expq2 (section 3.11)

Percentage Accurate: 38.2% → 100.0%

Time: 9.1s

Alternatives: 19

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: 38.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (exp x) (expm1 x)))

C

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

Java

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

Python

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

Julia

function code(x)
	return Float64(exp(x) / expm1(x))
end

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0.8:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot 0.5}{x} + x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.8)
   (/ 1.0 (+ 1.0 (/ -1.0 (exp x))))
   (+
    (/ (+ 1.0 (* x 0.5)) x)
    (* x (+ 0.08333333333333333 (* -0.001388888888888889 (* x x)))))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.8) {
		tmp = 1.0 / (1.0 + (-1.0 / exp(x)));
	} else {
		tmp = ((1.0 + (x * 0.5)) / x) + (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (exp(x) <= 0.8d0) then
        tmp = 1.0d0 / (1.0d0 + ((-1.0d0) / exp(x)))
    else
        tmp = ((1.0d0 + (x * 0.5d0)) / x) + (x * (0.08333333333333333d0 + ((-0.001388888888888889d0) * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.exp(x) <= 0.8) {
		tmp = 1.0 / (1.0 + (-1.0 / Math.exp(x)));
	} else {
		tmp = ((1.0 + (x * 0.5)) / x) + (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.exp(x) <= 0.8:
		tmp = 1.0 / (1.0 + (-1.0 / math.exp(x)))
	else:
		tmp = ((1.0 + (x * 0.5)) / x) + (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.8)
		tmp = Float64(1.0 / Float64(1.0 + Float64(-1.0 / exp(x))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x * 0.5)) / x) + Float64(x * Float64(0.08333333333333333 + Float64(-0.001388888888888889 * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (exp(x) <= 0.8)
		tmp = 1.0 / (1.0 + (-1.0 / exp(x)));
	else
		tmp = ((1.0 + (x * 0.5)) / x) + (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.8], N[(1.0 / N[(1.0 + N[(-1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(x * N[(0.08333333333333333 + N[(-0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.80000000000000004

   1. Initial program 100.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

      7. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   if 0.80000000000000004 < (exp.f64 x)

   1. Initial program 8.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}{\color{blue}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(x \cdot \frac{1}{2} + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(\frac{1}{2} \cdot x + \color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \frac{1}{x} \cdot \left(\left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x} + \color{blue}{\frac{1}{x}} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \frac{\color{blue}{1}}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \left(\frac{1}{x} \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} \]

      11. lft-mult-inverse N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + 1 \cdot \left(\color{blue}{x} \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right) \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + x \cdot \color{blue}{\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1 + \frac{1}{2} \cdot x}{x}\right), \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 98.9%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + 0.5\right) + x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{1 + \frac{1}{2} \cdot x}{x}\right)}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot x\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   10. Simplified 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0.8:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot 0.5}{x} + x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.75:\\ \;\;\;\;\frac{e^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot 0.5}{x} + x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.75)
   (/ (exp x) x)
   (+
    (/ (+ 1.0 (* x 0.5)) x)
    (* x (+ 0.08333333333333333 (* -0.001388888888888889 (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -3.75) {
		tmp = exp(x) / x;
	} else {
		tmp = ((1.0 + (x * 0.5)) / x) + (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.75d0)) then
        tmp = exp(x) / x
    else
        tmp = ((1.0d0 + (x * 0.5d0)) / x) + (x * (0.08333333333333333d0 + ((-0.001388888888888889d0) * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.75) {
		tmp = Math.exp(x) / x;
	} else {
		tmp = ((1.0 + (x * 0.5)) / x) + (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.75:
		tmp = math.exp(x) / x
	else:
		tmp = ((1.0 + (x * 0.5)) / x) + (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.75)
		tmp = Float64(exp(x) / x);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x * 0.5)) / x) + Float64(x * Float64(0.08333333333333333 + Float64(-0.001388888888888889 * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.75)
		tmp = exp(x) / x;
	else
		tmp = ((1.0 + (x * 0.5)) / x) + (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.75], N[(N[Exp[x], $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(x * N[(0.08333333333333333 + N[(-0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.75

   1. Initial program 100.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{x}\right) \]
   6. Step-by-step derivation

   7. Simplified 96.8%

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

   if -3.75 < x

   1. Initial program 8.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}{\color{blue}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(x \cdot \frac{1}{2} + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(\frac{1}{2} \cdot x + \color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \frac{1}{x} \cdot \left(\left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x} + \color{blue}{\frac{1}{x}} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \frac{\color{blue}{1}}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \left(\frac{1}{x} \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} \]

