expq2 (section 3.11)

Percentage Accurate: 37.1% → 100.0%

Time: 9.4s

Alternatives: 18

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: 37.1% 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 40.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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: 99.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return math.exp(x) / (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 99.0%

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

7. Simplified 99.0%

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

Alternative 3: 98.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 98.2%

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

Alternative 4: 96.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ t_1 := x \cdot t\_0\\ t_2 := t\_0 \cdot t\_0\\ \mathbf{if}\;x \leq -7 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\left(x \cdot \left(1 + t\_1\right)\right) \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{\frac{\frac{x \cdot \left(1 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(t\_2 \cdot t\_2\right)\right)}{1 + \left(x \cdot x\right) \cdot t\_2}}{1 - t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664)))))
        (t_1 (* x t_0))
        (t_2 (* t_0 t_0)))
   (if (<= x -7e+51)
     (/ 1.0 (* (* x (+ 1.0 t_1)) (- 1.0 x)))
     (/
      (+ x 1.0)
      (/
       (/
        (* x (- 1.0 (* (* x (* x (* x x))) (* t_2 t_2))))
        (+ 1.0 (* (* x x) t_2)))
       (- 1.0 t_1))))))

C

double code(double x) {
	double t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	double t_1 = x * t_0;
	double t_2 = t_0 * t_0;
	double tmp;
	if (x <= -7e+51) {
		tmp = 1.0 / ((x * (1.0 + t_1)) * (1.0 - x));
	} else {
		tmp = (x + 1.0) / (((x * (1.0 - ((x * (x * (x * x))) * (t_2 * t_2)))) / (1.0 + ((x * x) * t_2))) / (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) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0)))
    t_1 = x * t_0
    t_2 = t_0 * t_0
    if (x <= (-7d+51)) then
        tmp = 1.0d0 / ((x * (1.0d0 + t_1)) * (1.0d0 - x))
    else
        tmp = (x + 1.0d0) / (((x * (1.0d0 - ((x * (x * (x * x))) * (t_2 * t_2)))) / (1.0d0 + ((x * x) * t_2))) / (1.0d0 - t_1))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	double t_1 = x * t_0;
	double t_2 = t_0 * t_0;
	double tmp;
	if (x <= -7e+51) {
		tmp = 1.0 / ((x * (1.0 + t_1)) * (1.0 - x));
	} else {
		tmp = (x + 1.0) / (((x * (1.0 - ((x * (x * (x * x))) * (t_2 * t_2)))) / (1.0 + ((x * x) * t_2))) / (1.0 - t_1));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))
	t_1 = x * t_0
	t_2 = t_0 * t_0
	tmp = 0
	if x <= -7e+51:
		tmp = 1.0 / ((x * (1.0 + t_1)) * (1.0 - x))
	else:
		tmp = (x + 1.0) / (((x * (1.0 - ((x * (x * (x * x))) * (t_2 * t_2)))) / (1.0 + ((x * x) * t_2))) / (1.0 - t_1))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))
	t_1 = Float64(x * t_0)
	t_2 = Float64(t_0 * t_0)
	tmp = 0.0
	if (x <= -7e+51)
		tmp = Float64(1.0 / Float64(Float64(x * Float64(1.0 + t_1)) * Float64(1.0 - x)));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(Float64(Float64(x * Float64(1.0 - Float64(Float64(x * Float64(x * Float64(x * x))) * Float64(t_2 * t_2)))) / Float64(1.0 + Float64(Float64(x * x) * t_2))) / Float64(1.0 - t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	t_1 = x * t_0;
	t_2 = t_0 * t_0;
	tmp = 0.0;
	if (x <= -7e+51)
		tmp = 1.0 / ((x * (1.0 + t_1)) * (1.0 - x));
	else
		tmp = (x + 1.0) / (((x * (1.0 - ((x * (x * (x * x))) * (t_2 * t_2)))) / (1.0 + ((x * x) * t_2))) / (1.0 - t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$0), $MachinePrecision]}, If[LessEqual[x, -7e+51], N[(1.0 / N[(N[(x * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] / N[(N[(N[(x * N[(1.0 - N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -7e51

