expq2 (section 3.11)

Percentage Accurate: 38.0% → 100.0%

Time: 9.7s

Alternatives: 13

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: 38.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 94.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \frac{1}{x} + 0.5\\ t_2 := \left(x \cdot x\right) \cdot \left(x \cdot -0.001388888888888889\right)\\ t_3 := x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{1}{x \cdot x}}{\frac{1}{x} + \left(0.5 - t\_3\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\frac{t\_1 \cdot t\_1 - t\_3 \cdot t\_3}{\frac{1}{x} + \left(0.5 + \frac{t\_0 \cdot 0.0005787037037037037 + t\_0 \cdot \left(t\_0 \cdot \left(t\_0 \cdot -2.6791838134430728 \cdot 10^{-9}\right)\right)}{\left(\left(x \cdot 0.08333333333333333\right) \cdot t\_2 - t\_2 \cdot t\_2\right) - \left(x \cdot 0.08333333333333333\right) \cdot \left(x \cdot 0.08333333333333333\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (+ (/ 1.0 x) 0.5))
        (t_2 (* (* x x) (* x -0.001388888888888889)))
        (t_3 (* x (+ 0.08333333333333333 (* x (* x -0.001388888888888889))))))
   (if (<= x -2.1e+52)
     (/ (/ 1.0 (* x x)) (+ (/ 1.0 x) (- 0.5 t_3)))
     (if (<= x -5e+36)
       (/
        (- (* t_1 t_1) (* t_3 t_3))
        (+
         (/ 1.0 x)
         (+
          0.5
          (/
           (+
            (* t_0 0.0005787037037037037)
            (* t_0 (* t_0 (* t_0 -2.6791838134430728e-9))))
           (-
            (- (* (* x 0.08333333333333333) t_2) (* t_2 t_2))
            (* (* x 0.08333333333333333) (* x 0.08333333333333333)))))))
       (+
        t_1
        (* x (+ 0.08333333333333333 (* (* x x) -0.001388888888888889))))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (1.0 / x) + 0.5;
	double t_2 = (x * x) * (x * -0.001388888888888889);
	double t_3 = x * (0.08333333333333333 + (x * (x * -0.001388888888888889)));
	double tmp;
	if (x <= -2.1e+52) {
		tmp = (1.0 / (x * x)) / ((1.0 / x) + (0.5 - t_3));
	} else if (x <= -5e+36) {
		tmp = ((t_1 * t_1) - (t_3 * t_3)) / ((1.0 / x) + (0.5 + (((t_0 * 0.0005787037037037037) + (t_0 * (t_0 * (t_0 * -2.6791838134430728e-9)))) / ((((x * 0.08333333333333333) * t_2) - (t_2 * t_2)) - ((x * 0.08333333333333333) * (x * 0.08333333333333333))))));
	} else {
		tmp = t_1 + (x * (0.08333333333333333 + ((x * x) * -0.001388888888888889)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = (1.0d0 / x) + 0.5d0
    t_2 = (x * x) * (x * (-0.001388888888888889d0))
    t_3 = x * (0.08333333333333333d0 + (x * (x * (-0.001388888888888889d0))))
    if (x <= (-2.1d+52)) then
        tmp = (1.0d0 / (x * x)) / ((1.0d0 / x) + (0.5d0 - t_3))
    else if (x <= (-5d+36)) then
        tmp = ((t_1 * t_1) - (t_3 * t_3)) / ((1.0d0 / x) + (0.5d0 + (((t_0 * 0.0005787037037037037d0) + (t_0 * (t_0 * (t_0 * (-2.6791838134430728d-9))))) / ((((x * 0.08333333333333333d0) * t_2) - (t_2 * t_2)) - ((x * 0.08333333333333333d0) * (x * 0.08333333333333333d0))))))
    else
        tmp = t_1 + (x * (0.08333333333333333d0 + ((x * x) * (-0.001388888888888889d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (1.0 / x) + 0.5;
	double t_2 = (x * x) * (x * -0.001388888888888889);
	double t_3 = x * (0.08333333333333333 + (x * (x * -0.001388888888888889)));
	double tmp;
	if (x <= -2.1e+52) {
		tmp = (1.0 / (x * x)) / ((1.0 / x) + (0.5 - t_3));
