expq2 (section 3.11)

Percentage Accurate: 37.3% → 100.0%

Time: 9.8s

Alternatives: 17

Speedup: 68.3×

Specification

\[710 > x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.3% 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 39.3%

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0.95:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.95)
   (/ 1.0 (+ 1.0 (/ -1.0 (exp x))))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.95) {
		tmp = 1.0 / (1.0 + (-1.0 / exp(x)));
	} 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 (exp(x) <= 0.95d0) then
        tmp = 1.0d0 / (1.0d0 + ((-1.0d0) / exp(x)))
    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 (Math.exp(x) <= 0.95) {
		tmp = 1.0 / (1.0 + (-1.0 / Math.exp(x)));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.exp(x) <= 0.95:
		tmp = 1.0 / (1.0 + (-1.0 / math.exp(x)))
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.95)
		tmp = Float64(1.0 / Float64(1.0 + Float64(-1.0 / exp(x))));
	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 (exp(x) <= 0.95)
		tmp = 1.0 / (1.0 + (-1.0 / exp(x)));
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.95], N[(1.0 / N[(1.0 + N[(-1.0 / N[Exp[x], $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 (exp.f64 x) < 0.94999999999999996

   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 99.9%

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

   6. Applied egg-rr 99.9%

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

   if 0.94999999999999996 < (exp.f64 x)

   1. Initial program 7.5%

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

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

   7. Simplified 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0.95:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.8d0)) then
        tmp = exp(x) / x
    else
        tmp = ((1.0d0 / x) + (0.5d0 + (x * 0.08333333333333333d0))) + (x * (x * (x * (-0.001388888888888889d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.8) {
		tmp = Math.exp(x) / x;
	} else {
		tmp = ((1.0 / x) + (0.5 + (x * 0.08333333333333333))) + (x * (x * (x * -0.001388888888888889)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.8:
		tmp = math.exp(x) / x
	else:
		tmp = ((1.0 / x) + (0.5 + (x * 0.08333333333333333))) + (x * (x * (x * -0.001388888888888889)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.8)
		tmp = Float64(exp(x) / x);
	else
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(0.5 + Float64(x * 0.08333333333333333))) + Float64(x * Float64(x * Float64(x * -0.001388888888888889))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.8)
		tmp = exp(x) / x;
	else
		tmp = ((1.0 / x) + (0.5 + (x * 0.08333333333333333))) + (x * (x * (x * -0.001388888888888889)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.8], N[(N[Exp[x], $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998

   1. Initial program 100.0%

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

      1. /-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 100.0%

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

   if -3.7999999999999998 < x

   1. Initial program 8.0%

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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 99.0%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + 0.5\right) + x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. *-lowering-*.f64 99.0%

