *-rgt-identity 1.1%

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

      13. associate-*r/ 1.1%

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

      14. rgt-mult-inverse 100.0%

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

      15. distribute-frac-neg2 100.0%

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

      16. distribute-neg-frac 100.0%

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

      17. metadata-eval 100.0%

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

      18. expm1-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 44.5%

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

   6. Taylor expanded in x around inf 44.5%

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

      1. *-commutative 44.5%

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

   8. Simplified 44.5%

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

   if -1.75 < x

   1. Initial program 8.8%

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

      1. sub-neg 8.8%

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

      2. +-commutative 8.8%

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

      3. rgt-mult-inverse 8.7%

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

      4. exp-neg 8.7%

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

      5. distribute-rgt-neg-out 8.7%

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

      6. *-rgt-identity 8.7%

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

      7. distribute-lft-in 8.7%

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

      8. neg-sub0 8.7%

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

      9. associate-+l- 8.7%

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

      10. neg-sub0 8.3%

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

      11. associate-/r* 8.3%

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

      12. *-rgt-identity 8.3%

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

      13. associate-*r/ 8.3%

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

      14. rgt-mult-inverse 8.3%

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

      15. distribute-frac-neg2 8.3%

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

      16. distribute-neg-frac 8.3%

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

      17. metadata-eval 8.3%

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

      18. expm1-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}} \]

   6. Taylor expanded in x around 0 97.5%

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

      1. +-commutative 97.5%

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

      2. *-commutative 97.5%

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

      3. metadata-eval 97.5%

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

      4. sub-neg 97.5%

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

      5. div-sub 97.5%

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

      6. associate-*r/ 97.5%

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

      7. metadata-eval 97.5%

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

      8. associate-*r/ 97.5%

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

      9. *-commutative 97.5%

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

      10. associate-*r* 97.5%

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

      11. rgt-mult-inverse 97.5%

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

      12. metadata-eval 97.5%

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

   8. Simplified 97.5%

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

Alternative 9: 87.6% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x \cdot \left(-1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ -1.0 (* x (+ -1.0 (* x (* x -0.16666666666666666))))))

C

double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (x * -0.16666666666666666))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (x * ((-1.0d0) + (x * (x * (-0.16666666666666666d0)))))
end function

Java

public static double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (x * -0.16666666666666666))));
}

Python

def code(x):
	return -1.0 / (x * (-1.0 + (x * (x * -0.16666666666666666))))

Julia

function code(x)
	return Float64(-1.0 / Float64(x * Float64(-1.0 + Float64(x * Float64(x * -0.16666666666666666)))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / (x * (-1.0 + (x * (x * -0.16666666666666666))));
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(-1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.5%

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

   1. sub-neg 41.5%

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

   2. +-commutative 41.5%

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

   3. rgt-mult-inverse 6.0%

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

   4. exp-neg 6.0%

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

   5. distribute-rgt-neg-out 6.0%

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

   6. *-rgt-identity 6.0%

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

   7. distribute-lft-in 6.0%

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

   8. neg-sub0 6.0%

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

   9. associate-+l- 6.0%

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

   10. neg-sub0 5.7%

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

   11. associate-/r* 5.7%

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

   12. *-rgt-identity 5.7%

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

   13. associate-*r/ 5.7%

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

   14. rgt-mult-inverse 41.3%

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

   15. distribute-frac-neg2 41.3%

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

   16. distribute-neg-frac 41.3%

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

   17. metadata-eval 41.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 86.8%

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)}} \]

6. Taylor expanded in x around inf 85.6%

   \[\leadsto \frac{-1}{x \cdot \left(x \cdot \color{blue}{\left(-0.16666666666666666 \cdot x\right)} - 1\right)} \]
7. Step-by-step derivation

   1. *-commutative 85.6%

      \[\leadsto \frac{-1}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)} - 1\right)} \]

8. Simplified 85.6%

   \[\leadsto \frac{-1}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)} - 1\right)} \]

9. Final simplification 85.6%

   \[\leadsto \frac{-1}{x \cdot \left(-1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)} \]
10. Add Preprocessing

