expq2 (section 3.11)

Percentage Accurate: 37.7% → 100.0%

Time: 8.7s

Alternatives: 14

Speedup: 68.3×

Specification

\[710 > x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.7% 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 42.2%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0.002:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot \left(0.5 + x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.002)
   (/ 1.0 (+ 1.0 (/ -1.0 (exp x))))
   (/
    (+
     1.0
     (*
      x
      (+ 0.5 (* x (+ 0.08333333333333333 (* x (* x -0.001388888888888889)))))))
    x)))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.002) {
		tmp = 1.0 / (1.0 + (-1.0 / exp(x)));
	} else {
		tmp = (1.0 + (x * (0.5 + (x * (0.08333333333333333 + (x * (x * -0.001388888888888889))))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (exp(x) <= 0.002d0) then
        tmp = 1.0d0 / (1.0d0 + ((-1.0d0) / exp(x)))
    else
        tmp = (1.0d0 + (x * (0.5d0 + (x * (0.08333333333333333d0 + (x * (x * (-0.001388888888888889d0)))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.exp(x) <= 0.002) {
		tmp = 1.0 / (1.0 + (-1.0 / Math.exp(x)));
	} else {
		tmp = (1.0 + (x * (0.5 + (x * (0.08333333333333333 + (x * (x * -0.001388888888888889))))))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.exp(x) <= 0.002:
		tmp = 1.0 / (1.0 + (-1.0 / math.exp(x)))
	else:
		tmp = (1.0 + (x * (0.5 + (x * (0.08333333333333333 + (x * (x * -0.001388888888888889))))))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.002)
		tmp = Float64(1.0 / Float64(1.0 + Float64(-1.0 / exp(x))));
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.08333333333333333 + Float64(x * Float64(x * -0.001388888888888889))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (exp(x) <= 0.002)
		tmp = 1.0 / (1.0 + (-1.0 / exp(x)));
	else
		tmp = (1.0 + (x * (0.5 + (x * (0.08333333333333333 + (x * (x * -0.001388888888888889))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.002], N[(1.0 / N[(1.0 + N[(-1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.08333333333333333 + N[(x * N[(x * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 2e-3

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

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

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

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

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

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

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

      7. exp-lowering-exp.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 2e-3 < (exp.f64 x)

   1. Initial program 9.3%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. *-commutative N/A

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

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

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

      6. clear-num N/A

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

      7. flip-- N/A

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

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

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

      9. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{expm1}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      10. expm1-lowering-expm1.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{expm1.f64}\left(x\right)\right), \left(e^{x}\right)\right) \]

      11. exp-lowering-exp.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{expm1.f64}\left(x\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0.002:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot \left(0.5 + x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

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 * -0.001388888888888889))))))) / x;
	}
	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 * (-0.001388888888888889d0)))))))) / x
    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 * -0.001388888888888889))))))) / x;
	}
	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 * -0.001388888888888889))))))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.8)
		tmp = Float64(exp(x) / x);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.08333333333333333 + Float64(x * Float64(x * -0.001388888888888889))))))) / x);
	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 * -0.001388888888888889))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.8], N[(N[Exp[x], $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.08333333333333333 + N[(x * N[(x * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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 99.1%

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

   if -3.7999999999999998 < x

   1. Initial program 9.3%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. *-commutative N/A

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

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

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

      6. clear-num N/A

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

      7. flip-- N/A

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

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

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

      9. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{expm1}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      10. expm1-lowering-expm1.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{expm1.f64}\left(x\right)\right), \left(e^{x}\right)\right) \]

      11. exp-lowering-exp.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{expm1.f64}\left(x\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 98.8%

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

   9. Simplified 98.8%

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

Alternative 4: 94.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\\ t_1 := x \cdot \left(-1 - x \cdot 0.5\right)\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x \cdot \left(1 + x \cdot 0.5\right)\right) \cdot t\_1}{t\_0 \cdot \left(1 + t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          x
          (+
           1.0
           (*
            x
            (+
             0.5
             (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))
        (t_1 (* x (- -1.0 (* x 0.5)))))
   (if (<= x -1.65e+77)
     (/ 1.0 t_0)
     (/ (+ 1.0 (* (* x (+ 1.0 (* x 0.5))) t_1)) (* t_0 (+ 1.0 t_1))))))