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

   9. Applied egg-rr 99.0%

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

Alternative 4: 96.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot \left(x \cdot t\_0\right)\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot 0.16666666666666666\right)\right)}\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{+51}:\\ \;\;\;\;t\_0 \cdot \frac{x \cdot x}{t\_1}\\ \mathbf{elif}\;x \leq -3.9:\\ \;\;\;\;t\_1 \cdot \frac{x \cdot x}{t\_0 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\right) + x \cdot \left(x \cdot \left(x \cdot -0.001388888888888889\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (* (* x x) (* x t_0))))
   (if (<= x -5.6e+102)
     (/ 1.0 (* x (+ 1.0 (* x (+ -0.5 (* x 0.16666666666666666))))))
     (if (<= x -2.35e+51)
       (* t_0 (/ (* x x) t_1))
       (if (<= x -3.9)
         (* t_1 (/ (* x x) (* t_0 t_1)))
         (+
          (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))
          (* x (* x (* x -0.001388888888888889)))))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * (x * t_0);
	double tmp;
	if (x <= -5.6e+102) {
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))));
	} else if (x <= -2.35e+51) {
		tmp = t_0 * ((x * x) / t_1);
	} else if (x <= -3.9) {
		tmp = t_1 * ((x * x) / (t_0 * t_1));
	} else {
		tmp = ((1.0 / x) + (0.5 + (x * 0.08333333333333333))) + (x * (x * (x * -0.001388888888888889)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = (x * x) * (x * t_0)
    if (x <= (-5.6d+102)) then
        tmp = 1.0d0 / (x * (1.0d0 + (x * ((-0.5d0) + (x * 0.16666666666666666d0)))))
    else if (x <= (-2.35d+51)) then
        tmp = t_0 * ((x * x) / t_1)
    else if (x <= (-3.9d0)) then
        tmp = t_1 * ((x * x) / (t_0 * t_1))
    else
        tmp = ((1.0d0 / x) + (0.5d0 + (x * 0.08333333333333333d0))) + (x * (x * (x * (-0.001388888888888889d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * (x * t_0);
	double tmp;
	if (x <= -5.6e+102) {
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))));
	} else if (x <= -2.35e+51) {
		tmp = t_0 * ((x * x) / t_1);
	} else if (x <= -3.9) {
		tmp = t_1 * ((x * x) / (t_0 * t_1));
	} else {
		tmp = ((1.0 / x) + (0.5 + (x * 0.08333333333333333))) + (x * (x * (x * -0.001388888888888889)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = (x * x) * (x * t_0)
	tmp = 0
	if x <= -5.6e+102:
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))))
	elif x <= -2.35e+51:
		tmp = t_0 * ((x * x) / t_1)
	elif x <= -3.9:
		tmp = t_1 * ((x * x) / (t_0 * t_1))
	else:
		tmp = ((1.0 / x) + (0.5 + (x * 0.08333333333333333))) + (x * (x * (x * -0.001388888888888889)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(x * x) * Float64(x * t_0))
	tmp = 0.0
	if (x <= -5.6e+102)
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + Float64(x * Float64(-0.5 + Float64(x * 0.16666666666666666))))));
	elseif (x <= -2.35e+51)
		tmp = Float64(t_0 * Float64(Float64(x * x) / t_1));
	elseif (x <= -3.9)
		tmp = Float64(t_1 * Float64(Float64(x * x) / Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(0.5 + Float64(x * 0.08333333333333333))) + Float64(x * Float64(x * Float64(x * -0.001388888888888889))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = (x * x) * (x * t_0);
	tmp = 0.0;
	if (x <= -5.6e+102)
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))));
	elseif (x <= -2.35e+51)
		tmp = t_0 * ((x * x) / t_1);
	elseif (x <= -3.9)
		tmp = t_1 * ((x * x) / (t_0 * t_1));
	else
		tmp = ((1.0 / x) + (0.5 + (x * 0.08333333333333333))) + (x * (x * (x * -0.001388888888888889)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.6e+102], N[(1.0 / N[(x * N[(1.0 + N[(x * N[(-0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.35e+51], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.9], N[(t$95$1 * N[(N[(x * x), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.60000000000000037e102

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   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 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 1.8%

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

       1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       7. *-lowering-*.f64 1.8%

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

   12. Applied egg-rr 1.8%

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

   13. Taylor expanded in x around 0

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 100.0%

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

   15. Simplified 100.0%

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

   if -5.60000000000000037e102 < x < -2.3500000000000001e51

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 1.9%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{3} \cdot \left(\left(\frac{1}{720} + -1 \cdot \frac{\frac{1}{2} + \frac{1}{x}}{{x}^{3}}\right) - \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   10. Simplified 1.9%

       \[\leadsto \color{blue}{\left(0 - x\right) \cdot \left(\frac{\frac{-1}{x} + -0.5}{x} + \left(-0.08333333333333333 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \color{blue}{\left(\frac{-1}{{x}^{2}}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       3. *-lowering-*.f64 3.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 3.8%

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

   15. Applied egg-rr 100.0%

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

   if -2.3500000000000001e51 < x < -3.89999999999999991

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 2.5%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{3} \cdot \left(\left(\frac{1}{720} + -1 \cdot \frac{\frac{1}{2} + \frac{1}{x}}{{x}^{3}}\right) - \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   10. Simplified 2.5%

       \[\leadsto \color{blue}{\left(0 - x\right) \cdot \left(\frac{\frac{-1}{x} + -0.5}{x} + \left(-0.08333333333333333 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \color{blue}{\left(\frac{-1}{{x}^{2}}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       3. *-lowering-*.f64 3.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 3.4%

