expq2 (section 3.11)

Percentage Accurate: 38.1% → 100.0%

Time: 8.2s

Alternatives: 14

Speedup: 68.3×

Specification

\[710 > x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.1% 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 37.8%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 98.0% 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 37.8%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 98.7%

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

Alternative 3: 95.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{+103}:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(1 - t\_0 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)}{1 - x \cdot t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+ -0.5 (* x (+ 0.16666666666666666 (* x -0.041666666666666664))))))
   (if (<= x -4e+103)
     (/ 6.0 (* x (* x x)))
     (/ 1.0 (/ (* x (- 1.0 (* t_0 (* (* x x) t_0)))) (- 1.0 (* x t_0)))))))

C

double code(double x) {
	double t_0 = -0.5 + (x * (0.16666666666666666 + (x * -0.041666666666666664)));
	double tmp;
	if (x <= -4e+103) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = 1.0 / ((x * (1.0 - (t_0 * ((x * x) * t_0)))) / (1.0 - (x * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) + (x * (0.16666666666666666d0 + (x * (-0.041666666666666664d0))))
    if (x <= (-4d+103)) then
        tmp = 6.0d0 / (x * (x * x))
    else
        tmp = 1.0d0 / ((x * (1.0d0 - (t_0 * ((x * x) * t_0)))) / (1.0d0 - (x * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -0.5 + (x * (0.16666666666666666 + (x * -0.041666666666666664)));
	double tmp;
	if (x <= -4e+103) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = 1.0 / ((x * (1.0 - (t_0 * ((x * x) * t_0)))) / (1.0 - (x * t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = -0.5 + (x * (0.16666666666666666 + (x * -0.041666666666666664)))
	tmp = 0
	if x <= -4e+103:
		tmp = 6.0 / (x * (x * x))
	else:
		tmp = 1.0 / ((x * (1.0 - (t_0 * ((x * x) * t_0)))) / (1.0 - (x * t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(-0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * -0.041666666666666664))))
	tmp = 0.0
	if (x <= -4e+103)
		tmp = Float64(6.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(1.0 / Float64(Float64(x * Float64(1.0 - Float64(t_0 * Float64(Float64(x * x) * t_0)))) / Float64(1.0 - Float64(x * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -0.5 + (x * (0.16666666666666666 + (x * -0.041666666666666664)));
	tmp = 0.0;
	if (x <= -4e+103)
		tmp = 6.0 / (x * (x * x));
	else
		tmp = 1.0 / ((x * (1.0 - (t_0 * ((x * x) * t_0)))) / (1.0 - (x * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 + N[(x * N[(0.16666666666666666 + N[(x * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e+103], N[(6.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * N[(1.0 - N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e103

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

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

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

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

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

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

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

      7. exp-lowering-exp.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({x}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{6}{x \cdot \left(x \cdot x\right)}} \]

   if -4e103 < x

   1. Initial program 21.5%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

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

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

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

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

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

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

      7. exp-lowering-exp.f64 21.5%

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

   6. Applied egg-rr 21.5%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 89.7%

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

   9. Simplified 89.7%

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

       1. *-commutative N/A

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

       2. flip-+ N/A

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

       3. associate-*l/ N/A

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

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

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

   11. Applied egg-rr 95.2%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+103}:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(1 - \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)\right)\right)}{1 - x \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot -0.041666666666666664\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot \left(1 + x \cdot \left(-0.5 + \frac{x \cdot \left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot -7.233796296296296 \cdot 10^{-5}\right)}{0.027777777777777776 + x \cdot 0.006944444444444444}\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (*
   x
   (+
    1.0
    (*
     x
     (+
      -0.5
      (/
       (* x (+ 0.004629629629629629 (* (* x (* x x)) -7.233796296296296e-5)))
       (+ 0.027777777777777776 (* x 0.006944444444444444)))))))))

C

double code(double x) {
	return 1.0 / (x * (1.0 + (x * (-0.5 + ((x * (0.004629629629629629 + ((x * (x * x)) * -7.233796296296296e-5))) / (0.027777777777777776 + (x * 0.006944444444444444)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x * (1.0d0 + (x * ((-0.5d0) + ((x * (0.004629629629629629d0 + ((x * (x * x)) * (-7.233796296296296d-5)))) / (0.027777777777777776d0 + (x * 0.006944444444444444d0)))))))
end function