      11. lft-mult-inverse N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + 1 \cdot \left(\color{blue}{x} \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right) \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + x \cdot \color{blue}{\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1 + \frac{1}{2} \cdot x}{x}\right), \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 98.7%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + 0.5\right) + x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{1 + \frac{1}{2} \cdot x}{x}\right)}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot x\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 98.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   10. Simplified 98.7%

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

Alternative 4: 96.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\\ t_1 := t\_0 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\\ t_2 := x \cdot t\_0\\ \mathbf{if}\;x \leq -2 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.041666666666666664\right)\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(1 - t\_1\right)}{1 - t\_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(1 + t\_1 \cdot t\_2\right)}{1 + t\_2 \cdot \left(t\_2 + -1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+ -0.5 (* x (+ 0.16666666666666666 (* x -0.041666666666666664)))))
        (t_1 (* t_0 (* (* x x) t_0)))
        (t_2 (* x t_0)))
   (if (<= x -2e+103)
     (/ 1.0 (* x (* x (* (* x x) -0.041666666666666664))))
     (if (<= x -5e+51)
       (/ 1.0 (/ (* x (- 1.0 t_1)) (- 1.0 t_2)))
       (/ 1.0 (/ (* x (+ 1.0 (* t_1 t_2))) (+ 1.0 (* t_2 (+ t_2 -1.0)))))))))

C

double code(double x) {
	double t_0 = -0.5 + (x * (0.16666666666666666 + (x * -0.041666666666666664)));
	double t_1 = t_0 * ((x * x) * t_0);
	double t_2 = x * t_0;
	double tmp;
	if (x <= -2e+103) {
		tmp = 1.0 / (x * (x * ((x * x) * -0.041666666666666664)));
	} else if (x <= -5e+51) {
		tmp = 1.0 / ((x * (1.0 - t_1)) / (1.0 - t_2));
	} else {
		tmp = 1.0 / ((x * (1.0 + (t_1 * t_2))) / (1.0 + (t_2 * (t_2 + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (-0.5d0) + (x * (0.16666666666666666d0 + (x * (-0.041666666666666664d0))))
    t_1 = t_0 * ((x * x) * t_0)
    t_2 = x * t_0
    if (x <= (-2d+103)) then
        tmp = 1.0d0 / (x * (x * ((x * x) * (-0.041666666666666664d0))))
    else if (x <= (-5d+51)) then
        tmp = 1.0d0 / ((x * (1.0d0 - t_1)) / (1.0d0 - t_2))
    else
        tmp = 1.0d0 / ((x * (1.0d0 + (t_1 * t_2))) / (1.0d0 + (t_2 * (t_2 + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -0.5 + (x * (0.16666666666666666 + (x * -0.041666666666666664)));
	double t_1 = t_0 * ((x * x) * t_0);
	double t_2 = x * t_0;
	double tmp;
	if (x <= -2e+103) {
		tmp = 1.0 / (x * (x * ((x * x) * -0.041666666666666664)));
	} else if (x <= -5e+51) {
		tmp = 1.0 / ((x * (1.0 - t_1)) / (1.0 - t_2));
	} else {
		tmp = 1.0 / ((x * (1.0 + (t_1 * t_2))) / (1.0 + (t_2 * (t_2 + -1.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = -0.5 + (x * (0.16666666666666666 + (x * -0.041666666666666664)))
	t_1 = t_0 * ((x * x) * t_0)
	t_2 = x * t_0
	tmp = 0
	if x <= -2e+103:
		tmp = 1.0 / (x * (x * ((x * x) * -0.041666666666666664)))
	elif x <= -5e+51:
		tmp = 1.0 / ((x * (1.0 - t_1)) / (1.0 - t_2))
	else:
		tmp = 1.0 / ((x * (1.0 + (t_1 * t_2))) / (1.0 + (t_2 * (t_2 + -1.0))))
	return tmp