   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

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 88.0%

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

   10. Simplified 88.0%

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

       1. flip-+ N/A

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

       2. associate-/l/ N/A

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

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

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

       4. metadata-eval N/A

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

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

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

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

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

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

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

   12. Applied egg-rr 42.0%

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

   13. Taylor expanded in x around 0

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

   15. Simplified 95.4%

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

   if -7e51 < x

   1. Initial program 14.5%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 90.2%

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

   10. Simplified 90.2%

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

       1. *-commutative N/A

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

       2. flip-+ N/A

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

       3. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 91.3%

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

       1. flip-- N/A

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

       2. associate-*l/ N/A

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

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

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

   14. Applied egg-rr 96.1%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\left(x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right) \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{\frac{\frac{x \cdot \left(1 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(\left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(\left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}}{1 - x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\left(x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right) \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(1 + t\_1 \cdot \left(t\_0 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)\right)}{1 + t\_1 \cdot \left(t\_1 + -1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ -0.5 (* x (+ 0.25 (* x -0.125))))) (t_1 (* x t_0)))
   (if (<= x -4.7e+51)
     (/
      1.0
      (*
       (*
        x
        (+
         1.0
         (*
          x
          (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664)))))))
       (- 1.0 x)))
     (/
      1.0
      (/
       (* x (+ 1.0 (* t_1 (* t_0 (* (* x x) t_0)))))
       (+ 1.0 (* t_1 (+ t_1 -1.0))))))))

C

double code(double x) {
	double t_0 = -0.5 + (x * (0.25 + (x * -0.125)));
	double t_1 = x * t_0;
	double tmp;
	if (x <= -4.7e+51) {
		tmp = 1.0 / ((x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) * (1.0 - x));
	} else {
		tmp = 1.0 / ((x * (1.0 + (t_1 * (t_0 * ((x * x) * t_0))))) / (1.0 + (t_1 * (t_1 + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) + (x * (0.25d0 + (x * (-0.125d0))))
    t_1 = x * t_0
    if (x <= (-4.7d+51)) then
        tmp = 1.0d0 / ((x * (1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))))))) * (1.0d0 - x))
    else
        tmp = 1.0d0 / ((x * (1.0d0 + (t_1 * (t_0 * ((x * x) * t_0))))) / (1.0d0 + (t_1 * (t_1 + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -0.5 + (x * (0.25 + (x * -0.125)));
	double t_1 = x * t_0;
	double tmp;
	if (x <= -4.7e+51) {
		tmp = 1.0 / ((x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) * (1.0 - x));
	} else {
		tmp = 1.0 / ((x * (1.0 + (t_1 * (t_0 * ((x * x) * t_0))))) / (1.0 + (t_1 * (t_1 + -1.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = -0.5 + (x * (0.25 + (x * -0.125)))
	t_1 = x * t_0
	tmp = 0
	if x <= -4.7e+51:
		tmp = 1.0 / ((x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) * (1.0 - x))
	else:
		tmp = 1.0 / ((x * (1.0 + (t_1 * (t_0 * ((x * x) * t_0))))) / (1.0 + (t_1 * (t_1 + -1.0))))
	return tmp

Julia

function code(x)
	t_0 = Float64(-0.5 + Float64(x * Float64(0.25 + Float64(x * -0.125))))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= -4.7e+51)
		tmp = Float64(1.0 / Float64(Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))))) * Float64(1.0 - x)));
	else
		tmp = Float64(1.0 / Float64(Float64(x * Float64(1.0 + Float64(t_1 * Float64(t_0 * Float64(Float64(x * x) * t_0))))) / Float64(1.0 + Float64(t_1 * Float64(t_1 + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -0.5 + (x * (0.25 + (x * -0.125)));
	t_1 = x * t_0;
	tmp = 0.0;
	if (x <= -4.7e+51)
		tmp = 1.0 / ((x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) * (1.0 - x));
	else
		tmp = 1.0 / ((x * (1.0 + (t_1 * (t_0 * ((x * x) * t_0))))) / (1.0 + (t_1 * (t_1 + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 + N[(x * N[(0.25 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[x, -4.7e+51], N[(1.0 / N[(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] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * N[(1.0 + N[(t$95$1 * N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7000000000000002e51

   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

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 88.0%

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

   10. Simplified 88.0%

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

       1. flip-+ N/A

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

       2. associate-/l/ N/A

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

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

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

       4. metadata-eval N/A

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

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

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

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

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

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

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

   12. Applied egg-rr 42.0%

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

   13. Taylor expanded in x around 0

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

   15. Simplified 95.4%

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

   if -4.7000000000000002e51 < x

   1. Initial program 14.5%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
   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 90.2%