C

double code(double x) {
	double t_0 = x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
	double t_1 = x * (-1.0 - (x * 0.5));
	double tmp;
	if (x <= -1.65e+77) {
		tmp = 1.0 / t_0;
	} else {
		tmp = (1.0 + ((x * (1.0 + (x * 0.5))) * t_1)) / (t_0 * (1.0 + t_1));
	}
	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 * (1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))))))
    t_1 = x * ((-1.0d0) - (x * 0.5d0))
    if (x <= (-1.65d+77)) then
        tmp = 1.0d0 / t_0
    else
        tmp = (1.0d0 + ((x * (1.0d0 + (x * 0.5d0))) * t_1)) / (t_0 * (1.0d0 + t_1))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
	double t_1 = x * (-1.0 - (x * 0.5));
	double tmp;
	if (x <= -1.65e+77) {
		tmp = 1.0 / t_0;
	} else {
		tmp = (1.0 + ((x * (1.0 + (x * 0.5))) * t_1)) / (t_0 * (1.0 + t_1));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))
	t_1 = x * (-1.0 - (x * 0.5))
	tmp = 0
	if x <= -1.65e+77:
		tmp = 1.0 / t_0
	else:
		tmp = (1.0 + ((x * (1.0 + (x * 0.5))) * t_1)) / (t_0 * (1.0 + t_1))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664)))))))
	t_1 = Float64(x * Float64(-1.0 - Float64(x * 0.5)))
	tmp = 0.0
	if (x <= -1.65e+77)
		tmp = Float64(1.0 / t_0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * Float64(1.0 + Float64(x * 0.5))) * t_1)) / Float64(t_0 * Float64(1.0 + t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
	t_1 = x * (-1.0 - (x * 0.5));
	tmp = 0.0;
	if (x <= -1.65e+77)
		tmp = 1.0 / t_0;
	else
		tmp = (1.0 + ((x * (1.0 + (x * 0.5))) * t_1)) / (t_0 * (1.0 + t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$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]}, Block[{t$95$1 = N[(x * N[(-1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e+77], N[(1.0 / t$95$0), $MachinePrecision], N[(N[(1.0 + N[(N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6499999999999999e77

   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

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}, \mathsf{expm1.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(\color{blue}{x}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      5. *-lowering-*.f64 1.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   7. Simplified 1.7%

      \[\leadsto \frac{\color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)}}{\mathsf{expm1}\left(x\right)} \]

   8. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 35.8%

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

   10. Simplified 35.8%

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

   11. Taylor expanded in x around 0

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

   13. Simplified 98.9%

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

   if -1.6499999999999999e77 < x

   1. Initial program 20.5%

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

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}, \mathsf{expm1.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(\color{blue}{x}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      5. *-lowering-*.f64 86.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   7. Simplified 86.3%

      \[\leadsto \frac{\color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)}}{\mathsf{expm1}\left(x\right)} \]

   8. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 86.4%

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

   10. Simplified 86.4%

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

       1. flip-+ N/A

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

       2. associate-/l/ N/A

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

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

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

   12. Applied egg-rr 91.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x \cdot \left(1 + x \cdot 0.5\right)\right) \cdot \left(x \cdot \left(-1 - x \cdot 0.5\right)\right)}{\left(x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right) \cdot \left(1 + x \cdot \left(-1 - x \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot \left(0.5 + x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.2)
   (/
    1.0
    (*
     x
     (+
      1.0
      (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))
   (/
    (+
     1.0
     (*
      x
      (+ 0.5 (* x (+ 0.08333333333333333 (* x (* x -0.001388888888888889)))))))
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -5.2) {
		tmp = 1.0 / (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))));
	} else {
		tmp = (1.0 + (x * (0.5 + (x * (0.08333333333333333 + (x * (x * -0.001388888888888889))))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.2d0)) then
        tmp = 1.0d0 / (x * (1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0)))))))
    else
        tmp = (1.0d0 + (x * (0.5d0 + (x * (0.08333333333333333d0 + (x * (x * (-0.001388888888888889d0)))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.2) {
		tmp = 1.0 / (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))));
	} else {
		tmp = (1.0 + (x * (0.5 + (x * (0.08333333333333333 + (x * (x * -0.001388888888888889))))))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.2:
		tmp = 1.0 / (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))))
	else:
		tmp = (1.0 + (x * (0.5 + (x * (0.08333333333333333 + (x * (x * -0.001388888888888889))))))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.2)
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))))));
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.08333333333333333 + Float64(x * Float64(x * -0.001388888888888889))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.2)
		tmp = 1.0 / (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))));
	else
		tmp = (1.0 + (x * (0.5 + (x * (0.08333333333333333 + (x * (x * -0.001388888888888889))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.2], 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], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.08333333333333333 + N[(x * N[(x * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.20000000000000018