Julia

function code(x)
	t_0 = Float64(-0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * -0.041666666666666664))))
	t_1 = Float64(t_0 * Float64(Float64(x * x) * t_0))
	t_2 = Float64(x * t_0)
	tmp = 0.0
	if (x <= -2e+103)
		tmp = Float64(1.0 / Float64(x * Float64(x * Float64(Float64(x * x) * -0.041666666666666664))));
	elseif (x <= -5e+51)
		tmp = Float64(1.0 / Float64(Float64(x * Float64(1.0 - t_1)) / Float64(1.0 - t_2)));
	else
		tmp = Float64(1.0 / Float64(Float64(x * Float64(1.0 + Float64(t_1 * t_2))) / Float64(1.0 + Float64(t_2 * Float64(t_2 + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -0.5 + (x * (0.16666666666666666 + (x * -0.041666666666666664)));
	t_1 = t_0 * ((x * x) * t_0);
	t_2 = x * t_0;
	tmp = 0.0;
	if (x <= -2e+103)
		tmp = 1.0 / (x * (x * ((x * x) * -0.041666666666666664)));
	elseif (x <= -5e+51)
		tmp = 1.0 / ((x * (1.0 - t_1)) / (1.0 - t_2));
	else
		tmp = 1.0 / ((x * (1.0 + (t_1 * t_2))) / (1.0 + (t_2 * (t_2 + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 + N[(x * N[(0.16666666666666666 + N[(x * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[x, -2e+103], N[(1.0 / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e+51], N[(1.0 / N[(N[(x * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * N[(1.0 + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 * N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e103

   1. Initial program 100.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

      7. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 100.0%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{24} \cdot {x}^{3}\right)}\right)\right) \]
   11. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{24} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{24} \cdot \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{24} \cdot x\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{-1}{24}} \cdot x\right)\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{24} \cdot x\right)\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{-1}{24} \cdot x\right)\right)}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{-1}{24}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{24}}\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{24}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{24}\right)\right)\right)\right) \]

   12. Simplified 100.0%

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

   if -2e103 < x < -5e51

   1. Initial program 100.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

      7. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 32.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 32.5%

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(1 + x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot \color{blue}{x}\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 \cdot 1 - \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)}{1 - x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)} \cdot x\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left(1 \cdot 1 - \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)\right) \cdot x}{\color{blue}{1 - x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)}}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)\right) \cdot x\right), \color{blue}{\left(1 - x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)}\right)\right) \]

   11. Applied egg-rr 100.0%

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

   if -5e51 < x

   1. Initial program 17.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

      7. exp-lowering-exp.f64 17.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 17.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 89.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 89.3%

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(1 + x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot \color{blue}{x}\right)\right) \]

       2. flip3-+ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{{1}^{3} + {\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) - 1 \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)\right)} \cdot x\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left({1}^{3} + {\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)}^{3}\right) \cdot x}{\color{blue}{1 \cdot 1 + \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) - 1 \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)\right)}}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left({1}^{3} + {\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)}^{3}\right) \cdot x\right), \color{blue}{\left(1 \cdot 1 + \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) - 1 \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)\right)\right)}\right)\right) \]

   11. Applied egg-rr 92.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(1 + \left(x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)\right) \cdot \left(\left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right) \cdot \left(\left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot x}{1 + \left(x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right) - 1\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.041666666666666664\right)\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(1 - \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)\right)\right)}{1 - x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(1 + \left(\left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)\right)\right) \cdot \left(x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)\right)\right)}{1 + \left(x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right) + -1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.041666666666666664\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(1 - t\_0 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)}{1 - x \cdot t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+ -0.5 (* x (+ 0.16666666666666666 (* x -0.041666666666666664))))))
   (if (<= x -2e+103)
     (/ 1.0 (* x (* x (* (* x x) -0.041666666666666664))))
     (/ 1.0 (/ (* x (- 1.0 (* t_0 (* (* x x) t_0)))) (- 1.0 (* x t_0)))))))

C

double code(double x) {
	double t_0 = -0.5 + (x * (0.16666666666666666 + (x * -0.041666666666666664)));
	double tmp;
	if (x <= -2e+103) {
		tmp = 1.0 / (x * (x * ((x * x) * -0.041666666666666664)));
	} else {
		tmp = 1.0 / ((x * (1.0 - (t_0 * ((x * x) * t_0)))) / (1.0 - (x * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) + (x * (0.16666666666666666d0 + (x * (-0.041666666666666664d0))))
    if (x <= (-2d+103)) then
        tmp = 1.0d0 / (x * (x * ((x * x) * (-0.041666666666666664d0))))
    else
        tmp = 1.0d0 / ((x * (1.0d0 - (t_0 * ((x * x) * t_0)))) / (1.0d0 - (x * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -0.5 + (x * (0.16666666666666666 + (x * -0.041666666666666664)));
	double tmp;
	if (x <= -2e+103) {
		tmp = 1.0 / (x * (x * ((x * x) * -0.041666666666666664)));
	} else {
		tmp = 1.0 / ((x * (1.0 - (t_0 * ((x * x) * t_0)))) / (1.0 - (x * t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = -0.5 + (x * (0.16666666666666666 + (x * -0.041666666666666664)))
	tmp = 0
	if x <= -2e+103:
		tmp = 1.0 / (x * (x * ((x * x) * -0.041666666666666664)))
	else:
		tmp = 1.0 / ((x * (1.0 - (t_0 * ((x * x) * t_0)))) / (1.0 - (x * t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(-0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * -0.041666666666666664))))
	tmp = 0.0
	if (x <= -2e+103)
		tmp = Float64(1.0 / Float64(x * Float64(x * Float64(Float64(x * x) * -0.041666666666666664))));
	else
		tmp = Float64(1.0 / Float64(Float64(x * Float64(1.0 - Float64(t_0 * Float64(Float64(x * x) * t_0)))) / Float64(1.0 - Float64(x * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -0.5 + (x * (0.16666666666666666 + (x * -0.041666666666666664)));
	tmp = 0.0;
	if (x <= -2e+103)
		tmp = 1.0 / (x * (x * ((x * x) * -0.041666666666666664)));
	else
		tmp = 1.0 / ((x * (1.0 - (t_0 * ((x * x) * t_0)))) / (1.0 - (x * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 + N[(x * N[(0.16666666666666666 + N[(x * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e+103], N[(1.0 / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * N[(1.0 - N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2e103