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

   7. Simplified 90.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1 \cdot \frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{\frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\frac{1}{2}}^{3} + \color{blue}{{\left(\frac{1}{x}\right)}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\left({\frac{1}{2}}^{3}\right), \color{blue}{\left({\left(\frac{1}{x}\right)}^{3}\right)}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left({\color{blue}{\left(\frac{1}{x}\right)}}^{3}\right)\right)\right)\right) \]

      18. cube-div N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{{1}^{3}}{\color{blue}{{x}^{3}}}\right)\right)\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{1}{{\color{blue}{x}}^{3}}\right)\right)\right)\right) \]

   9. Applied egg-rr 34.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{x \cdot x} + \left(0.25 - \frac{0.5}{x}\right)}{0.125 + \frac{1}{x \cdot \left(x \cdot x\right)}}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 90.3%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 90.3%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right)}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(1 + x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right) \cdot \color{blue}{x}\right)\right) \]

       2. flip3-+ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{{1}^{3} + {\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right) - 1 \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right)\right)} \cdot x\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left({1}^{3} + {\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right)}^{3}\right) \cdot x}{\color{blue}{1 \cdot 1 + \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right) - 1 \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right)\right)}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left({1}^{3} + {\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right)}^{3}\right) \cdot x\right), \color{blue}{\left(1 \cdot 1 + \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right) - 1 \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right)\right)\right)}\right)\right) \]

   14. Applied egg-rr 95.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(1 + \left(x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right) \cdot \left(\left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right) \cdot \left(\left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot x}{1 + \left(x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right) \cdot \left(x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right) - 1\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\left(x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right) \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(1 + \left(x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right) \cdot \left(\left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right)\right)\right)}{1 + \left(x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right) \cdot \left(x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right) + -1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\left(x \cdot t\_0\right) \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\left(x \cdot \left(t\_0 \cdot \left(1 - x\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+
          1.0
          (*
           x
           (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))
   (if (<= x -2.4e+51)
     (/ 1.0 (* (* x t_0) (- 1.0 x)))
     (/
      (- 1.0 (* (* x x) (* x (* x (* x x)))))
      (* (* x (* t_0 (- 1.0 x))) (+ 1.0 (* (* x x) (+ 1.0 (* x x)))))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))));
	double tmp;
	if (x <= -2.4e+51) {
		tmp = 1.0 / ((x * t_0) * (1.0 - x));
	} else {
		tmp = (1.0 - ((x * x) * (x * (x * (x * x))))) / ((x * (t_0 * (1.0 - x))) * (1.0 + ((x * x) * (1.0 + (x * x)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0)))))
    if (x <= (-2.4d+51)) then
        tmp = 1.0d0 / ((x * t_0) * (1.0d0 - x))
    else
        tmp = (1.0d0 - ((x * x) * (x * (x * (x * x))))) / ((x * (t_0 * (1.0d0 - x))) * (1.0d0 + ((x * x) * (1.0d0 + (x * x)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))));
	double tmp;
	if (x <= -2.4e+51) {
		tmp = 1.0 / ((x * t_0) * (1.0 - x));
	} else {
		tmp = (1.0 - ((x * x) * (x * (x * (x * x))))) / ((x * (t_0 * (1.0 - x))) * (1.0 + ((x * x) * (1.0 + (x * x)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))
	tmp = 0
	if x <= -2.4e+51:
		tmp = 1.0 / ((x * t_0) * (1.0 - x))
	else:
		tmp = (1.0 - ((x * x) * (x * (x * (x * x))))) / ((x * (t_0 * (1.0 - x))) * (1.0 + ((x * x) * (1.0 + (x * x)))))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))))
	tmp = 0.0
	if (x <= -2.4e+51)
		tmp = Float64(1.0 / Float64(Float64(x * t_0) * Float64(1.0 - x)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x))))) / Float64(Float64(x * Float64(t_0 * Float64(1.0 - x))) * Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(x * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))));
	tmp = 0.0;
	if (x <= -2.4e+51)
		tmp = 1.0 / ((x * t_0) * (1.0 - x));
	else
		tmp = (1.0 - ((x * x) * (x * (x * (x * x))))) / ((x * (t_0 * (1.0 - x))) * (1.0 + ((x * x) * (1.0 + (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4e+51], N[(1.0 / N[(N[(x * t$95$0), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(t$95$0 * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3999999999999999e51