Alternative 10: 82.7% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 - \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.75) (/ (/ -2.0 x) x) (- 0.5 (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = (-2.0 / x) / x;
	} 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 <= (-1.75d0)) then
        tmp = ((-2.0d0) / x) / x
    else
        tmp = 0.5d0 - ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = (-2.0 / x) / x;
	} else {
		tmp = 0.5 - (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.75:
		tmp = (-2.0 / x) / x
	else:
		tmp = 0.5 - (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.75)
		tmp = Float64(Float64(-2.0 / x) / x);
	else
		tmp = Float64(0.5 - Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.75)
		tmp = (-2.0 / x) / x;
	else
		tmp = 0.5 - (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.75], N[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(0.5 - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. rgt-mult-inverse 1.1%

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

      4. exp-neg 1.1%

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

      5. distribute-rgt-neg-out 1.1%

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

      6. *-rgt-identity 1.1%

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

      7. distribute-lft-in 1.1%

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

      8. neg-sub0 1.1%

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

      9. associate-+l- 1.1%

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

      10. neg-sub0 1.1%

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

      11. associate-/r* 1.1%

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

      12. *-rgt-identity 1.1%

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

      13. associate-*r/ 1.1%

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

      14. rgt-mult-inverse 100.0%

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

      15. distribute-frac-neg2 100.0%

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

      16. distribute-neg-frac 100.0%

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

      17. metadata-eval 100.0%

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

      18. expm1-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 44.5%

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

      1. *-un-lft-identity 44.5%

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

      2. associate-/r* 43.2%

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

      3. *-commutative 43.2%

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

      4. fma-neg 43.2%

         \[\leadsto 1 \cdot \frac{\frac{-1}{x}}{\color{blue}{\mathsf{fma}\left(x, 0.5, -1\right)}} \]

      5. metadata-eval 43.2%

         \[\leadsto 1 \cdot \frac{\frac{-1}{x}}{\mathsf{fma}\left(x, 0.5, \color{blue}{-1}\right)} \]

   7. Applied egg-rr 43.2%

      \[\leadsto \color{blue}{1 \cdot \frac{\frac{-1}{x}}{\mathsf{fma}\left(x, 0.5, -1\right)}} \]
   8. Step-by-step derivation

      1. *-lft-identity 43.2%

         \[\leadsto \color{blue}{\frac{\frac{-1}{x}}{\mathsf{fma}\left(x, 0.5, -1\right)}} \]

      2. associate-/l/ 44.5%

         \[\leadsto \color{blue}{\frac{-1}{\mathsf{fma}\left(x, 0.5, -1\right) \cdot x}} \]

      3. associate-/r* 43.2%

         \[\leadsto \color{blue}{\frac{\frac{-1}{\mathsf{fma}\left(x, 0.5, -1\right)}}{x}} \]

   9. Simplified 43.2%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\mathsf{fma}\left(x, 0.5, -1\right)}}{x}} \]

   10. Taylor expanded in x around inf 43.2%

       \[\leadsto \frac{\color{blue}{\frac{-2}{x}}}{x} \]

   if -1.75 < x

   1. Initial program 8.8%

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

      1. sub-neg 8.8%

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

      2. +-commutative 8.8%

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

      3. rgt-mult-inverse 8.7%

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

      4. exp-neg 8.7%

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

      5. distribute-rgt-neg-out 8.7%

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

      6. *-rgt-identity 8.7%

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

      7. distribute-lft-in 8.7%

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

      8. neg-sub0 8.7%

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

      9. associate-+l- 8.7%

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

      10. neg-sub0 8.3%

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

      11. associate-/r* 8.3%

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

      12. *-rgt-identity 8.3%

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

      13. associate-*r/ 8.3%

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

      14. rgt-mult-inverse 8.3%

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

      15. distribute-frac-neg2 8.3%

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

      16. distribute-neg-frac 8.3%

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

      17. metadata-eval 8.3%

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

      18. expm1-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}} \]

