Kahan's exp quotient

Percentage Accurate: 53.2% → 100.0%

Time: 14.0s

Alternatives: 14

Speedup: 10.5×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

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

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

   2. accelerator-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 75.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 + x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - x \cdot \left(t\_0 \cdot t\_1\right)\right)}{1 + t\_1}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+ -0.5 (* x (+ (* x -0.041666666666666664) -0.16666666666666666))))
        (t_1 (* x t_0)))
   (if (<= x -1.55)
     (/ (/ x (+ 1.0 (* x -0.5))) x)
     (if (<= x 1.65e+103)
       (/ (/ (* x (- 1.0 (* x (* t_0 t_1)))) (+ 1.0 t_1)) x)
       (* x (* 0.041666666666666664 (* x x)))))))

C

double code(double x) {
	double t_0 = -0.5 + (x * ((x * -0.041666666666666664) + -0.16666666666666666));
	double t_1 = x * t_0;
	double tmp;
	if (x <= -1.55) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else if (x <= 1.65e+103) {
		tmp = ((x * (1.0 - (x * (t_0 * t_1)))) / (1.0 + t_1)) / x;
	} else {
		tmp = x * (0.041666666666666664 * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) + (x * ((x * (-0.041666666666666664d0)) + (-0.16666666666666666d0)))
    t_1 = x * t_0
    if (x <= (-1.55d0)) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    else if (x <= 1.65d+103) then
        tmp = ((x * (1.0d0 - (x * (t_0 * t_1)))) / (1.0d0 + t_1)) / x
    else
        tmp = x * (0.041666666666666664d0 * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -0.5 + (x * ((x * -0.041666666666666664) + -0.16666666666666666));
	double t_1 = x * t_0;
	double tmp;
	if (x <= -1.55) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else if (x <= 1.65e+103) {
		tmp = ((x * (1.0 - (x * (t_0 * t_1)))) / (1.0 + t_1)) / x;
	} else {
		tmp = x * (0.041666666666666664 * (x * x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = -0.5 + (x * ((x * -0.041666666666666664) + -0.16666666666666666))
	t_1 = x * t_0
	tmp = 0
	if x <= -1.55:
		tmp = (x / (1.0 + (x * -0.5))) / x
	elif x <= 1.65e+103:
		tmp = ((x * (1.0 - (x * (t_0 * t_1)))) / (1.0 + t_1)) / x
	else:
		tmp = x * (0.041666666666666664 * (x * x))
	return tmp

Julia

function code(x)
	t_0 = Float64(-0.5 + Float64(x * Float64(Float64(x * -0.041666666666666664) + -0.16666666666666666)))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	elseif (x <= 1.65e+103)
		tmp = Float64(Float64(Float64(x * Float64(1.0 - Float64(x * Float64(t_0 * t_1)))) / Float64(1.0 + t_1)) / x);
	else
		tmp = Float64(x * Float64(0.041666666666666664 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -0.5 + (x * ((x * -0.041666666666666664) + -0.16666666666666666));
	t_1 = x * t_0;
	tmp = 0.0;
	if (x <= -1.55)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	elseif (x <= 1.65e+103)
		tmp = ((x * (1.0 - (x * (t_0 * t_1)))) / (1.0 + t_1)) / x;
	else
		tmp = x * (0.041666666666666664 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 + N[(x * N[(N[(x * -0.041666666666666664), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[x, -1.55], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.65e+103], N[(N[(N[(x * N[(1.0 - N[(x * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 100.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. cancel-sign-sub N/A

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

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

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

      4. *-rgt-identity N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 1.2%

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

      1. flip3-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 1.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 18.8%

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

   12. Simplified 18.8%

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

   if -1.55000000000000004 < x < 1.65000000000000004e103

   1. Initial program 22.3%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. cancel-sign-sub N/A

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

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

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

      4. *-rgt-identity N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 85.8%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 95.6%