	} else if (x <= -5e+36) {
		tmp = ((t_1 * t_1) - (t_3 * t_3)) / ((1.0 / x) + (0.5 + (((t_0 * 0.0005787037037037037) + (t_0 * (t_0 * (t_0 * -2.6791838134430728e-9)))) / ((((x * 0.08333333333333333) * t_2) - (t_2 * t_2)) - ((x * 0.08333333333333333) * (x * 0.08333333333333333))))));
	} else {
		tmp = t_1 + (x * (0.08333333333333333 + ((x * x) * -0.001388888888888889)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = (1.0 / x) + 0.5
	t_2 = (x * x) * (x * -0.001388888888888889)
	t_3 = x * (0.08333333333333333 + (x * (x * -0.001388888888888889)))
	tmp = 0
	if x <= -2.1e+52:
		tmp = (1.0 / (x * x)) / ((1.0 / x) + (0.5 - t_3))
	elif x <= -5e+36:
		tmp = ((t_1 * t_1) - (t_3 * t_3)) / ((1.0 / x) + (0.5 + (((t_0 * 0.0005787037037037037) + (t_0 * (t_0 * (t_0 * -2.6791838134430728e-9)))) / ((((x * 0.08333333333333333) * t_2) - (t_2 * t_2)) - ((x * 0.08333333333333333) * (x * 0.08333333333333333))))))
	else:
		tmp = t_1 + (x * (0.08333333333333333 + ((x * x) * -0.001388888888888889)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(1.0 / x) + 0.5)
	t_2 = Float64(Float64(x * x) * Float64(x * -0.001388888888888889))
	t_3 = Float64(x * Float64(0.08333333333333333 + Float64(x * Float64(x * -0.001388888888888889))))
	tmp = 0.0
	if (x <= -2.1e+52)
		tmp = Float64(Float64(1.0 / Float64(x * x)) / Float64(Float64(1.0 / x) + Float64(0.5 - t_3)));
	elseif (x <= -5e+36)
		tmp = Float64(Float64(Float64(t_1 * t_1) - Float64(t_3 * t_3)) / Float64(Float64(1.0 / x) + Float64(0.5 + Float64(Float64(Float64(t_0 * 0.0005787037037037037) + Float64(t_0 * Float64(t_0 * Float64(t_0 * -2.6791838134430728e-9)))) / Float64(Float64(Float64(Float64(x * 0.08333333333333333) * t_2) - Float64(t_2 * t_2)) - Float64(Float64(x * 0.08333333333333333) * Float64(x * 0.08333333333333333)))))));
	else
		tmp = Float64(t_1 + Float64(x * Float64(0.08333333333333333 + Float64(Float64(x * x) * -0.001388888888888889))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = (1.0 / x) + 0.5;
	t_2 = (x * x) * (x * -0.001388888888888889);
	t_3 = x * (0.08333333333333333 + (x * (x * -0.001388888888888889)));
	tmp = 0.0;
	if (x <= -2.1e+52)
		tmp = (1.0 / (x * x)) / ((1.0 / x) + (0.5 - t_3));
	elseif (x <= -5e+36)
		tmp = ((t_1 * t_1) - (t_3 * t_3)) / ((1.0 / x) + (0.5 + (((t_0 * 0.0005787037037037037) + (t_0 * (t_0 * (t_0 * -2.6791838134430728e-9)))) / ((((x * 0.08333333333333333) * t_2) - (t_2 * t_2)) - ((x * 0.08333333333333333) * (x * 0.08333333333333333))))));
	else
		tmp = t_1 + (x * (0.08333333333333333 + ((x * x) * -0.001388888888888889)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * N[(x * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[(0.08333333333333333 + N[(x * N[(x * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.1e+52], N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e+36], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(N[(N[(t$95$0 * 0.0005787037037037037), $MachinePrecision] + N[(t$95$0 * N[(t$95$0 * N[(t$95$0 * -2.6791838134430728e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(x * 0.08333333333333333), $MachinePrecision] * t$95$2), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 0.08333333333333333), $MachinePrecision] * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(x * N[(0.08333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1e52