   1. Initial program 100.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

      7. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 100.0%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{24} \cdot {x}^{3}\right)}\right)\right) \]
   11. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{24} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{24} \cdot \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{24} \cdot x\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{-1}{24}} \cdot x\right)\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{24} \cdot x\right)\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{-1}{24} \cdot x\right)\right)}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{-1}{24}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{24}}\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{24}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{24}\right)\right)\right)\right) \]

   12. Simplified 100.0%

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

   if -2e103 < x

   1. Initial program 21.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

      7. exp-lowering-exp.f64 21.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 21.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 86.2%

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(1 + x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot \color{blue}{x}\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 \cdot 1 - \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)}{1 - x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)} \cdot x\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left(1 \cdot 1 - \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)\right) \cdot x}{\color{blue}{1 - x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)}}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)\right) \cdot x\right), \color{blue}{\left(1 - x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right)}\right)\right) \]

   11. Applied egg-rr 90.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(1 - \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right) \cdot \left(\left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right) \cdot \left(x \cdot x\right)\right)\right) \cdot x}{1 - x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.041666666666666664\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(1 - \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)\right)\right)}{1 - x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.4% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot 0.5}{x} + x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.5)
   (/
    1.0
    (*
     x
     (+ 1.0 (* (* x x) (+ 0.16666666666666666 (* x -0.041666666666666664))))))
   (+
    (/ (+ 1.0 (* x 0.5)) x)
    (* x (+ 0.08333333333333333 (* -0.001388888888888889 (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -3.5) {
		tmp = 1.0 / (x * (1.0 + ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664)))));
	} else {
		tmp = ((1.0 + (x * 0.5)) / x) + (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.5d0)) then
        tmp = 1.0d0 / (x * (1.0d0 + ((x * x) * (0.16666666666666666d0 + (x * (-0.041666666666666664d0))))))
    else
        tmp = ((1.0d0 + (x * 0.5d0)) / x) + (x * (0.08333333333333333d0 + ((-0.001388888888888889d0) * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.5) {
		tmp = 1.0 / (x * (1.0 + ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664)))));
	} else {
		tmp = ((1.0 + (x * 0.5)) / x) + (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.5:
		tmp = 1.0 / (x * (1.0 + ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664)))))
	else:
		tmp = ((1.0 + (x * 0.5)) / x) + (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.5)
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * -0.041666666666666664))))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x * 0.5)) / x) + Float64(x * Float64(0.08333333333333333 + Float64(-0.001388888888888889 * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.5)
		tmp = 1.0 / (x * (1.0 + ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664)))));
	else
		tmp = ((1.0 + (x * 0.5)) / x) + (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.5], N[(1.0 / N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(x * N[(0.08333333333333333 + N[(-0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5

   1. Initial program 100.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

      7. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 69.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 69.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} - \frac{1}{24}\right)\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{1}{6} \cdot \frac{1}{x}} - \frac{1}{24}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{1}{x}} - \frac{1}{24}\right)\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} - \frac{1}{24}\right)\right)}\right)\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} + \frac{-1}{24}\right)\right)\right)\right)\right)\right) \]

       6. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \color{blue}{{x}^{2} \cdot \frac{-1}{24}}\right)\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \left(x \cdot x\right) \cdot \frac{-1}{24}\right)\right)\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + x \cdot \color{blue}{\left(x \cdot \frac{-1}{24}\right)}\right)\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + x \cdot \left(\frac{-1}{24} \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right) + \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right) \]