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}, \mathsf{expm1.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(\color{blue}{x}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      5. *-lowering-*.f64 2.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   7. Simplified 2.0%

      \[\leadsto \frac{\color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)}}{\mathsf{expm1}\left(x\right)} \]

   8. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 27.9%

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

   10. Simplified 27.9%

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

   11. Taylor expanded in x around 0

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

   13. Simplified 75.8%

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

   if -5.20000000000000018 < x

   1. Initial program 9.3%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. *-commutative N/A

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

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

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

      6. clear-num N/A

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

      7. flip-- N/A

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

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

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

      9. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{expm1}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      10. expm1-lowering-expm1.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{expm1.f64}\left(x\right)\right), \left(e^{x}\right)\right) \]

      11. exp-lowering-exp.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{expm1.f64}\left(x\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 98.8%

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

   9. Simplified 98.8%

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

Alternative 6: 91.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right) + \left(0.5 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.2)
   (/
    1.0
    (*
     x
     (+
      1.0
      (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))
   (+
    (* x (+ 0.08333333333333333 (* x (* x -0.001388888888888889))))
    (+ 0.5 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5.2) {
		tmp = 1.0 / (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))));
	} else {
		tmp = (x * (0.08333333333333333 + (x * (x * -0.001388888888888889)))) + (0.5 + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.2d0)) then
        tmp = 1.0d0 / (x * (1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0)))))))
    else
        tmp = (x * (0.08333333333333333d0 + (x * (x * (-0.001388888888888889d0))))) + (0.5d0 + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.2) {
		tmp = 1.0 / (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))));
	} else {
		tmp = (x * (0.08333333333333333 + (x * (x * -0.001388888888888889)))) + (0.5 + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.2:
		tmp = 1.0 / (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))))
	else:
		tmp = (x * (0.08333333333333333 + (x * (x * -0.001388888888888889)))) + (0.5 + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.2)
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))))));
	else
		tmp = Float64(Float64(x * Float64(0.08333333333333333 + Float64(x * Float64(x * -0.001388888888888889)))) + Float64(0.5 + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.2)
		tmp = 1.0 / (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))));
	else
		tmp = (x * (0.08333333333333333 + (x * (x * -0.001388888888888889)))) + (0.5 + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.2], 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], N[(N[(x * N[(0.08333333333333333 + N[(x * N[(x * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.20000000000000018

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}, \mathsf{expm1.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(\color{blue}{x}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      5. *-lowering-*.f64 2.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   7. Simplified 2.0%

      \[\leadsto \frac{\color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)}}{\mathsf{expm1}\left(x\right)} \]

   8. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 27.9%

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

   10. Simplified 27.9%

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

   11. Taylor expanded in x around 0

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

   13. Simplified 75.8%

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

   if -5.20000000000000018 < x

   1. Initial program 9.3%

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

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. lft-mult-inverse N/A

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

      12. *-lft-identity N/A

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

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

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

   7. Simplified 98.8%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right) + \left(0.5 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.9% accurate, 10.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -5.5999999999999996

   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

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}, \mathsf{expm1.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(\color{blue}{x}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      5. *-lowering-*.f64 2.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   7. Simplified 2.0%

      \[\leadsto \frac{\color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)}}{\mathsf{expm1}\left(x\right)} \]

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 1.6%

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

   10. Simplified 1.6%

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

   11. Taylor expanded in x around 0

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

   13. Simplified 50.2%

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

   if -5.5999999999999996 < x

   1. Initial program 9.3%

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

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. lft-mult-inverse N/A

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

      12. *-lft-identity N/A

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

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

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

   7. Simplified 98.8%

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

4. Final simplification 81.1%

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

Alternative 8: 83.8% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + x \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -700.0)
   (/ 1.0 (* x (+ 1.0 (* x 0.5))))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -700.0) {
		tmp = 1.0 / (x * (1.0 + (x * 0.5)));
	} 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 <= (-700.0d0)) then
        tmp = 1.0d0 / (x * (1.0d0 + (x * 0.5d0)))
    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 <= -700.0) {
		tmp = 1.0 / (x * (1.0 + (x * 0.5)));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -700.0:
		tmp = 1.0 / (x * (1.0 + (x * 0.5)))
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + Float64(x * 0.5))));
	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 <= -700.0)
		tmp = 1.0 / (x * (1.0 + (x * 0.5)));
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -700.0], N[(1.0 / N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   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

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}, \mathsf{expm1.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(\color{blue}{x}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      5. *-lowering-*.f64 1.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   7. Simplified 1.9%

      \[\leadsto \frac{\color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)}}{\mathsf{expm1}\left(x\right)} \]