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

   15. Applied egg-rr 48.8%

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

   if -3.89999999999999991 < x

   1. Initial program 8.0%

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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 99.0%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + 0.5\right) + x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. *-lowering-*.f64 99.0%

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

   9. Applied egg-rr 99.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot 0.16666666666666666\right)\right)}\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{+51}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{x \cdot x}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{elif}\;x \leq -3.9:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \frac{x \cdot x}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\right) + x \cdot \left(x \cdot \left(x \cdot -0.001388888888888889\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot 0.16666666666666666\right)\right)}\\ \mathbf{elif}\;x \leq -3.9:\\ \;\;\;\;\frac{x \cdot \left(x \cdot x\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\right) + x \cdot \left(x \cdot \left(x \cdot -0.001388888888888889\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.6e+102)
   (/ 1.0 (* x (+ 1.0 (* x (+ -0.5 (* x 0.16666666666666666))))))
   (if (<= x -3.9)
     (/ (* x (* x x)) (* (* x x) (* x x)))
     (+
      (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))
      (* x (* x (* x -0.001388888888888889)))))))

C

double code(double x) {
	double tmp;
	if (x <= -5.6e+102) {
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))));
	} else if (x <= -3.9) {
		tmp = (x * (x * x)) / ((x * x) * (x * x));
	} else {
		tmp = ((1.0 / x) + (0.5 + (x * 0.08333333333333333))) + (x * (x * (x * -0.001388888888888889)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.6d+102)) then
        tmp = 1.0d0 / (x * (1.0d0 + (x * ((-0.5d0) + (x * 0.16666666666666666d0)))))
    else if (x <= (-3.9d0)) then
        tmp = (x * (x * x)) / ((x * x) * (x * x))
    else
        tmp = ((1.0d0 / x) + (0.5d0 + (x * 0.08333333333333333d0))) + (x * (x * (x * (-0.001388888888888889d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.6e+102) {
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))));
	} else if (x <= -3.9) {
		tmp = (x * (x * x)) / ((x * x) * (x * x));
	} else {
		tmp = ((1.0 / x) + (0.5 + (x * 0.08333333333333333))) + (x * (x * (x * -0.001388888888888889)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.6e+102:
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))))
	elif x <= -3.9:
		tmp = (x * (x * x)) / ((x * x) * (x * x))
	else:
		tmp = ((1.0 / x) + (0.5 + (x * 0.08333333333333333))) + (x * (x * (x * -0.001388888888888889)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.6e+102)
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + Float64(x * Float64(-0.5 + Float64(x * 0.16666666666666666))))));
	elseif (x <= -3.9)
		tmp = Float64(Float64(x * Float64(x * x)) / Float64(Float64(x * x) * Float64(x * x)));
	else
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(0.5 + Float64(x * 0.08333333333333333))) + Float64(x * Float64(x * Float64(x * -0.001388888888888889))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.6e+102)
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))));
	elseif (x <= -3.9)
		tmp = (x * (x * x)) / ((x * x) * (x * x));
	else
		tmp = ((1.0 / x) + (0.5 + (x * 0.08333333333333333))) + (x * (x * (x * -0.001388888888888889)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.6e+102], N[(1.0 / N[(x * N[(1.0 + N[(x * N[(-0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.9], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.60000000000000037e102

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   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 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 1.8%

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

       1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       7. *-lowering-*.f64 1.8%