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

   if 1.65000000000000004e103 < x

   1. Initial program 100.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. cancel-sign-sub N/A

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

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

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

      4. *-rgt-identity N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3}} \]
   9. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - x \cdot \left(\left(-0.5 + x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right) \cdot \left(x \cdot \left(-0.5 + x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right)\right)\right)}{1 + x \cdot \left(-0.5 + x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \frac{t\_0}{-24}\\ \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 + 7.233796296296296 \cdot 10^{-5} \cdot \left(t\_0 \cdot \left(t\_0 \cdot t\_0\right)\right)}{1 + t\_1 \cdot \left(1 + t\_1\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot \left(7.233796296296296 \cdot 10^{-5} \cdot t\_0 + 0.004629629629629629\right)}{0.027777777777777776 + \left(x \cdot 0.041666666666666664\right) \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (/ t_0 -24.0)))
   (if (<= x 1.65)
     (/ (/ x (+ 1.0 (* x -0.5))) x)
     (if (<= x 7e+51)
       (/
        (+ 1.0 (* 7.233796296296296e-5 (* t_0 (* t_0 t_0))))
        (+ 1.0 (* t_1 (+ 1.0 t_1))))
       (if (<= x 2e+154)
         (/
          (/
           (* t_0 (+ (* 7.233796296296296e-5 t_0) 0.004629629629629629))
           (+
            0.027777777777777776
            (*
             (* x 0.041666666666666664)
             (- (* x 0.041666666666666664) 0.16666666666666666))))
          x)
         (* x (* x 0.16666666666666666)))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = t_0 / -24.0;
	double tmp;
	if (x <= 1.65) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else if (x <= 7e+51) {
		tmp = (1.0 + (7.233796296296296e-5 * (t_0 * (t_0 * t_0)))) / (1.0 + (t_1 * (1.0 + t_1)));
	} else if (x <= 2e+154) {
		tmp = ((t_0 * ((7.233796296296296e-5 * t_0) + 0.004629629629629629)) / (0.027777777777777776 + ((x * 0.041666666666666664) * ((x * 0.041666666666666664) - 0.16666666666666666)))) / x;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = t_0 / (-24.0d0)
    if (x <= 1.65d0) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    else if (x <= 7d+51) then
        tmp = (1.0d0 + (7.233796296296296d-5 * (t_0 * (t_0 * t_0)))) / (1.0d0 + (t_1 * (1.0d0 + t_1)))
    else if (x <= 2d+154) then
        tmp = ((t_0 * ((7.233796296296296d-5 * t_0) + 0.004629629629629629d0)) / (0.027777777777777776d0 + ((x * 0.041666666666666664d0) * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))) / x
    else
        tmp = x * (x * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = t_0 / -24.0;
	double tmp;
	if (x <= 1.65) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else if (x <= 7e+51) {
		tmp = (1.0 + (7.233796296296296e-5 * (t_0 * (t_0 * t_0)))) / (1.0 + (t_1 * (1.0 + t_1)));
	} else if (x <= 2e+154) {
		tmp = ((t_0 * ((7.233796296296296e-5 * t_0) + 0.004629629629629629)) / (0.027777777777777776 + ((x * 0.041666666666666664) * ((x * 0.041666666666666664) - 0.16666666666666666)))) / x;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = t_0 / -24.0
	tmp = 0
	if x <= 1.65:
		tmp = (x / (1.0 + (x * -0.5))) / x
	elif x <= 7e+51:
		tmp = (1.0 + (7.233796296296296e-5 * (t_0 * (t_0 * t_0)))) / (1.0 + (t_1 * (1.0 + t_1)))
	elif x <= 2e+154:
		tmp = ((t_0 * ((7.233796296296296e-5 * t_0) + 0.004629629629629629)) / (0.027777777777777776 + ((x * 0.041666666666666664) * ((x * 0.041666666666666664) - 0.16666666666666666)))) / x
	else:
		tmp = x * (x * 0.16666666666666666)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(t_0 / -24.0)
	tmp = 0.0
	if (x <= 1.65)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	elseif (x <= 7e+51)
		tmp = Float64(Float64(1.0 + Float64(7.233796296296296e-5 * Float64(t_0 * Float64(t_0 * t_0)))) / Float64(1.0 + Float64(t_1 * Float64(1.0 + t_1))));
	elseif (x <= 2e+154)
		tmp = Float64(Float64(Float64(t_0 * Float64(Float64(7.233796296296296e-5 * t_0) + 0.004629629629629629)) / Float64(0.027777777777777776 + Float64(Float64(x * 0.041666666666666664) * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))) / x);
	else
		tmp = Float64(x * Float64(x * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = t_0 / -24.0;
	tmp = 0.0;
	if (x <= 1.65)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	elseif (x <= 7e+51)
		tmp = (1.0 + (7.233796296296296e-5 * (t_0 * (t_0 * t_0)))) / (1.0 + (t_1 * (1.0 + t_1)));
	elseif (x <= 2e+154)
		tmp = ((t_0 * ((7.233796296296296e-5 * t_0) + 0.004629629629629629)) / (0.027777777777777776 + ((x * 0.041666666666666664) * ((x * 0.041666666666666664) - 0.16666666666666666)))) / x;
	else
		tmp = x * (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / -24.0), $MachinePrecision]}, If[LessEqual[x, 1.65], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7e+51], N[(N[(1.0 + N[(7.233796296296296e-5 * N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+154], N[(N[(N[(t$95$0 * N[(N[(7.233796296296296e-5 * t$95$0), $MachinePrecision] + 0.004629629629629629), $MachinePrecision]), $MachinePrecision] / N[(0.027777777777777776 + N[(N[(x * 0.041666666666666664), $MachinePrecision] * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.6499999999999999

   1. Initial program 37.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. cancel-sign-sub N/A

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

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

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

      4. *-rgt-identity N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 66.2%

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

      1. flip3-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 66.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 72.2%

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

   12. Simplified 72.2%

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

   if 1.6499999999999999 < x < 7e51

   1. Initial program 100.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. cancel-sign-sub N/A

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

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

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

      3. *-rgt-identity N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. distribute-lft-neg-out N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