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 1.6%

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

   10. Simplified 1.6%

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

   11. Taylor expanded in x around 0

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

   13. Simplified 50.7%

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

   if -700 < x

   1. Initial program 9.8%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 98.2%

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

   7. Simplified 98.2%

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

Alternative 9: 83.8% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{12}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -700.0)
   (/ 12.0 (* x x))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -700.0) {
		tmp = 12.0 / (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 <= (-700.0d0)) then
        tmp = 12.0d0 / (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 <= -700.0) {
		tmp = 12.0 / (x * x);
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -700.0:
		tmp = 12.0 / (x * x)
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(12.0 / 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 <= -700.0)
		tmp = 12.0 / (x * x);
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -700.0], N[(12.0 / 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 < -700

   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

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}, \mathsf{expm1.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(\color{blue}{x}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      5. *-lowering-*.f64 1.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   7. Simplified 1.9%

      \[\leadsto \frac{\color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)}}{\mathsf{expm1}\left(x\right)} \]

   8. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 28.2%

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

   10. Simplified 28.2%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{12}{{x}^{2}}} \]
   12. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 50.7%

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

   13. Simplified 50.7%

       \[\leadsto \color{blue}{\frac{12}{x \cdot x}} \]

   if -700 < x

   1. Initial program 9.8%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 98.2%

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

   7. Simplified 98.2%

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

Alternative 10: 83.4% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6:\\ \;\;\;\;\frac{12}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6.0) (/ 12.0 (* x x)) (+ 0.5 (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -6.0) {
		tmp = 12.0 / (x * x);
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-6.0d0)) then
        tmp = 12.0d0 / (x * x)
    else
        tmp = 0.5d0 + (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -6.0) {
		tmp = 12.0 / (x * x);
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -6.0:
		tmp = 12.0 / (x * x)
	else:
		tmp = 0.5 + (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -6.0)
		tmp = Float64(12.0 / Float64(x * x));
	else
		tmp = Float64(0.5 + Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -6.0)
		tmp = 12.0 / (x * x);
	else
		tmp = 0.5 + (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -6.0], N[(12.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(1.0 / x), $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

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}, \mathsf{expm1.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(\color{blue}{x}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot x\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

      5. *-lowering-*.f64 2.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   7. Simplified 2.0%

      \[\leadsto \frac{\color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)}}{\mathsf{expm1}\left(x\right)} \]

   8. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 27.9%

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

   10. Simplified 27.9%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{12}{{x}^{2}}} \]
   12. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 50.1%

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

   13. Simplified 50.1%

       \[\leadsto \color{blue}{\frac{12}{x \cdot x}} \]

   if -6 < x

   1. Initial program 9.3%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 97.7%

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

   7. Simplified 97.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6:\\ \;\;\;\;\frac{12}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.8% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.2%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*l* N/A

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

   5. rgt-mult-inverse N/A

      \[\leadsto 1 \cdot \frac{1}{x} + \frac{1}{2} \cdot 1 \]

   6. metadata-eval N/A

      \[\leadsto 1 \cdot \frac{1}{x} + \frac{1}{2} \]

   7. *-lft-identity N/A

      \[\leadsto \frac{1}{x} + \frac{1}{2} \]

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

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

   9. /-lowering-/.f64 63.3%

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

7. Simplified 63.3%

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

8. Final simplification 63.3%

   \[\leadsto 0.5 + \frac{1}{x} \]
9. Add Preprocessing

Alternative 12: 66.8% 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 42.2%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 62.9%

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

7. Simplified 62.9%

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

Alternative 13: 3.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 42.2%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 96.9%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + x\right)}, x\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 62.1%

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

10. Simplified 62.1%

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

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 14: 3.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 42.2%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-/l* N/A

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

   3. associate-*l/ N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-*r* N/A

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

   7. lft-mult-inverse N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

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

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 63.7%

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

7. Simplified 63.7%

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

8. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. rgt-mult-inverse N/A

      \[\leadsto 1 \cdot \frac{1}{2} + x \cdot \frac{1}{12} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{2} + \color{blue}{x} \cdot \frac{1}{12} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{2} + x \cdot \left(\mathsf{neg}\left(\frac{-1}{12}\right)\right) \]

   8. metadata-eval N/A

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

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

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

   10. unsub-neg N/A

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

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

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

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

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

   13. metadata-eval 3.2%

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

10. Simplified 3.2%

    \[\leadsto \color{blue}{0.5 - x \cdot -0.08333333333333333} \]

11. Taylor expanded in x around 0

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

13. Simplified 3.5%

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

Developer Target 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024145 
(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)))