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. lft-mult-inverse N/A

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

      12. *-lft-identity N/A

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

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

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

   7. Simplified 1.6%

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

      1. flip-+ N/A

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

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

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

   9. Applied egg-rr 0.3%

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

   10. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 95.4%

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

   12. Simplified 95.4%

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

   if -2.1e52 < x < -4.99999999999999977e36

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. lft-mult-inverse N/A

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

      12. *-lft-identity N/A

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

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

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

   7. Simplified 2.3%

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

      1. flip-+ N/A

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

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

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

   9. Applied egg-rr 2.3%

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

   11. Applied egg-rr 100.0%

       \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)\right) \cdot \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)\right)}{\frac{1}{x} + \left(0.5 - \color{blue}{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.0005787037037037037 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot -2.6791838134430728 \cdot 10^{-9}\right)\right)}{\left(x \cdot 0.08333333333333333\right) \cdot \left(x \cdot 0.08333333333333333\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot -0.001388888888888889\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.001388888888888889\right)\right) - \left(x \cdot 0.08333333333333333\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.001388888888888889\right)\right)\right)}}\right)} \]

   if -4.99999999999999977e36 < x

   1. Initial program 10.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. lft-mult-inverse N/A

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

      12. *-lft-identity N/A

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

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

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

   7. Simplified 95.1%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{1}{x \cdot x}}{\frac{1}{x} + \left(0.5 - x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)\right) \cdot \left(x \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)\right)}{\frac{1}{x} + \left(0.5 + \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.0005787037037037037 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot -2.6791838134430728 \cdot 10^{-9}\right)\right)}{\left(\left(x \cdot 0.08333333333333333\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.001388888888888889\right)\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.001388888888888889\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.001388888888888889\right)\right)\right) - \left(x \cdot 0.08333333333333333\right) \cdot \left(x \cdot 0.08333333333333333\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + 0.5\right) + x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% 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 39.6%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 98.2%

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

Alternative 4: 93.6% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -5.20000000000000018

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. lft-mult-inverse N/A

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

      12. *-lft-identity N/A

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

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

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

   7. Simplified 1.8%

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

      1. flip-+ N/A

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

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

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

   9. Applied egg-rr 0.6%

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

   10. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 82.4%

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

   12. Simplified 82.4%

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

   if -5.20000000000000018 < x

   1. Initial program 6.2%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. lft-mult-inverse N/A

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

      12. *-lft-identity N/A

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

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

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

   7. Simplified 99.0%

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

4. Final simplification 93.1%

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

Alternative 5: 84.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55:\\ \;\;\;\;\frac{-2 + \frac{-4}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(0.5 + x \cdot 0.08333333333333333\right) + 1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.55)
   (/ (+ -2.0 (/ -4.0 x)) (* x x))
   (/ (+ (* x (+ 0.5 (* x 0.08333333333333333))) 1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -2.55) {
		tmp = (-2.0 + (-4.0 / x)) / (x * x);
	} else {
		tmp = ((x * (0.5 + (x * 0.08333333333333333))) + 1.0) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.55d0)) then
        tmp = ((-2.0d0) + ((-4.0d0) / x)) / (x * x)
    else
        tmp = ((x * (0.5d0 + (x * 0.08333333333333333d0))) + 1.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.55) {
		tmp = (-2.0 + (-4.0 / x)) / (x * x);
	} else {
		tmp = ((x * (0.5 + (x * 0.08333333333333333))) + 1.0) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.55:
		tmp = (-2.0 + (-4.0 / x)) / (x * x)
	else:
		tmp = ((x * (0.5 + (x * 0.08333333333333333))) + 1.0) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.55)
		tmp = Float64(Float64(-2.0 + Float64(-4.0 / x)) / Float64(x * x));
	else
		tmp = Float64(Float64(Float64(x * Float64(0.5 + Float64(x * 0.08333333333333333))) + 1.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.55)
		tmp = (-2.0 + (-4.0 / x)) / (x * x);
	else
		tmp = ((x * (0.5 + (x * 0.08333333333333333))) + 1.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.55], N[(N[(-2.0 + N[(-4.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5499999999999998