   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

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 88.0%

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

   10. Simplified 88.0%

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

       1. flip-+ N/A

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

       2. associate-/l/ N/A

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

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

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

       4. metadata-eval N/A

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

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

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

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

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

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

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

   12. Applied egg-rr 42.0%

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

   13. Taylor expanded in x around 0

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

   15. Simplified 95.4%

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

   if -2.3999999999999999e51 < x

   1. Initial program 14.5%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 90.2%

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

   10. Simplified 90.2%

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

       1. flip-+ N/A

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

       2. associate-/l/ N/A

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

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

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

       4. metadata-eval N/A

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

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

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

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

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

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

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

   12. Applied egg-rr 90.2%

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

       1. flip3-- N/A

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

       2. associate-/l/ N/A

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

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

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

   14. Applied egg-rr 95.1%

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

Alternative 7: 94.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{-8}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\left(x \cdot \left(\left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 - x\right)\right)\right) \cdot \left(1 + x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x -1.2e+77)
     (/ -8.0 t_0)
     (/
      (- 1.0 t_0)
      (*
       (*
        x
        (*
         (+
          1.0
          (*
           x
           (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))
         (- 1.0 x)))
       (+ 1.0 (* x x)))))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -1.2e+77) {
		tmp = -8.0 / t_0;
	} else {
		tmp = (1.0 - t_0) / ((x * ((1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))) * (1.0 - x))) * (1.0 + (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    if (x <= (-1.2d+77)) then
        tmp = (-8.0d0) / t_0
    else
        tmp = (1.0d0 - t_0) / ((x * ((1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0)))))) * (1.0d0 - x))) * (1.0d0 + (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -1.2e+77) {
		tmp = -8.0 / t_0;
	} else {
		tmp = (1.0 - t_0) / ((x * ((1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))) * (1.0 - x))) * (1.0 + (x * x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * (x * x))
	tmp = 0
	if x <= -1.2e+77:
		tmp = -8.0 / t_0
	else:
		tmp = (1.0 - t_0) / ((x * ((1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))) * (1.0 - x))) * (1.0 + (x * x)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= -1.2e+77)
		tmp = Float64(-8.0 / t_0);
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(Float64(x * Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664)))))) * Float64(1.0 - x))) * Float64(1.0 + Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * (x * x));
	tmp = 0.0;
	if (x <= -1.2e+77)
		tmp = -8.0 / t_0;
	else
		tmp = (1.0 - t_0) / ((x * ((1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))) * (1.0 - x))) * (1.0 + (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e+77], N[(-8.0 / t$95$0), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(N[(x * N[(N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1999999999999999e77

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
   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 3.1%

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

   7. Simplified 3.1%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1 \cdot \frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{\frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\frac{1}{2}}^{3} + \color{blue}{{\left(\frac{1}{x}\right)}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\left({\frac{1}{2}}^{3}\right), \color{blue}{\left({\left(\frac{1}{x}\right)}^{3}\right)}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left({\color{blue}{\left(\frac{1}{x}\right)}}^{3}\right)\right)\right)\right) \]

      18. cube-div N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{{1}^{3}}{\color{blue}{{x}^{3}}}\right)\right)\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{1}{{\color{blue}{x}}^{3}}\right)\right)\right)\right) \]

   9. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{x \cdot x} + \left(0.25 - \frac{0.5}{x}\right)}{0.125 + \frac{1}{x \cdot \left(x \cdot x\right)}}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 100.0%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right)}} \]

   13. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-8}{{x}^{4}}} \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-8, \color{blue}{\left({x}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-8, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-8, \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{x}\right)\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), x\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), x\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right)\right) \]

       9. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right)\right) \]

   15. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{-8}{\left(x \cdot \left(x \cdot x\right)\right) \cdot x}} \]

   if -1.1999999999999999e77 < x

   1. Initial program 19.0%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 85.6%

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

   10. Simplified 85.6%

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

       1. flip-+ N/A

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

       2. associate-/l/ N/A

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

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

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

       4. metadata-eval N/A

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

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

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

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

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

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

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

   12. Applied egg-rr 88.7%

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

       1. flip-- N/A

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

       2. associate-/l/ N/A

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

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

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

       4. metadata-eval N/A

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

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

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

       6. associate-*l* N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

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

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

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

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

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

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

   14. Applied egg-rr 91.7%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{-8}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\left(x \cdot \left(\left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 - x\right)\right)\right) \cdot \left(1 + x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.1% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in x around 0