   12. Simplified 69.4%

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

   if -3.5 < x

   1. Initial program 8.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}{\color{blue}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(x \cdot \frac{1}{2} + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(\frac{1}{2} \cdot x + \color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \frac{1}{x} \cdot \left(\left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x} + \color{blue}{\frac{1}{x}} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \frac{\color{blue}{1}}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \left(\frac{1}{x} \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} \]

      11. lft-mult-inverse N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + 1 \cdot \left(\color{blue}{x} \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right) \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + x \cdot \color{blue}{\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1 + \frac{1}{2} \cdot x}{x}\right), \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 98.7%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + 0.5\right) + x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{1 + \frac{1}{2} \cdot x}{x}\right)}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot x\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 98.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   10. Simplified 98.7%

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

Alternative 7: 91.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right) + \left(0.5 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.5)
   (/
    1.0
    (*
     x
     (+ 1.0 (* (* x x) (+ 0.16666666666666666 (* x -0.041666666666666664))))))
   (+
    (* x (+ 0.08333333333333333 (* -0.001388888888888889 (* x x))))
    (+ 0.5 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.5) {
		tmp = 1.0 / (x * (1.0 + ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664)))));
	} else {
		tmp = (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x)))) + (0.5 + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.5d0)) then
        tmp = 1.0d0 / (x * (1.0d0 + ((x * x) * (0.16666666666666666d0 + (x * (-0.041666666666666664d0))))))
    else
        tmp = (x * (0.08333333333333333d0 + ((-0.001388888888888889d0) * (x * x)))) + (0.5d0 + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.5) {
		tmp = 1.0 / (x * (1.0 + ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664)))));
	} else {
		tmp = (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x)))) + (0.5 + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.5:
		tmp = 1.0 / (x * (1.0 + ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664)))))
	else:
		tmp = (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x)))) + (0.5 + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.5)
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * -0.041666666666666664))))));
	else
		tmp = Float64(Float64(x * Float64(0.08333333333333333 + Float64(-0.001388888888888889 * Float64(x * x)))) + Float64(0.5 + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.5)
		tmp = 1.0 / (x * (1.0 + ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664)))));
	else
		tmp = (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x)))) + (0.5 + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.5], N[(1.0 / N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(0.08333333333333333 + N[(-0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5

   1. Initial program 100.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

      7. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 69.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 69.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} - \frac{1}{24}\right)\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{1}{6} \cdot \frac{1}{x}} - \frac{1}{24}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{1}{x}} - \frac{1}{24}\right)\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} - \frac{1}{24}\right)\right)}\right)\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} + \frac{-1}{24}\right)\right)\right)\right)\right)\right) \]

       6. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \color{blue}{{x}^{2} \cdot \frac{-1}{24}}\right)\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \left(x \cdot x\right) \cdot \frac{-1}{24}\right)\right)\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + x \cdot \color{blue}{\left(x \cdot \frac{-1}{24}\right)}\right)\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + x \cdot \left(\frac{-1}{24} \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right) + \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right) \]

   12. Simplified 69.4%

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

   if -3.5 < x

   1. Initial program 8.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}{\color{blue}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(x \cdot \frac{1}{2} + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(\frac{1}{2} \cdot x + \color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \frac{1}{x} \cdot \left(\left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x} + \color{blue}{\frac{1}{x}} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \frac{\color{blue}{1}}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \left(\frac{1}{x} \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} \]

      11. lft-mult-inverse N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + 1 \cdot \left(\color{blue}{x} \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right) \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + x \cdot \color{blue}{\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1 + \frac{1}{2} \cdot x}{x}\right), \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 98.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right) + \left(0.5 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right) + \left(0.5 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.6)
   (/
    1.0
    (* x (* (* x x) (+ 0.16666666666666666 (* x -0.041666666666666664)))))
   (+
    (* x (+ 0.08333333333333333 (* -0.001388888888888889 (* x x))))
    (+ 0.5 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = 1.0 / (x * ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664))));
	} else {
		tmp = (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x)))) + (0.5 + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.6d0)) then
        tmp = 1.0d0 / (x * ((x * x) * (0.16666666666666666d0 + (x * (-0.041666666666666664d0)))))
    else
        tmp = (x * (0.08333333333333333d0 + ((-0.001388888888888889d0) * (x * x)))) + (0.5d0 + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = 1.0 / (x * ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664))));
	} else {
		tmp = (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x)))) + (0.5 + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.6:
		tmp = 1.0 / (x * ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664))))
	else:
		tmp = (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x)))) + (0.5 + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.6)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * -0.041666666666666664)))));
	else
		tmp = Float64(Float64(x * Float64(0.08333333333333333 + Float64(-0.001388888888888889 * Float64(x * x)))) + Float64(0.5 + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.6)
		tmp = 1.0 / (x * ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664))));
	else
		tmp = (x * (0.08333333333333333 + (-0.001388888888888889 * (x * x)))) + (0.5 + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.6], N[(1.0 / N[(x * N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(0.08333333333333333 + N[(-0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009