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. rgt-mult-inverse N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

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

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

      9. /-lowering-/.f64 3.1%

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

   7. Simplified 3.1%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. sub-neg N/A

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

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

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

      13. sub-neg N/A

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

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

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

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

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

      16. metadata-eval 3.1%

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

   9. Applied egg-rr 3.1%

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

   10. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 51.2%

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

   12. Simplified 51.2%

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

   13. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{{x}^{2}}} \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(2 + 4 \cdot \frac{1}{x}\right)\right)}{{\color{blue}{x}}^{2}} \]

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

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

       4. distribute-neg-in N/A

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

       5. metadata-eval N/A

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-frac N/A

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

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(4\right)\right), x\right)\right), \left({x}^{2}\right)\right) \]

       11. metadata-eval N/A

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 51.2%

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

   15. Simplified 51.2%

       \[\leadsto \color{blue}{\frac{-2 + \frac{-4}{x}}{x \cdot x}} \]

   if -2.5499999999999998 < x

   1. Initial program 6.2%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. div-inv N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. *-commutative N/A

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

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

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

      6. clear-num N/A

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

      7. flip3-- N/A

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

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

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

      9. expm1-define N/A

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

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

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

      11. exp-lowering-exp.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 98.9%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55:\\ \;\;\;\;\frac{-2 + \frac{-4}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(0.5 + x \cdot 0.08333333333333333\right) + 1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55:\\ \;\;\;\;\frac{-2 + \frac{-4}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.55)
   (/ (+ -2.0 (/ -4.0 x)) (* x x))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -2.55) {
		tmp = (-2.0 + (-4.0 / 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 <= (-2.55d0)) then
        tmp = ((-2.0d0) + ((-4.0d0) / 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 <= -2.55) {
		tmp = (-2.0 + (-4.0 / x)) / (x * x);
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.55:
		tmp = (-2.0 + (-4.0 / x)) / (x * x)
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.55)
		tmp = Float64(Float64(-2.0 + Float64(-4.0 / 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 <= -2.55)
		tmp = (-2.0 + (-4.0 / x)) / (x * x);
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.55], N[(N[(-2.0 + N[(-4.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5499999999999998

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. rgt-mult-inverse N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

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

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

      9. /-lowering-/.f64 3.1%

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

   7. Simplified 3.1%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. sub-neg N/A

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

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

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

      13. sub-neg N/A

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

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

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

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

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

      16. metadata-eval 3.1%

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

   9. Applied egg-rr 3.1%

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

   10. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 51.2%

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

   12. Simplified 51.2%

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

   13. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{{x}^{2}}} \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(2 + 4 \cdot \frac{1}{x}\right)\right)}{{\color{blue}{x}}^{2}} \]

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

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

       4. distribute-neg-in N/A

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

       5. metadata-eval N/A

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-frac N/A

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

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(4\right)\right), x\right)\right), \left({x}^{2}\right)\right) \]

       11. metadata-eval N/A

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 51.2%

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

   15. Simplified 51.2%

       \[\leadsto \color{blue}{\frac{-2 + \frac{-4}{x}}{x \cdot x}} \]

   if -2.5499999999999998 < x

   1. Initial program 6.2%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 98.9%

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

   7. Simplified 98.9%

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

Alternative 7: 84.2% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.5)
   (/ -2.0 (* x x))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.5) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.5d0)) then
        tmp = (-2.0d0) / (x * x)
    else
        tmp = (1.0d0 / x) + (0.5d0 + (x * 0.08333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.5) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.5:
		tmp = -2.0 / (x * x)
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.5)
		tmp = Float64(-2.0 / Float64(x * x));
	else
		tmp = Float64(Float64(1.0 / x) + Float64(0.5 + Float64(x * 0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.5)
		tmp = -2.0 / (x * x);
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.5], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. rgt-mult-inverse N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