Java

public static double code(double x) {
	return 1.0 / (x * (1.0 + (x * (-0.5 + ((x * (0.004629629629629629 + ((x * (x * x)) * -7.233796296296296e-5))) / (0.027777777777777776 + (x * 0.006944444444444444)))))));
}

Python

def code(x):
	return 1.0 / (x * (1.0 + (x * (-0.5 + ((x * (0.004629629629629629 + ((x * (x * x)) * -7.233796296296296e-5))) / (0.027777777777777776 + (x * 0.006944444444444444)))))))

Julia

function code(x)
	return Float64(1.0 / Float64(x * Float64(1.0 + Float64(x * Float64(-0.5 + Float64(Float64(x * Float64(0.004629629629629629 + Float64(Float64(x * Float64(x * x)) * -7.233796296296296e-5))) / Float64(0.027777777777777776 + Float64(x * 0.006944444444444444))))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x * (1.0 + (x * (-0.5 + ((x * (0.004629629629629629 + ((x * (x * x)) * -7.233796296296296e-5))) / (0.027777777777777776 + (x * 0.006944444444444444)))))));
end

Wolfram

code[x_] := N[(1.0 / N[(x * N[(1.0 + N[(x * N[(-0.5 + N[(N[(x * N[(0.004629629629629629 + N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * -7.233796296296296e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.027777777777777776 + N[(x * 0.006944444444444444), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

   1. clear-num N/A

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

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

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

   3. div-sub N/A

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

   4. *-inverses N/A

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

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

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

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

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

   7. exp-lowering-exp.f64 37.7%

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

6. Applied egg-rr 37.7%

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

7. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

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

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

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

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

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

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 91.8%

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

9. Simplified 91.8%

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

    1. *-commutative N/A

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

    2. flip3-+ N/A

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

    3. associate-*l/ N/A

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

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

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

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

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

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

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

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \left({\left(x \cdot \frac{-1}{24}\right)}^{3}\right)\right), x\right), \left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \frac{-1}{24}\right) \cdot \left(x \cdot \frac{-1}{24}\right) - \frac{1}{6} \cdot \left(x \cdot \frac{-1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    8. unpow-prod-down N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \left({x}^{3} \cdot {\frac{-1}{24}}^{3}\right)\right), x\right), \left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \frac{-1}{24}\right) \cdot \left(x \cdot \frac{-1}{24}\right) - \frac{1}{6} \cdot \left(x \cdot \frac{-1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\left({x}^{3}\right), \left({\frac{-1}{24}}^{3}\right)\right)\right), x\right), \left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \frac{-1}{24}\right) \cdot \left(x \cdot \frac{-1}{24}\right) - \frac{1}{6} \cdot \left(x \cdot \frac{-1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    10. cube-mult N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left({\frac{-1}{24}}^{3}\right)\right)\right), x\right), \left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \frac{-1}{24}\right) \cdot \left(x \cdot \frac{-1}{24}\right) - \frac{1}{6} \cdot \left(x \cdot \frac{-1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left({\frac{-1}{24}}^{3}\right)\right)\right), x\right), \left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \frac{-1}{24}\right) \cdot \left(x \cdot \frac{-1}{24}\right) - \frac{1}{6} \cdot \left(x \cdot \frac{-1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({\frac{-1}{24}}^{3}\right)\right)\right), x\right), \left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \frac{-1}{24}\right) \cdot \left(x \cdot \frac{-1}{24}\right) - \frac{1}{6} \cdot \left(x \cdot \frac{-1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    13. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{-1}{13824}\right)\right), x\right), \left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \frac{-1}{24}\right) \cdot \left(x \cdot \frac{-1}{24}\right) - \frac{1}{6} \cdot \left(x \cdot \frac{-1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{-1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\left(\frac{1}{6} \cdot \frac{1}{6}\right), \color{blue}{\left(\left(x \cdot \frac{-1}{24}\right) \cdot \left(x \cdot \frac{-1}{24}\right) - \frac{1}{6} \cdot \left(x \cdot \frac{-1}{24}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

    15. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{-1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \left(\color{blue}{\left(x \cdot \frac{-1}{24}\right) \cdot \left(x \cdot \frac{-1}{24}\right)} - \frac{1}{6} \cdot \left(x \cdot \frac{-1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    16. distribute-rgt-out-- N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{-1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \left(\left(x \cdot \frac{-1}{24}\right) \cdot \color{blue}{\left(x \cdot \frac{-1}{24} - \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{-1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\left(x \cdot \frac{-1}{24}\right), \color{blue}{\left(x \cdot \frac{-1}{24} - \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{-1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \left(\color{blue}{x \cdot \frac{-1}{24}} - \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    19. --lowering--.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{-1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \mathsf{\_.f64}\left(\left(x \cdot \frac{-1}{24}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    20. *-lowering-*.f64 77.4%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{-1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