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

   12. Applied egg-rr 1.8%

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

   13. Taylor expanded in x around 0

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 100.0%

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

   15. Simplified 100.0%

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

   if -5.60000000000000037e102 < x < -3.89999999999999991

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 2.2%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{3} \cdot \left(\left(\frac{1}{720} + -1 \cdot \frac{\frac{1}{2} + \frac{1}{x}}{{x}^{3}}\right) - \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   10. Simplified 2.2%

       \[\leadsto \color{blue}{\left(0 - x\right) \cdot \left(\frac{\frac{-1}{x} + -0.5}{x} + \left(-0.08333333333333333 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \color{blue}{\left(\frac{-1}{{x}^{2}}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       3. *-lowering-*.f64 3.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 3.6%

       \[\leadsto \left(0 - x\right) \cdot \color{blue}{\frac{-1}{x \cdot x}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

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

       2. flip3-- N/A

          \[\leadsto \frac{-1}{x \cdot x} \cdot \frac{{0}^{3} - {x}^{3}}{\color{blue}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{-1}{x \cdot x} \cdot \frac{0 - {x}^{3}}{\color{blue}{0} \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

       4. cube-unmult N/A

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

       5. frac-times N/A

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

       6. neg-mul-1 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(0 - x \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right)} \]

       7. sub0-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\left(\color{blue}{x} \cdot x\right) \cdot \left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right)} \]

       8. remove-double-neg N/A

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

       9. metadata-eval N/A

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

       10. +-lft-identity N/A

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

       11. distribute-rgt-out N/A

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

       12. +-commutative N/A

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

       13. +-lft-identity N/A

           \[\leadsto \frac{x \cdot \left(x \cdot x\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \]

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

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

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

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

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

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

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

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

       19. *-lowering-*.f64 31.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   15. Applied egg-rr 31.9%

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

   if -3.89999999999999991 < x

   1. Initial program 8.0%

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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 99.0%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + 0.5\right) + x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. *-lowering-*.f64 99.0%

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

   9. Applied egg-rr 99.0%

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

Alternative 6: 91.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot 0.16666666666666666\right)\right)}\\ \mathbf{elif}\;x \leq -3.9:\\ \;\;\;\;\frac{x \cdot \left(x \cdot x\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + 0.5\right) + x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.6e+102)
   (/ 1.0 (* x (+ 1.0 (* x (+ -0.5 (* x 0.16666666666666666))))))
   (if (<= x -3.9)
     (/ (* x (* x x)) (* (* x x) (* x x)))
     (+
      (+ (/ 1.0 x) 0.5)
      (* x (+ 0.08333333333333333 (* x (* x -0.001388888888888889))))))))

C

double code(double x) {
	double tmp;
	if (x <= -5.6e+102) {
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))));
	} else if (x <= -3.9) {
		tmp = (x * (x * x)) / ((x * x) * (x * x));
	} else {
		tmp = ((1.0 / x) + 0.5) + (x * (0.08333333333333333 + (x * (x * -0.001388888888888889))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.6d+102)) then
        tmp = 1.0d0 / (x * (1.0d0 + (x * ((-0.5d0) + (x * 0.16666666666666666d0)))))
    else if (x <= (-3.9d0)) then
        tmp = (x * (x * x)) / ((x * x) * (x * x))
    else
        tmp = ((1.0d0 / x) + 0.5d0) + (x * (0.08333333333333333d0 + (x * (x * (-0.001388888888888889d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.6e+102) {
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))));
	} else if (x <= -3.9) {
		tmp = (x * (x * x)) / ((x * x) * (x * x));
	} else {
		tmp = ((1.0 / x) + 0.5) + (x * (0.08333333333333333 + (x * (x * -0.001388888888888889))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.6e+102:
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))))
	elif x <= -3.9:
		tmp = (x * (x * x)) / ((x * x) * (x * x))
	else:
		tmp = ((1.0 / x) + 0.5) + (x * (0.08333333333333333 + (x * (x * -0.001388888888888889))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.6e+102)
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + Float64(x * Float64(-0.5 + Float64(x * 0.16666666666666666))))));
	elseif (x <= -3.9)
		tmp = Float64(Float64(x * Float64(x * x)) / Float64(Float64(x * x) * Float64(x * x)));
	else
		tmp = Float64(Float64(Float64(1.0 / x) + 0.5) + Float64(x * Float64(0.08333333333333333 + Float64(x * Float64(x * -0.001388888888888889)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.6e+102)
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))));
	elseif (x <= -3.9)
		tmp = (x * (x * x)) / ((x * x) * (x * x));
	else
		tmp = ((1.0 / x) + 0.5) + (x * (0.08333333333333333 + (x * (x * -0.001388888888888889))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.6e+102], N[(1.0 / N[(x * N[(1.0 + N[(x * N[(-0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.9], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision] + N[(x * N[(0.08333333333333333 + N[(x * N[(x * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.60000000000000037e102