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

   1. +-lowering-+.f64 89.5%

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

10. Simplified 89.5%

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

    1. flip-+ N/A

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

    2. associate-/l/ N/A

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

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

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

    4. metadata-eval N/A

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

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

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

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

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

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

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

12. Applied egg-rr 75.3%

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

13. Taylor expanded in x around 0

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

15. Simplified 91.7%

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

Alternative 9: 91.8% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1 + x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)}{\frac{1}{x}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (/ (+ 1.0 (* x (+ -0.5 (* x (+ 0.25 (* x -0.125)))))) (/ 1.0 x))))

C

double code(double x) {
	return 1.0 / ((1.0 + (x * (-0.5 + (x * (0.25 + (x * -0.125)))))) / (1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((1.0d0 + (x * ((-0.5d0) + (x * (0.25d0 + (x * (-0.125d0))))))) / (1.0d0 / x))
end function

Java

public static double code(double x) {
	return 1.0 / ((1.0 + (x * (-0.5 + (x * (0.25 + (x * -0.125)))))) / (1.0 / x));
}

Python

def code(x):
	return 1.0 / ((1.0 + (x * (-0.5 + (x * (0.25 + (x * -0.125)))))) / (1.0 / x))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(1.0 + Float64(x * Float64(-0.5 + Float64(x * Float64(0.25 + Float64(x * -0.125)))))) / Float64(1.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((1.0 + (x * (-0.5 + (x * (0.25 + (x * -0.125)))))) / (1.0 / x));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(1.0 + N[(x * N[(-0.5 + N[(x * N[(0.25 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.9%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

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

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

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

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

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

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

   6. frac-times N/A

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

   7. metadata-eval N/A

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

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

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

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

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

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

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

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

   12. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1 \cdot \frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{\frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\frac{1}{2}}^{3} + \color{blue}{{\left(\frac{1}{x}\right)}^{3}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\left({\frac{1}{2}}^{3}\right), \color{blue}{\left({\left(\frac{1}{x}\right)}^{3}\right)}\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left({\color{blue}{\left(\frac{1}{x}\right)}}^{3}\right)\right)\right)\right) \]

   18. cube-div N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{{1}^{3}}{\color{blue}{{x}^{3}}}\right)\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{1}{{\color{blue}{x}}^{3}}\right)\right)\right)\right) \]

9. Applied egg-rr 25.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{x \cdot x} + \left(0.25 - \frac{0.5}{x}\right)}{0.125 + \frac{1}{x \cdot \left(x \cdot x\right)}}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

    11. *-lowering-*.f64 89.7%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

12. Simplified 89.7%

    \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right)}} \]
13. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{1 + x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)}} \]

    2. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{1 + x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)}{\frac{1}{x}}}} \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)}{\frac{1}{x}}\right)}\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right), \color{blue}{\left(\frac{1}{x}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right)\right), \left(\frac{\color{blue}{1}}{x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right)\right), \left(\frac{1}{x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right)\right)\right), \left(\frac{1}{x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right)\right)\right), \left(\frac{1}{x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \frac{-1}{8}\right)\right)\right)\right)\right)\right), \left(\frac{1}{x}\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \frac{-1}{8}\right)\right)\right)\right)\right)\right), \left(\frac{1}{x}\right)\right)\right) \]

    11. /-lowering-/.f64 89.7%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \frac{-1}{8}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{x}\right)\right)\right) \]

14. Applied egg-rr 89.7%

    \[\leadsto \color{blue}{\frac{1}{\frac{1 + x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)}{\frac{1}{x}}}} \]
15. Add Preprocessing

Alternative 10: 92.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3:\\ \;\;\;\;\frac{-8 + \frac{-16}{x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.0)
   (/ (+ -8.0 (/ -16.0 x)) (* x (* x (* x x))))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.0) {
		tmp = (-8.0 + (-16.0 / x)) / (x * (x * (x * x)));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.0d0)) then
        tmp = ((-8.0d0) + ((-16.0d0) / x)) / (x * (x * (x * x)))
    else
        tmp = (1.0d0 / x) + (0.5d0 + (x * 0.08333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.0) {
		tmp = (-8.0 + (-16.0 / x)) / (x * (x * (x * x)));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.0:
		tmp = (-8.0 + (-16.0 / x)) / (x * (x * (x * x)))
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.0)
		tmp = Float64(Float64(-8.0 + Float64(-16.0 / x)) / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(1.0 / x) + Float64(0.5 + Float64(x * 0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.0)
		tmp = (-8.0 + (-16.0 / x)) / (x * (x * (x * x)));
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.0], N[(N[(-8.0 + N[(-16.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -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. 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 3.1%