11. Applied egg-rr 77.4%

    \[\leadsto \frac{1}{x \cdot \left(1 + x \cdot \left(-0.5 + \color{blue}{\frac{\left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot -7.233796296296296 \cdot 10^{-5}\right) \cdot x}{0.027777777777777776 + \left(x \cdot -0.041666666666666664\right) \cdot \left(x \cdot -0.041666666666666664 - 0.16666666666666666\right)}}\right)\right)} \]

12. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{-1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \color{blue}{\left(\frac{1}{144} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{-1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \left(x \cdot \color{blue}{\frac{1}{144}}\right)\right)\right)\right)\right)\right)\right)\right) \]

    2. *-lowering-*.f64 93.3%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{-1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{144}}\right)\right)\right)\right)\right)\right)\right)\right) \]

14. Simplified 93.3%

    \[\leadsto \frac{1}{x \cdot \left(1 + x \cdot \left(-0.5 + \frac{\left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot -7.233796296296296 \cdot 10^{-5}\right) \cdot x}{0.027777777777777776 + \color{blue}{x \cdot 0.006944444444444444}}\right)\right)} \]

15. Final simplification 93.3%

    \[\leadsto \frac{1}{x \cdot \left(1 + x \cdot \left(-0.5 + \frac{x \cdot \left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot -7.233796296296296 \cdot 10^{-5}\right)}{0.027777777777777776 + x \cdot 0.006944444444444444}\right)\right)} \]
16. Add Preprocessing

Alternative 5: 91.6% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 / 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]

Derivation

1. Initial program 37.8%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

   1. clear-num N/A

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

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

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

   3. div-sub N/A

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

   4. *-inverses N/A

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

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

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

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

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

   7. exp-lowering-exp.f64 37.7%

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

6. Applied egg-rr 37.7%

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

7. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

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

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

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

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

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

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 91.8%

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

9. Simplified 91.8%

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

    1. inv-pow N/A

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

    2. *-commutative N/A

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

    3. unpow-prod-down N/A

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

    4. inv-pow N/A

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

    5. inv-pow N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    14. /-lowering-/.f64 91.9%

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

11. Applied egg-rr 91.9%

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

Alternative 6: 91.7% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

   1. clear-num N/A

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

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

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

   3. div-sub N/A

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

   4. *-inverses N/A

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

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

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

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

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

   7. exp-lowering-exp.f64 37.7%

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

6. Applied egg-rr 37.7%

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

7. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

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

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

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

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

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

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 91.8%

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

9. Simplified 91.8%

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

Alternative 7: 93.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{-96}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -700.0)
   (/ -96.0 (* x (* x (* x (* x x)))))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -700.0) {
		tmp = -96.0 / (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 <= (-700.0d0)) then
        tmp = (-96.0d0) / (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 <= -700.0) {
		tmp = -96.0 / (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 <= -700.0:
		tmp = -96.0 / (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 <= -700.0)
		tmp = Float64(-96.0 / Float64(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 <= -700.0)
		tmp = -96.0 / (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, -700.0], N[(-96.0 / N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

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

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

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

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

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

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

      7. exp-lowering-exp.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 76.9%

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

   9. Simplified 76.9%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{24 + 96 \cdot \frac{1}{x}}{{x}^{4}}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(24 + 96 \cdot \frac{1}{x}\right)}{\color{blue}{{x}^{4}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(24 + 96 \cdot \frac{1}{x}\right)\right), \color{blue}{\left({x}^{4}\right)}\right) \]

       3. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot 24 + -1 \cdot \left(96 \cdot \frac{1}{x}\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-24 + -1 \cdot \left(96 \cdot \frac{1}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(-1 \cdot \left(96 \cdot \frac{1}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       6. neg-mul-1 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(\mathsf{neg}\left(96 \cdot \frac{1}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(\mathsf{neg}\left(\frac{96 \cdot 1}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(\mathsf{neg}\left(\frac{96}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(\frac{\mathsf{neg}\left(96\right)}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(96\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \left({x}^{4}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       13. pow-plus N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \left(x \cdot \color{blue}{{x}^{3}}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       16. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       20. *-lowering-*.f64 76.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   12. Simplified 76.9%

       \[\leadsto \color{blue}{\frac{-24 + \frac{-96}{x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-96}{{x}^{5}}} \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-96, \color{blue}{\left({x}^{5}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-96, \left({x}^{\left(4 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-96, \left({x}^{4} \cdot \color{blue}{x}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-96, \left(x \cdot \color{blue}{{x}^{4}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-96, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{4}\right)}\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-96, \mathsf{*.f64}\left(x, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right) \]