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   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 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 1.8%

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

       1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       7. *-lowering-*.f64 1.8%

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

   12. Applied egg-rr 1.8%

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

   13. Taylor expanded in x around 0

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 100.0%

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

   15. Simplified 100.0%

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

   if -5.60000000000000037e102 < x < -3.89999999999999991

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 2.2%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{3} \cdot \left(\left(\frac{1}{720} + -1 \cdot \frac{\frac{1}{2} + \frac{1}{x}}{{x}^{3}}\right) - \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   10. Simplified 2.2%

       \[\leadsto \color{blue}{\left(0 - x\right) \cdot \left(\frac{\frac{-1}{x} + -0.5}{x} + \left(-0.08333333333333333 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \color{blue}{\left(\frac{-1}{{x}^{2}}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       3. *-lowering-*.f64 3.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 3.6%

       \[\leadsto \left(0 - x\right) \cdot \color{blue}{\frac{-1}{x \cdot x}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

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

       2. flip3-- N/A

          \[\leadsto \frac{-1}{x \cdot x} \cdot \frac{{0}^{3} - {x}^{3}}{\color{blue}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{-1}{x \cdot x} \cdot \frac{0 - {x}^{3}}{\color{blue}{0} \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

       4. cube-unmult N/A

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

       5. frac-times N/A

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

       6. neg-mul-1 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(0 - x \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right)} \]

       7. sub0-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\left(\color{blue}{x} \cdot x\right) \cdot \left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right)} \]

       8. remove-double-neg N/A

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

       9. metadata-eval N/A

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

       10. +-lft-identity N/A

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

       11. distribute-rgt-out N/A

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

       12. +-commutative N/A

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

       13. +-lft-identity N/A

           \[\leadsto \frac{x \cdot \left(x \cdot x\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \]

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

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

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

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

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

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

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

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

       19. *-lowering-*.f64 31.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   15. Applied egg-rr 31.9%

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

   if -3.89999999999999991 < x

   1. Initial program 8.0%

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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 99.0%

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

Alternative 7: 91.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot 0.16666666666666666\right)\right)}\\ \mathbf{elif}\;x \leq -6:\\ \;\;\;\;\frac{x \cdot \left(x \cdot x\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.6e+102)
   (/ 1.0 (* x (+ 1.0 (* x (+ -0.5 (* x 0.16666666666666666))))))
   (if (<= x -6.0)
     (/ (* x (* x x)) (* (* x x) (* x x)))
     (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333))))))

C

double code(double x) {
	double tmp;
	if (x <= -5.6e+102) {
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))));
	} else if (x <= -6.0) {
		tmp = (x * (x * x)) / ((x * x) * (x * x));
	} 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 <= (-5.6d+102)) then
        tmp = 1.0d0 / (x * (1.0d0 + (x * ((-0.5d0) + (x * 0.16666666666666666d0)))))
    else if (x <= (-6.0d0)) then
        tmp = (x * (x * x)) / ((x * x) * (x * x))
    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 <= -5.6e+102) {
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))));
	} else if (x <= -6.0) {
		tmp = (x * (x * x)) / ((x * x) * (x * x));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.6e+102:
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))))
	elif x <= -6.0:
		tmp = (x * (x * x)) / ((x * x) * (x * x))
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.6e+102)
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + Float64(x * Float64(-0.5 + Float64(x * 0.16666666666666666))))));
	elseif (x <= -6.0)
		tmp = Float64(Float64(x * Float64(x * x)) / Float64(Float64(x * x) * Float64(x * x)));
	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 <= -5.6e+102)
		tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * 0.16666666666666666)))));
	elseif (x <= -6.0)
		tmp = (x * (x * x)) / ((x * x) * (x * x));
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.6e+102], N[(1.0 / N[(x * N[(1.0 + N[(x * N[(-0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.0], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.60000000000000037e102