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

   7. Simplified 3.1%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1 \cdot \frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{\frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\frac{1}{2}}^{3} + \color{blue}{{\left(\frac{1}{x}\right)}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\left({\frac{1}{2}}^{3}\right), \color{blue}{\left({\left(\frac{1}{x}\right)}^{3}\right)}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left({\color{blue}{\left(\frac{1}{x}\right)}}^{3}\right)\right)\right)\right) \]

      18. cube-div N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{{1}^{3}}{\color{blue}{{x}^{3}}}\right)\right)\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{1}{{\color{blue}{x}}^{3}}\right)\right)\right)\right) \]

   9. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{x \cdot x} + \left(0.25 - \frac{0.5}{x}\right)}{0.125 + \frac{1}{x \cdot \left(x \cdot x\right)}}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 74.3%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 74.3%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right)}} \]

   13. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{8 + 16 \cdot \frac{1}{x}}{{x}^{4}}} \]
   14. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{8 + 16 \cdot \frac{1}{x}}{{x}^{4}}\right) \]

       2. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(8 + 16 \cdot \frac{1}{x}\right)\right)}{\color{blue}{{x}^{4}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(8 + 16 \cdot \frac{1}{x}\right)\right)\right), \color{blue}{\left({x}^{4}\right)}\right) \]

       4. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(8\right)\right) + \left(\mathsf{neg}\left(16 \cdot \frac{1}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-8 + \left(\mathsf{neg}\left(16 \cdot \frac{1}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \left(\mathsf{neg}\left(16 \cdot \frac{1}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \left(\mathsf{neg}\left(\frac{16 \cdot 1}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \left(\mathsf{neg}\left(\frac{16}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \left(\frac{\mathsf{neg}\left(16\right)}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(16\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \mathsf{/.f64}\left(-16, x\right)\right), \left({x}^{4}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \mathsf{/.f64}\left(-16, x\right)\right), \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       13. pow-plus N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \mathsf{/.f64}\left(-16, x\right)\right), \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \mathsf{/.f64}\left(-16, x\right)\right), \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{x}\right)\right) \]

       15. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \mathsf{/.f64}\left(-16, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \mathsf{/.f64}\left(-16, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), x\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \mathsf{/.f64}\left(-16, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), x\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \mathsf{/.f64}\left(-16, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right)\right) \]

       19. *-lowering-*.f64 74.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-8, \mathsf{/.f64}\left(-16, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right)\right) \]

   15. Simplified 74.3%

       \[\leadsto \color{blue}{\frac{-8 + \frac{-16}{x}}{\left(x \cdot \left(x \cdot x\right)\right) \cdot x}} \]

   if -3 < x

   1. Initial program 6.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3:\\ \;\;\;\;\frac{-8 + \frac{-16}{x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + \left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ x (* (* x x) (+ -0.5 (* x (+ 0.25 (* x -0.125))))))))

C

double code(double x) {
	return 1.0 / (x + ((x * x) * (-0.5 + (x * (0.25 + (x * -0.125))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x + ((x * x) * ((-0.5d0) + (x * (0.25d0 + (x * (-0.125d0)))))))
end function

Java

public static double code(double x) {
	return 1.0 / (x + ((x * x) * (-0.5 + (x * (0.25 + (x * -0.125))))));
}

Python

def code(x):
	return 1.0 / (x + ((x * x) * (-0.5 + (x * (0.25 + (x * -0.125))))))

Julia

function code(x)
	return Float64(1.0 / Float64(x + Float64(Float64(x * x) * Float64(-0.5 + Float64(x * Float64(0.25 + Float64(x * -0.125)))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x + ((x * x) * (-0.5 + (x * (0.25 + (x * -0.125))))));
end

Wolfram

code[x_] := N[(1.0 / N[(x + N[(N[(x * x), $MachinePrecision] * N[(-0.5 + N[(x * N[(0.25 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.9%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

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

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

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

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

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

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

   6. frac-times N/A

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

   7. metadata-eval N/A

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

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

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

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

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

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

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

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

   12. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1 \cdot \frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{\frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\frac{1}{2}}^{3} + \color{blue}{{\left(\frac{1}{x}\right)}^{3}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\left({\frac{1}{2}}^{3}\right), \color{blue}{\left({\left(\frac{1}{x}\right)}^{3}\right)}\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left({\color{blue}{\left(\frac{1}{x}\right)}}^{3}\right)\right)\right)\right) \]