       7. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-96, \mathsf{*.f64}\left(x, \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-96, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-96, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(-96, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-96, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(-96, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-96, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 81.4%

           \[\leadsto \mathsf{/.f64}\left(-96, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   15. Simplified 81.4%

       \[\leadsto \color{blue}{\frac{-96}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}} \]

   if -700 < x

   1. Initial program 6.3%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 99.5%

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

   7. Simplified 99.5%

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

Alternative 8: 91.8% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.0)
   (/ -24.0 (* x (* x (* x x))))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.0) {
		tmp = -24.0 / (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 <= (-4.0d0)) then
        tmp = (-24.0d0) / (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 <= -4.0) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.0:
		tmp = -24.0 / (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 <= -4.0)
		tmp = Float64(-24.0 / 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 <= -4.0)
		tmp = -24.0 / (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, -4.0], N[(-24.0 / 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 < -4

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

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

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

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

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

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

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

      7. exp-lowering-exp.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 76.1%

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

   9. Simplified 76.1%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-24}{{x}^{4}}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-24, \color{blue}{\left({x}^{4}\right)}\right) \]

       2. metadata-eval N/A

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

       3. pow-plus N/A

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

       4. *-commutative N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 76.1%

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

   12. Simplified 76.1%

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

   if -4 < x

   1. Initial program 5.7%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 100.0%

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

   7. Simplified 100.0%

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

Alternative 9: 89.1% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.0)
   (/ 6.0 (* x (* x x)))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.0) {
		tmp = 6.0 / (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 <= (-4.0d0)) then
        tmp = 6.0d0 / (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 <= -4.0) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.0:
		tmp = 6.0 / (x * (x * x))
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(6.0 / 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 <= -4.0)
		tmp = 6.0 / (x * (x * x));
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.0], N[(6.0 / 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 < -4

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

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

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

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

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

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

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

      7. exp-lowering-exp.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 63.2%

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

   9. Simplified 63.2%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({x}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 63.2%

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

   12. Simplified 63.2%

       \[\leadsto \color{blue}{\frac{6}{x \cdot \left(x \cdot x\right)}} \]

   if -4 < x

   1. Initial program 5.7%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 100.0%

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

   7. Simplified 100.0%

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

Alternative 10: 88.8% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.85) (/ 6.0 (* x (* x x))) (+ (/ 1.0 x) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.85d0)) then
        tmp = 6.0d0 / (x * (x * x))
    else
        tmp = (1.0d0 / x) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.85:
		tmp = 6.0 / (x * (x * x))
	else:
		tmp = (1.0 / x) + 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.85)
		tmp = Float64(6.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 / x) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.85)
		tmp = 6.0 / (x * (x * x));
	else
		tmp = (1.0 / x) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.85], N[(6.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8500000000000001

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

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

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

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

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

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

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

      7. exp-lowering-exp.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 63.2%

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

   9. Simplified 63.2%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({x}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 63.2%

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

   12. Simplified 63.2%

       \[\leadsto \color{blue}{\frac{6}{x \cdot \left(x \cdot x\right)}} \]

   if -1.8500000000000001 < x

   1. Initial program 5.7%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 99.9%

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

   7. Simplified 99.9%

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

Alternative 11: 83.7% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.75d0)) then
        tmp = (-2.0d0) / (x * x)
    else
        tmp = (1.0d0 / x) + 0.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.75

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

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

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

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

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

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

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

      7. exp-lowering-exp.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 46.7%

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

   9. Simplified 46.7%

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

   10. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 46.7%

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

   12. Simplified 46.7%

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

   if -1.75 < x

   1. Initial program 5.7%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 99.9%

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

   7. Simplified 99.9%

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

Alternative 12: 66.6% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
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 67.0%

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

7. Simplified 67.0%

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

Alternative 13: 66.6% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 66.8%

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

7. Simplified 66.8%

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

Alternative 14: 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 37.8%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
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 67.0%

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

7. Simplified 67.0%

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

8. Taylor expanded in x around inf

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

10. Simplified 3.2%

    \[\leadsto \color{blue}{0.5} \]
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 2024150 
(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)))