   6. Taylor expanded in x around 0 97.5%

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

      1. +-commutative 97.5%

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

      2. *-commutative 97.5%

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

      3. metadata-eval 97.5%

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

      4. sub-neg 97.5%

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

      5. div-sub 97.5%

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

      6. associate-*r/ 97.5%

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

      7. metadata-eval 97.5%

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

      8. associate-*r/ 97.5%

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

      9. *-commutative 97.5%

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

      10. associate-*r* 97.5%

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

      11. rgt-mult-inverse 97.5%

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

      12. metadata-eval 97.5%

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

   8. Simplified 97.5%

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

Alternative 11: 66.8% 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 41.5%

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

   1. sub-neg 41.5%

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

   2. +-commutative 41.5%

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

   3. rgt-mult-inverse 6.0%

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

   4. exp-neg 6.0%

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

   5. distribute-rgt-neg-out 6.0%

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

   6. *-rgt-identity 6.0%

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

   7. distribute-lft-in 6.0%

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

   8. neg-sub0 6.0%

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

   9. associate-+l- 6.0%

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

   10. neg-sub0 5.7%

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

   11. associate-/r* 5.7%

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

   12. *-rgt-identity 5.7%

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

   13. associate-*r/ 5.7%

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

   14. rgt-mult-inverse 41.3%

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

   15. distribute-frac-neg2 41.3%

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

   16. distribute-neg-frac 41.3%

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

   17. metadata-eval 41.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 88.6%

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}} \]

6. Taylor expanded in x around 0 63.6%

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

   1. +-commutative 63.6%

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

   2. *-commutative 63.6%

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

   3. metadata-eval 63.6%

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

   4. sub-neg 63.6%

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

   5. div-sub 63.6%

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

   6. associate-*r/ 63.6%

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

   7. metadata-eval 63.6%

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

   8. associate-*r/ 63.6%

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

   9. *-commutative 63.6%

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

   10. associate-*r* 63.6%

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

   11. rgt-mult-inverse 63.6%

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

   12. metadata-eval 63.6%

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

8. Simplified 63.6%

   \[\leadsto \color{blue}{0.5 - \frac{-1}{x}} \]
9. Add Preprocessing

Alternative 12: 66.9% 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 41.5%

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

   1. sub-neg 41.5%

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

   2. +-commutative 41.5%

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

   3. rgt-mult-inverse 6.0%

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

   4. exp-neg 6.0%

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

   5. distribute-rgt-neg-out 6.0%

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

   6. *-rgt-identity 6.0%

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

   7. distribute-lft-in 6.0%

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

   8. neg-sub0 6.0%

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

   9. associate-+l- 6.0%

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

   10. neg-sub0 5.7%

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

   11. associate-/r* 5.7%

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

   12. *-rgt-identity 5.7%

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

   13. associate-*r/ 5.7%

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

   14. rgt-mult-inverse 41.3%

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

   15. distribute-frac-neg2 41.3%

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

   16. distribute-neg-frac 41.3%

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

   17. metadata-eval 41.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 63.4%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
6. Add Preprocessing

Alternative 13: 3.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 41.5%

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

   1. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 97.1%

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

6. Taylor expanded in x around 0 62.7%

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

7. Taylor expanded in x around inf 3.4%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Alternative 14: 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 41.5%

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

   1. sub-neg 41.5%

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

   2. +-commutative 41.5%

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

   3. rgt-mult-inverse 6.0%

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

   4. exp-neg 6.0%

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

   5. distribute-rgt-neg-out 6.0%

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

   6. *-rgt-identity 6.0%

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

   7. distribute-lft-in 6.0%

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

   8. neg-sub0 6.0%

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

   9. associate-+l- 6.0%

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

   10. neg-sub0 5.7%

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

   11. associate-/r* 5.7%

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

   12. *-rgt-identity 5.7%

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

   13. associate-*r/ 5.7%

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

   14. rgt-mult-inverse 41.3%

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

   15. distribute-frac-neg2 41.3%

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

   16. distribute-neg-frac 41.3%

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

   17. metadata-eval 41.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 63.6%

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

   1. *-commutative 63.6%

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

7. Simplified 63.6%

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

8. Taylor expanded in x around inf 3.4%

   \[\leadsto \color{blue}{0.5} \]
9. Add Preprocessing

Developer Target 1: 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 2024118 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

  :alt
  (! :herbie-platform default (/ (- 1) (expm1 (- x))))

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