   1. Initial program 100.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

      7. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 69.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 69.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} - \frac{1}{24}\right)\right)}\right)\right) \]
   11. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{1}{6} \cdot \frac{1}{x}} - \frac{1}{24}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{1}{x}} - \frac{1}{24}\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} - \frac{1}{24}\right)\right)}\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} + \frac{-1}{24}\right)\right)\right)\right)\right) \]

       6. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \color{blue}{{x}^{2} \cdot \frac{-1}{24}}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \left(x \cdot x\right) \cdot \frac{-1}{24}\right)\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + x \cdot \color{blue}{\left(x \cdot \frac{-1}{24}\right)}\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + x \cdot \left(\frac{-1}{24} \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right) + \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{24} \cdot x\right)\right)}\right)\right)\right) \]

   12. Simplified 69.4%

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

   if -3.60000000000000009 < x

   1. Initial program 8.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}{\color{blue}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(x \cdot \frac{1}{2} + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(\frac{1}{2} \cdot x + \color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \frac{1}{x} \cdot \left(\left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x} + \color{blue}{\frac{1}{x}} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \frac{\color{blue}{1}}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \left(\frac{1}{x} \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} \]

      11. lft-mult-inverse N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + 1 \cdot \left(\color{blue}{x} \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right) \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + x \cdot \color{blue}{\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1 + \frac{1}{2} \cdot x}{x}\right), \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 98.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right) + \left(0.5 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.45:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.45)
   (/
    1.0
    (* x (* (* x x) (+ 0.16666666666666666 (* x -0.041666666666666664)))))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.45) {
		tmp = 1.0 / (x * ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664))));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.45d0)) then
        tmp = 1.0d0 / (x * ((x * x) * (0.16666666666666666d0 + (x * (-0.041666666666666664d0)))))
    else
        tmp = (1.0d0 / x) + (0.5d0 + (x * 0.08333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.45) {
		tmp = 1.0 / (x * ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664))));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.45:
		tmp = 1.0 / (x * ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664))))
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.45)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * -0.041666666666666664)))));
	else
		tmp = Float64(Float64(1.0 / x) + Float64(0.5 + Float64(x * 0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.45)
		tmp = 1.0 / (x * ((x * x) * (0.16666666666666666 + (x * -0.041666666666666664))));
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.45], N[(1.0 / N[(x * N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4500000000000002

   1. Initial program 100.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

      7. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 69.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 69.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} - \frac{1}{24}\right)\right)}\right)\right) \]
   11. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{1}{6} \cdot \frac{1}{x}} - \frac{1}{24}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{1}{x}} - \frac{1}{24}\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} - \frac{1}{24}\right)\right)}\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x} + \frac{-1}{24}\right)\right)\right)\right)\right) \]

       6. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \color{blue}{{x}^{2} \cdot \frac{-1}{24}}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \left(x \cdot x\right) \cdot \frac{-1}{24}\right)\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + x \cdot \color{blue}{\left(x \cdot \frac{-1}{24}\right)}\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + x \cdot \left(\frac{-1}{24} \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right) + \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{24} \cdot x\right)\right)}\right)\right)\right) \]

   12. Simplified 69.4%

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

   if -3.4500000000000002 < x

   1. Initial program 8.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)}{\color{blue}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot 1 + \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{1}{x} + \color{blue}{\frac{1}{x}} \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{1}{x} + \left(\frac{1}{x} \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)} \]

      7. lft-mult-inverse N/A

         \[\leadsto \frac{1}{x} + 1 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{12} \cdot x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \frac{1}{x} + \left(\frac{1}{2} + \color{blue}{\frac{1}{12} \cdot x}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{12} \cdot x\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{12} \cdot x\right)}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{1}{12}}\right)\right)\right) \]

      13. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{12}}\right)\right)\right) \]

   7. Simplified 98.2%

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

Alternative 10: 91.3% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

   3. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

   4. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

   7. exp-lowering-exp.f64 37.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

6. Applied egg-rr 37.1%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 89.0%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 89.0%