   18. cube-div N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{{1}^{3}}{\color{blue}{{x}^{3}}}\right)\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{1}{{\color{blue}{x}}^{3}}\right)\right)\right)\right) \]

9. Applied egg-rr 25.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{x \cdot x} + \left(0.25 - \frac{0.5}{x}\right)}{0.125 + \frac{1}{x \cdot \left(x \cdot x\right)}}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

    11. *-lowering-*.f64 89.7%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

12. Simplified 89.7%

    \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right)}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right) + \color{blue}{1}\right)\right)\right) \]

    2. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right) \cdot x + \color{blue}{1 \cdot x}\right)\right) \]

    3. *-lft-identity N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right) \cdot x + x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right) \cdot x\right), \color{blue}{x}\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\left(\left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right) \cdot x\right) \cdot x\right), x\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right) \cdot \left(x \cdot x\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right), \left(x \cdot x\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right), \left(x \cdot x\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{4} + x \cdot \frac{-1}{8}\right)\right)\right), \left(x \cdot x\right)\right), x\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \frac{-1}{8}\right)\right)\right)\right), \left(x \cdot x\right)\right), x\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \frac{-1}{8}\right)\right)\right)\right), \left(x \cdot x\right)\right), x\right)\right) \]

    12. *-lowering-*.f64 89.7%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \frac{-1}{8}\right)\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), x\right)\right) \]

14. Applied egg-rr 89.7%

    \[\leadsto \frac{1}{\color{blue}{\left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right) \cdot \left(x \cdot x\right) + x}} \]

15. Final simplification 89.7%

    \[\leadsto \frac{1}{x + \left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)} \]
16. Add Preprocessing

Alternative 12: 91.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (* x (+ 1.0 (* x (+ -0.5 (* x (+ 0.25 (* x -0.125)))))))))

C

double code(double x) {
	return 1.0 / (x * (1.0 + (x * (-0.5 + (x * (0.25 + (x * -0.125)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x * (1.0d0 + (x * ((-0.5d0) + (x * (0.25d0 + (x * (-0.125d0))))))))
end function

Java

public static double code(double x) {
	return 1.0 / (x * (1.0 + (x * (-0.5 + (x * (0.25 + (x * -0.125)))))));
}

Python

def code(x):
	return 1.0 / (x * (1.0 + (x * (-0.5 + (x * (0.25 + (x * -0.125)))))))

Julia

function code(x)
	return Float64(1.0 / Float64(x * Float64(1.0 + Float64(x * Float64(-0.5 + Float64(x * Float64(0.25 + Float64(x * -0.125))))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x * (1.0 + (x * (-0.5 + (x * (0.25 + (x * -0.125)))))));
end

Wolfram

code[x_] := N[(1.0 / N[(x * N[(1.0 + N[(x * N[(-0.5 + N[(x * N[(0.25 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.9%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

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

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

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

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

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

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

   6. frac-times N/A

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

   7. metadata-eval N/A

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

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

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

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

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

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

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

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

   12. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1 \cdot \frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{\frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\frac{1}{2}}^{3} + \color{blue}{{\left(\frac{1}{x}\right)}^{3}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\left({\frac{1}{2}}^{3}\right), \color{blue}{\left({\left(\frac{1}{x}\right)}^{3}\right)}\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left({\color{blue}{\left(\frac{1}{x}\right)}}^{3}\right)\right)\right)\right) \]

   18. cube-div N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{{1}^{3}}{\color{blue}{{x}^{3}}}\right)\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{1}{{\color{blue}{x}}^{3}}\right)\right)\right)\right) \]

9. Applied egg-rr 25.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{x \cdot x} + \left(0.25 - \frac{0.5}{x}\right)}{0.125 + \frac{1}{x \cdot \left(x \cdot x\right)}}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

    11. *-lowering-*.f64 89.7%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

12. Simplified 89.7%

    \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right)}} \]
13. Add Preprocessing

Alternative 13: 92.2% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1:\\ \;\;\;\;\frac{-8}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.1)
   (/ -8.0 (* x (* x (* x x))))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.1) {
		tmp = -8.0 / (x * (x * (x * x)));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.1d0)) then
        tmp = (-8.0d0) / (x * (x * (x * x)))
    else
        tmp = (1.0d0 / x) + (0.5d0 + (x * 0.08333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.1) {
		tmp = -8.0 / (x * (x * (x * x)));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.1:
		tmp = -8.0 / (x * (x * (x * x)))
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.1)
		tmp = Float64(-8.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(1.0 / x) + Float64(0.5 + Float64(x * 0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.1)
		tmp = -8.0 / (x * (x * (x * x)));
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.1], N[(-8.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.10000000000000009