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   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 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 1.8%

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

       1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       7. *-lowering-*.f64 1.8%

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

   12. Applied egg-rr 1.8%

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

   13. Taylor expanded in x around 0

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 100.0%

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

   15. Simplified 100.0%

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

   if -5.60000000000000037e102 < x < -6

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 2.2%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{3} \cdot \left(\left(\frac{1}{720} + -1 \cdot \frac{\frac{1}{2} + \frac{1}{x}}{{x}^{3}}\right) - \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   10. Simplified 2.2%

       \[\leadsto \color{blue}{\left(0 - x\right) \cdot \left(\frac{\frac{-1}{x} + -0.5}{x} + \left(-0.08333333333333333 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \color{blue}{\left(\frac{-1}{{x}^{2}}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       3. *-lowering-*.f64 3.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 3.6%

       \[\leadsto \left(0 - x\right) \cdot \color{blue}{\frac{-1}{x \cdot x}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

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

       2. flip3-- N/A

          \[\leadsto \frac{-1}{x \cdot x} \cdot \frac{{0}^{3} - {x}^{3}}{\color{blue}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{-1}{x \cdot x} \cdot \frac{0 - {x}^{3}}{\color{blue}{0} \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

       4. cube-unmult N/A

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

       5. frac-times N/A

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

       6. neg-mul-1 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(0 - x \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right)} \]

       7. sub0-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\left(\color{blue}{x} \cdot x\right) \cdot \left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right)} \]

       8. remove-double-neg N/A

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

       9. metadata-eval N/A

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

       10. +-lft-identity N/A

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

       11. distribute-rgt-out N/A

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

       12. +-commutative N/A

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

       13. +-lft-identity N/A

           \[\leadsto \frac{x \cdot \left(x \cdot x\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \]

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

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

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

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

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

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

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

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

       19. *-lowering-*.f64 31.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   15. Applied egg-rr 31.9%

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

   if -6 < x

   1. Initial program 8.0%

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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   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 98.9%

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

   7. Simplified 98.9%

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

Alternative 8: 88.9% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \mathbf{elif}\;x \leq -6:\\ \;\;\;\;\frac{x \cdot x}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.35e+154)
   (/ x (* x x))
   (if (<= x -6.0)
     (/ (* x x) (* x (* x x)))
     (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = x / (x * x);
	} else if (x <= -6.0) {
		tmp = (x * x) / (x * (x * x));
	} 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 <= (-1.35d+154)) then
        tmp = x / (x * x)
    else if (x <= (-6.0d0)) then
        tmp = (x * x) / (x * (x * x))
    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 <= -1.35e+154) {
		tmp = x / (x * x);
	} else if (x <= -6.0) {
		tmp = (x * x) / (x * (x * x));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.35e+154:
		tmp = x / (x * x)
	elif x <= -6.0:
		tmp = (x * x) / (x * (x * x))
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(x / Float64(x * x));
	elseif (x <= -6.0)
		tmp = Float64(Float64(x * x) / Float64(x * Float64(x * x)));
	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 <= -1.35e+154)
		tmp = x / (x * x);
	elseif (x <= -6.0)
		tmp = (x * x) / (x * (x * x));
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.35e+154], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.0], N[(N[(x * x), $MachinePrecision] / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 1.6%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{3} \cdot \left(\left(\frac{1}{720} + -1 \cdot \frac{\frac{1}{2} + \frac{1}{x}}{{x}^{3}}\right) - \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   10. Simplified 1.6%

       \[\leadsto \color{blue}{\left(0 - x\right) \cdot \left(\frac{\frac{-1}{x} + -0.5}{x} + \left(-0.08333333333333333 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \color{blue}{\left(\frac{-1}{{x}^{2}}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       3. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \left(0 - x\right) \cdot \color{blue}{\frac{-1}{x \cdot x}} \]
   14. Step-by-step derivation