   \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)\right)}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right) + \color{blue}{1}\right)\right)\right) \]

    2. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot x + \color{blue}{1 \cdot x}\right)\right) \]

    3. *-lft-identity N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot x + x\right)\right) \]

    4. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right) \cdot x\right), \color{blue}{x}\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\left(\left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right) \cdot x\right) \cdot x\right), x\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right) \cdot \left(x \cdot x\right)\right), x\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right), \left(x \cdot x\right)\right), x\right)\right) \]

    8. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right), \left(x \cdot x\right)\right), x\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{6} + x \cdot \frac{-1}{24}\right)\right)\right), \left(x \cdot x\right)\right), x\right)\right) \]

    10. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \frac{-1}{24}\right)\right)\right)\right), \left(x \cdot x\right)\right), x\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{-1}{24}\right)\right)\right)\right), \left(x \cdot x\right)\right), x\right)\right) \]

    12. *-lowering-*.f64 89.0%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{-1}{24}\right)\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), x\right)\right) \]

11. Applied egg-rr 89.0%

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

12. Final simplification 89.0%

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

Alternative 11: 91.3% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

   3. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

   4. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

   7. exp-lowering-exp.f64 37.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

6. Applied egg-rr 37.1%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 89.0%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 89.0%

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

Alternative 12: 91.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.041666666666666664\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2)
   (/ 1.0 (* x (* x (* (* x x) -0.041666666666666664))))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = 1.0 / (x * (x * ((x * x) * -0.041666666666666664)));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.2d0)) then
        tmp = 1.0d0 / (x * (x * ((x * x) * (-0.041666666666666664d0))))
    else
        tmp = (1.0d0 / x) + (0.5d0 + (x * 0.08333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = 1.0 / (x * (x * ((x * x) * -0.041666666666666664)));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = 1.0 / (x * (x * ((x * x) * -0.041666666666666664)))
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(1.0 / Float64(x * Float64(x * Float64(Float64(x * x) * -0.041666666666666664))));
	else
		tmp = Float64(Float64(1.0 / x) + Float64(0.5 + Float64(x * 0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = 1.0 / (x * (x * ((x * x) * -0.041666666666666664)));
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(1.0 / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018

   1. Initial program 100.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

      7. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 69.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 69.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{24} \cdot {x}^{3}\right)}\right)\right) \]
   11. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{24} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{24} \cdot \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{24} \cdot x\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{24} \cdot x\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{-1}{24}} \cdot x\right)\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{24} \cdot x\right)\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{-1}{24} \cdot x\right)\right)}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{-1}{24}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{24}}\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{24}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 69.4%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{24}\right)\right)\right)\right) \]

   12. Simplified 69.4%

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

   if -4.20000000000000018 < x

   1. Initial program 8.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)}{\color{blue}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot 1 + \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{1}{x} + \color{blue}{\frac{1}{x}} \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{1}{x} + \left(\frac{1}{x} \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)} \]

      7. lft-mult-inverse N/A

         \[\leadsto \frac{1}{x} + 1 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{12} \cdot x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \frac{1}{x} + \left(\frac{1}{2} + \color{blue}{\frac{1}{12} \cdot x}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{12} \cdot x\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{12} \cdot x\right)}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{1}{12}}\right)\right)\right) \]

      13. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{12}}\right)\right)\right) \]

   7. Simplified 98.2%

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

Alternative 13: 88.7% 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 37.2%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

   3. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

   4. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

   7. exp-lowering-exp.f64 37.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

6. Applied egg-rr 37.1%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right)}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot x - \frac{1}{2}\right)\right)}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x - \frac{1}{2}\right)}\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot x + \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{6} \cdot x\right), \color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \frac{1}{6}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 87.8%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{6}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

9. Simplified 87.8%

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

10. Final simplification 87.8%

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

Alternative 14: 83.1% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x} - 1}{e^{x}}\right)}\right) \]

   3. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

   4. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{e^{x}}\right)}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{x}\right)}\right)\right)\right) \]

   7. exp-lowering-exp.f64 37.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right) \]

6. Applied egg-rr 37.1%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + \frac{-1}{2} \cdot x\right)\right)}\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 83.7%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

9. Simplified 83.7%

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

Alternative 15: 66.3% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

   \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
6. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto 1 \cdot \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)}{\color{blue}{x}} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)} \]

   4. distribute-lft-in N/A

      \[\leadsto \frac{1}{x} \cdot 1 + \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)} \]