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

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

      9. /-lowering-/.f64 3.1%

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

   7. Simplified 3.1%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. sub-neg N/A

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

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

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

      13. sub-neg N/A

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

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

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

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

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

      16. metadata-eval 3.1%

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

   9. Applied egg-rr 3.1%

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

   10. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 51.2%

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

   12. Simplified 51.2%

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

   13. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]
   14. 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 51.2%

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

   15. Simplified 51.2%

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

   if -4.5 < x

   1. Initial program 6.2%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 98.9%

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

   7. Simplified 98.9%

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

Alternative 8: 83.2% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
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 64.2%

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

7. Simplified 64.2%

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

   1. flip-+ N/A

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

   2. div-inv N/A

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

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

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

   4. sub-neg N/A

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

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

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

   6. frac-times N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

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

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

   13. sub-neg N/A

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

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

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

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

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

   16. metadata-eval 34.6%

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

9. Applied egg-rr 34.6%

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

10. Taylor expanded in x around 0

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

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

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

    2. unpow2 N/A

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

    3. *-lowering-*.f64 51.6%

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

12. Simplified 51.6%

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

    1. associate-*l/ N/A

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

    2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

    8. /-lowering-/.f64 80.6%

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

14. Applied egg-rr 80.6%

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

15. Final simplification 80.6%

    \[\leadsto \frac{1}{x} \cdot \frac{\frac{-1}{\frac{-1}{x} - -0.5}}{x} \]
16. Add Preprocessing

Alternative 9: 83.2% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
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 64.2%

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

7. Simplified 64.2%

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

   1. flip-+ N/A

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

   2. div-inv N/A

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

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

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

   4. sub-neg N/A

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

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

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

   6. frac-times N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

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

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

   13. sub-neg N/A

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

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

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

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

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

   16. metadata-eval 34.6%

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

9. Applied egg-rr 34.6%

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

10. Taylor expanded in x around 0

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

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

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

    2. unpow2 N/A

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

    3. *-lowering-*.f64 51.6%

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

12. Simplified 51.6%

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

    1. associate-*l/ N/A

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

    2. div-inv N/A

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

    3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

    8. /-lowering-/.f64 80.6%

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

14. Applied egg-rr 80.6%

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

15. Final simplification 80.6%

    \[\leadsto \frac{\frac{\frac{-1}{\frac{-1}{x} - -0.5}}{x}}{x} \]
16. Add Preprocessing

Alternative 10: 83.9% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.78:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.78) (/ -2.0 (* x x)) (+ (/ 1.0 x) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= -1.78) {
		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 <= (-1.78d0)) 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 <= -1.78) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.78:
		tmp = -2.0 / (x * x)
	else:
		tmp = (1.0 / x) + 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.78)
		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 <= -1.78)
		tmp = -2.0 / (x * x);
	else
		tmp = (1.0 / x) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.78], 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 < -1.78000000000000003

   1. Initial program 100.0%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. rgt-mult-inverse N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

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

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

      9. /-lowering-/.f64 3.1%

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

   7. Simplified 3.1%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. sub-neg N/A

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

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

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

      13. sub-neg N/A

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

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

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

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

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

      16. metadata-eval 3.1%

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

   9. Applied egg-rr 3.1%

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

   10. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 51.2%

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

   12. Simplified 51.2%

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

   13. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]
   14. 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 51.2%

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

   15. Simplified 51.2%

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

   if -1.78000000000000003 < x

   1. Initial program 6.2%

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. rgt-mult-inverse N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

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

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

      9. /-lowering-/.f64 97.9%

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

   7. Simplified 97.9%

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

Alternative 11: 66.5% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 64.5%

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

7. Simplified 64.5%

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

Alternative 12: 3.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 39.6%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 98.2%

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

8. Taylor expanded in x around 0

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

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

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

   2. +-lowering-+.f64 63.9%

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

10. Simplified 63.9%

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

11. Taylor expanded in x around inf

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

13. Simplified 3.6%

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

Alternative 13: 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 39.6%

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

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

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

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

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

   3. expm1-define N/A

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

   4. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
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 64.2%

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

7. Simplified 64.2%

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

8. Taylor expanded in x around inf

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

10. Simplified 3.3%

    \[\leadsto \color{blue}{0.5} \]
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 2024154 
(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)))