       1. frac-2neg N/A

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

       2. metadata-eval N/A

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

       3. un-div-inv N/A

          \[\leadsto \frac{0 - x}{\color{blue}{\mathsf{neg}\left(x \cdot x\right)}} \]

       4. neg-sub0 N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\color{blue}{x \cdot x}\right)} \]

       5. frac-2neg N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(x \cdot x\right)}\right) \]

       7. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   15. Applied egg-rr 100.0%

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

   if -1.35000000000000003e154 < x < -6

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 2.0%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{3} \cdot \left(\left(\frac{1}{720} + -1 \cdot \frac{\frac{1}{2} + \frac{1}{x}}{{x}^{3}}\right) - \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   10. Simplified 2.0%

       \[\leadsto \color{blue}{\left(0 - x\right) \cdot \left(\frac{\frac{-1}{x} + -0.5}{x} + \left(-0.08333333333333333 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \color{blue}{\left(\frac{-1}{{x}^{2}}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       3. *-lowering-*.f64 3.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 3.8%

       \[\leadsto \left(0 - x\right) \cdot \color{blue}{\frac{-1}{x \cdot x}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

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

       2. flip-- N/A

          \[\leadsto \frac{-1}{x \cdot x} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{0 + x}} \]

       3. +-lft-identity N/A

          \[\leadsto \frac{-1}{x \cdot x} \cdot \frac{0 \cdot 0 - x \cdot x}{x} \]

       4. times-frac N/A

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

       5. associate-*r* N/A

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

       6. metadata-eval N/A

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

       7. neg-sub0 N/A

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

       8. distribute-rgt-neg-in N/A

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

       9. neg-sub0 N/A

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

       10. associate-*l* N/A

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

       11. neg-mul-1 N/A

           \[\leadsto \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(0 - x\right)}{x \cdot \left(x \cdot x\right)} \]

       12. neg-sub0 N/A

           \[\leadsto \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}{x \cdot \left(x \cdot x\right)} \]

       13. sqr-neg N/A

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

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right) \]

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

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

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

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

       17. *-lowering-*.f64 31.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   15. Applied egg-rr 31.7%

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

   if -6 < x

   1. Initial program 8.0%

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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   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 98.9%

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

   7. Simplified 98.9%

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

Alternative 9: 91.5% 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 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 66.1%

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

7. Simplified 66.1%

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

8. Taylor expanded in x around 0

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

10. Simplified 66.0%

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

    1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

    7. *-lowering-*.f64 66.0%

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

12. Applied egg-rr 66.0%

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

13. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
14. 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 90.3%

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

15. Simplified 90.3%

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

Alternative 10: 83.6% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6.0)
   (/ x (* x x))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -6.0) {
		tmp = x / (x * x);
	} 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 <= (-6.0d0)) then
        tmp = x / (x * x)
    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 <= -6.0) {
		tmp = x / (x * x);
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -6.0:
		tmp = x / (x * x)
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -6.0)
		tmp = Float64(x / Float64(x * x));
	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 <= -6.0)
		tmp = x / (x * x);
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -6.0], N[(x / N[(x * x), $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 < -6

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 1.8%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{3} \cdot \left(\left(\frac{1}{720} + -1 \cdot \frac{\frac{1}{2} + \frac{1}{x}}{{x}^{3}}\right) - \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   10. Simplified 1.8%

       \[\leadsto \color{blue}{\left(0 - x\right) \cdot \left(\frac{\frac{-1}{x} + -0.5}{x} + \left(-0.08333333333333333 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \color{blue}{\left(\frac{-1}{{x}^{2}}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       3. *-lowering-*.f64 46.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 46.9%

       \[\leadsto \left(0 - x\right) \cdot \color{blue}{\frac{-1}{x \cdot x}} \]
   14. Step-by-step derivation

       1. frac-2neg N/A

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

       2. metadata-eval N/A

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

       3. un-div-inv N/A

          \[\leadsto \frac{0 - x}{\color{blue}{\mathsf{neg}\left(x \cdot x\right)}} \]