   5. *-rgt-identity N/A

      \[\leadsto \frac{1}{x} + \color{blue}{\frac{1}{x}} \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \frac{1}{x} + \left(\frac{1}{x} \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)} \]

   7. lft-mult-inverse N/A

      \[\leadsto \frac{1}{x} + 1 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{12} \cdot x\right) \]

   8. *-lft-identity N/A

      \[\leadsto \frac{1}{x} + \left(\frac{1}{2} + \color{blue}{\frac{1}{12} \cdot x}\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{12} \cdot x\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{12} \cdot x\right)}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{1}{12}}\right)\right)\right) \]

   13. *-lowering-*.f64 68.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{12}}\right)\right)\right) \]

7. Simplified 68.3%

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

Alternative 16: 66.1% 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 37.2%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

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

   1. *-lft-identity N/A

      \[\leadsto \frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto 1 \cdot \frac{1}{x} + \color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot \frac{1}{x}} \]

   4. associate-*l* N/A

      \[\leadsto 1 \cdot \frac{1}{x} + \frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{1}{x}\right)} \]

   5. rgt-mult-inverse N/A

      \[\leadsto 1 \cdot \frac{1}{x} + \frac{1}{2} \cdot 1 \]

   6. metadata-eval N/A

      \[\leadsto 1 \cdot \frac{1}{x} + \frac{1}{2} \]

   7. *-lft-identity N/A

      \[\leadsto \frac{1}{x} + \frac{1}{2} \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{1}{2}}\right) \]

   9. /-lowering-/.f64 68.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{1}{2}\right) \]

7. Simplified 68.0%

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

8. Final simplification 68.0%

   \[\leadsto 0.5 + \frac{1}{x} \]
9. Add Preprocessing

Alternative 17: 66.2% 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 37.2%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

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

   1. /-lowering-/.f64 67.7%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

7. Simplified 67.7%

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

Alternative 18: 3.3% 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 37.2%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

   \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{x}\right) \]
6. Step-by-step derivation

7. Simplified 96.3%

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

8. Taylor expanded in x around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. rgt-mult-inverse N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. associate-/r* N/A

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

   8. unpow2 N/A

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

   9. associate-*r/ N/A

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

   10. +-commutative N/A

       \[\leadsto \frac{\left(x + 1\right) \cdot x}{{x}^{2}} \]

   11. distribute-lft1-in N/A

       \[\leadsto \frac{x \cdot x + x}{{\color{blue}{x}}^{2}} \]

   12. unpow2 N/A

       \[\leadsto \frac{{x}^{2} + x}{{x}^{2}} \]

   13. remove-double-neg N/A

       \[\leadsto \frac{{x}^{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}{{x}^{2}} \]

   14. mul-1-neg N/A

       \[\leadsto \frac{{x}^{2} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}{{x}^{2}} \]

   15. unsub-neg N/A

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

   16. div-sub N/A

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

10. Simplified 67.1%

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

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{1} \]
12. Step-by-step derivation

13. Simplified 3.7%

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

Alternative 19: 3.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 37.2%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

   \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
6. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto 1 \cdot \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}{\color{blue}{x}} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

   4. distribute-lft-in N/A

      \[\leadsto \frac{1}{x} \cdot \left(1 + \left(x \cdot \frac{1}{2} + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{x} \cdot \left(1 + \left(\frac{1}{2} \cdot x + \color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

   6. associate-+r+ N/A

      \[\leadsto \frac{1}{x} \cdot \left(\left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right) \]

   7. distribute-lft-in N/A

      \[\leadsto \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

   8. associate-*l/ N/A

      \[\leadsto \frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x} + \color{blue}{\frac{1}{x}} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

   9. *-lft-identity N/A

      \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \frac{\color{blue}{1}}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \left(\frac{1}{x} \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} \]

   11. lft-mult-inverse N/A

       \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + 1 \cdot \left(\color{blue}{x} \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right) \]

   12. *-lft-identity N/A

       \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + x \cdot \color{blue}{\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)} \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\left(\frac{1 + \frac{1}{2} \cdot x}{x}\right), \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right) \]

7. Simplified 68.4%

   \[\leadsto \color{blue}{\left(\frac{1}{x} + 0.5\right) + x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right)} \]

8. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot x\right), \color{blue}{x}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right), x\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right), x\right) \]

   4. *-lowering-*.f64 67.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), x\right) \]

10. Simplified 67.9%

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

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{1}{2}} \]
12. Step-by-step derivation

13. Simplified 3.3%

    \[\leadsto \color{blue}{0.5} \]
14. 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 2024161 
(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)))