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
   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 3.1%

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

   7. Simplified 3.1%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1 \cdot \frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{\frac{1}{2}}{x}\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left({\frac{1}{2}}^{3} + \color{blue}{{\left(\frac{1}{x}\right)}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\left({\frac{1}{2}}^{3}\right), \color{blue}{\left({\left(\frac{1}{x}\right)}^{3}\right)}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left({\color{blue}{\left(\frac{1}{x}\right)}}^{3}\right)\right)\right)\right) \]

      18. cube-div N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{{1}^{3}}{\color{blue}{{x}^{3}}}\right)\right)\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{1}{{\color{blue}{x}}^{3}}\right)\right)\right)\right) \]

   9. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{x \cdot x} + \left(0.25 - \frac{0.5}{x}\right)}{0.125 + \frac{1}{x \cdot \left(x \cdot x\right)}}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} + \frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{8} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 74.3%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 74.3%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot \left(0.25 + x \cdot -0.125\right)\right)\right)}} \]

   13. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-8}{{x}^{4}}} \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-8, \color{blue}{\left({x}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-8, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-8, \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{x}\right)\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), x\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), x\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right)\right) \]

       9. *-lowering-*.f64 74.3%

          \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right)\right) \]

   15. Simplified 74.3%

       \[\leadsto \color{blue}{\frac{-8}{\left(x \cdot \left(x \cdot x\right)\right) \cdot x}} \]

   if -3.10000000000000009 < x

   1. Initial program 6.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1:\\ \;\;\;\;\frac{-8}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.8% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.5)
   (/ 24.0 (* x (* x x)))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -5.5) {
		tmp = 24.0 / (x * (x * x));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.5d0)) then
        tmp = 24.0d0 / (x * (x * x))
    else
        tmp = (1.0d0 / x) + (0.5d0 + (x * 0.08333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.5) {
		tmp = 24.0 / (x * (x * x));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.5:
		tmp = 24.0 / (x * (x * x))
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.5)
		tmp = Float64(24.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 / x) + Float64(0.5 + Float64(x * 0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.5)
		tmp = 24.0 / (x * (x * x));
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.5], N[(24.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 74.0%

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

   10. Simplified 74.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{24}{{x}^{3}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(24, \color{blue}{\left({x}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 67.2%

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

   13. Simplified 67.2%

       \[\leadsto \color{blue}{\frac{24}{x \cdot \left(x \cdot x\right)}} \]

   if -5.5 < x

   1. Initial program 6.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 99.1%

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

   7. Simplified 99.1%

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

Alternative 15: 89.4% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.95) (/ 24.0 (* x (* x x))) (+ 0.5 (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.95) {
		tmp = 24.0 / (x * (x * x));
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.95d0)) then
        tmp = 24.0d0 / (x * (x * x))
    else
        tmp = 0.5d0 + (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.95) {
		tmp = 24.0 / (x * (x * x));
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.95:
		tmp = 24.0 / (x * (x * x))
	else:
		tmp = 0.5 + (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.95)
		tmp = Float64(24.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(0.5 + Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.95)
		tmp = 24.0 / (x * (x * x));
	else
		tmp = 0.5 + (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.95], N[(24.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.94999999999999996

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 74.0%

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

   10. Simplified 74.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{24}{{x}^{3}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(24, \color{blue}{\left({x}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 67.2%

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

   13. Simplified 67.2%

       \[\leadsto \color{blue}{\frac{24}{x \cdot \left(x \cdot x\right)}} \]

   if -1.94999999999999996 < x

   1. Initial program 6.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. rgt-mult-inverse N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

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

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

      9. /-lowering-/.f64 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.3% 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 40.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 63.4%

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

7. Simplified 63.4%

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

Alternative 17: 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 40.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 98.2%

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

8. Taylor expanded in x around 0

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

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

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

   2. +-lowering-+.f64 62.7%

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

10. Simplified 62.7%

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

11. Taylor expanded in x around inf

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

13. Simplified 3.8%

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

Alternative 18: 3.3% 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 40.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Taylor expanded in x around inf

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

10. Simplified 3.4%

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

Developer Target 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024139 
(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)))