       4. neg-sub0 N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\color{blue}{x \cdot x}\right)} \]

       5. frac-2neg N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(x \cdot x\right)}\right) \]

       7. *-lowering-*.f64 46.9%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   15. Applied egg-rr 46.9%

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

   if -6 < x

   1. Initial program 8.0%

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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   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 98.9%

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

   7. Simplified 98.9%

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

Alternative 11: 88.9% 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 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 66.1%

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

7. Simplified 66.1%

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

8. Taylor expanded in x around 0

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

10. Simplified 66.0%

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

    1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

    7. *-lowering-*.f64 66.0%

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

12. Applied egg-rr 66.0%

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

13. Taylor expanded in x around 0

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

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

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

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

    8. *-commutative N/A

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

    9. *-lowering-*.f64 86.5%

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

15. Simplified 86.5%

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

Alternative 12: 83.2% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.7) (/ x (* x x)) (+ (/ 1.0 x) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= -1.7) {
		tmp = x / (x * x);
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.7d0)) then
        tmp = x / (x * x)
    else
        tmp = (1.0d0 / x) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.7) {
		tmp = x / (x * x);
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.7:
		tmp = x / (x * x)
	else:
		tmp = (1.0 / x) + 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.7)
		tmp = Float64(x / Float64(x * x));
	else
		tmp = Float64(Float64(1.0 / x) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.7)
		tmp = x / (x * x);
	else
		tmp = (1.0 / x) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.7], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999996

   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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 1.8%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{3} \cdot \left(\left(\frac{1}{720} + -1 \cdot \frac{\frac{1}{2} + \frac{1}{x}}{{x}^{3}}\right) - \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   10. Simplified 1.8%

       \[\leadsto \color{blue}{\left(0 - x\right) \cdot \left(\frac{\frac{-1}{x} + -0.5}{x} + \left(-0.08333333333333333 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \color{blue}{\left(\frac{-1}{{x}^{2}}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       3. *-lowering-*.f64 46.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 46.9%

       \[\leadsto \left(0 - x\right) \cdot \color{blue}{\frac{-1}{x \cdot x}} \]
   14. Step-by-step derivation

       1. frac-2neg N/A

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

       2. metadata-eval N/A

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

       3. un-div-inv N/A

          \[\leadsto \frac{0 - x}{\color{blue}{\mathsf{neg}\left(x \cdot x\right)}} \]

       4. neg-sub0 N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\color{blue}{x \cdot x}\right)} \]

       5. frac-2neg N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(x \cdot x\right)}\right) \]

       7. *-lowering-*.f64 46.9%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   15. Applied egg-rr 46.9%

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

   if -1.69999999999999996 < x

   1. Initial program 8.0%

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

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
   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 98.3%

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

   7. Simplified 98.3%

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

Alternative 13: 83.3% 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 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 66.1%

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

7. Simplified 66.1%

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

8. Taylor expanded in x around 0

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

10. Simplified 66.0%

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

    1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

    7. *-lowering-*.f64 66.0%

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

12. Applied egg-rr 66.0%

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

13. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + \frac{-1}{2} \cdot x\right)\right)}\right) \]
14. 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 81.0%

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

15. Simplified 81.0%

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

Alternative 14: 67.0% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 66.0%

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

7. Simplified 66.0%

   \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
8. Add Preprocessing

Alternative 15: 67.1% 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 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 65.4%

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

7. Simplified 65.4%

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

Alternative 16: 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 39.3%

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

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

8. Taylor expanded in x around 0

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

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

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

   2. +-lowering-+.f64 64.8%

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

10. Simplified 64.8%

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

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{1} \]
12. Step-by-step derivation

13. Simplified 3.8%

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

Alternative 17: 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 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 66.0%

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

7. Simplified 66.0%

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

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2}} \]
9. Step-by-step derivation

10. Simplified 3.4%

    \[\leadsto \color{blue}{0.5} \]
11. 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 2024288 
(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)))