   7. Simplified 4.1%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

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

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

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

      7. sub-neg N/A

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

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

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

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

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

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

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

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

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

      12. metadata-eval N/A

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

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

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

   9. Applied egg-rr 4.1%

      \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + 0.5}{\left(x \cdot \left(\left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right) + -0.25\right) \cdot x}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-24}{{x}^{3}}\right)}\right)\right) \]
   11. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 4.1%

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

   12. Simplified 4.1%

       \[\leadsto 1 - \frac{1}{\color{blue}{\frac{-24}{x \cdot \left(x \cdot x\right)}}} \]
   13. Step-by-step derivation

       1. flip3-- N/A

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

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

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

   14. Applied egg-rr 43.8%

       \[\leadsto \color{blue}{\frac{1 + 7.233796296296296 \cdot 10^{-5} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{1 + \frac{x \cdot \left(x \cdot x\right)}{-24} \cdot \left(1 + \frac{x \cdot \left(x \cdot x\right)}{-24}\right)}} \]

   if 7e51 < x < 2.00000000000000007e154

   1. Initial program 100.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. cancel-sign-sub N/A

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

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

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

      4. *-rgt-identity N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 56.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({x}^{4} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right) \cdot {x}^{4}\right), x\right) \]

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. *-lowering-*.f64 56.6%

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

   10. Simplified 56.6%

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

       1. *-commutative N/A

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

       2. flip3-+ N/A

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

       3. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 100.0%

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

   if 2.00000000000000007e154 < x

   1. Initial program 100.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. cancel-sign-sub N/A

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

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

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

      3. *-rgt-identity N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.16666666666666666\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 + 7.233796296296296 \cdot 10^{-5} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{1 + \frac{x \cdot \left(x \cdot x\right)}{-24} \cdot \left(1 + \frac{x \cdot \left(x \cdot x\right)}{-24}\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(7.233796296296296 \cdot 10^{-5} \cdot \left(x \cdot \left(x \cdot x\right)\right) + 0.004629629629629629\right)}{0.027777777777777776 + \left(x \cdot 0.041666666666666664\right) \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot \left(7.233796296296296 \cdot 10^{-5} \cdot t\_0 + 0.004629629629629629\right)}{0.027777777777777776 + \left(x \cdot 0.041666666666666664\right) \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x 1.8)
     (/ (/ x (+ 1.0 (* x -0.5))) x)
     (if (<= x 2e+154)
       (/
        (/
         (* t_0 (+ (* 7.233796296296296e-5 t_0) 0.004629629629629629))
         (+
          0.027777777777777776
          (*
           (* x 0.041666666666666664)
           (- (* x 0.041666666666666664) 0.16666666666666666))))
        x)
       (* x (* x 0.16666666666666666))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 1.8) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else if (x <= 2e+154) {
		tmp = ((t_0 * ((7.233796296296296e-5 * t_0) + 0.004629629629629629)) / (0.027777777777777776 + ((x * 0.041666666666666664) * ((x * 0.041666666666666664) - 0.16666666666666666)))) / x;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= 1.8d0) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    else if (x <= 2d+154) then
        tmp = ((t_0 * ((7.233796296296296d-5 * t_0) + 0.004629629629629629d0)) / (0.027777777777777776d0 + ((x * 0.041666666666666664d0) * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))) / x
    else
        tmp = x * (x * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 1.8) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else if (x <= 2e+154) {
		tmp = ((t_0 * ((7.233796296296296e-5 * t_0) + 0.004629629629629629)) / (0.027777777777777776 + ((x * 0.041666666666666664) * ((x * 0.041666666666666664) - 0.16666666666666666)))) / x;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	tmp = 0
	if x <= 1.8:
		tmp = (x / (1.0 + (x * -0.5))) / x
	elif x <= 2e+154:
		tmp = ((t_0 * ((7.233796296296296e-5 * t_0) + 0.004629629629629629)) / (0.027777777777777776 + ((x * 0.041666666666666664) * ((x * 0.041666666666666664) - 0.16666666666666666)))) / x
	else:
		tmp = x * (x * 0.16666666666666666)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	elseif (x <= 2e+154)
		tmp = Float64(Float64(Float64(t_0 * Float64(Float64(7.233796296296296e-5 * t_0) + 0.004629629629629629)) / Float64(0.027777777777777776 + Float64(Float64(x * 0.041666666666666664) * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))) / x);
	else
		tmp = Float64(x * Float64(x * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= 1.8)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	elseif (x <= 2e+154)
		tmp = ((t_0 * ((7.233796296296296e-5 * t_0) + 0.004629629629629629)) / (0.027777777777777776 + ((x * 0.041666666666666664) * ((x * 0.041666666666666664) - 0.16666666666666666)))) / x;
	else
		tmp = x * (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.8], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2e+154], N[(N[(N[(t$95$0 * N[(N[(7.233796296296296e-5 * t$95$0), $MachinePrecision] + 0.004629629629629629), $MachinePrecision]), $MachinePrecision] / N[(0.027777777777777776 + N[(N[(x * 0.041666666666666664), $MachinePrecision] * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.80000000000000004

   1. Initial program 37.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. cancel-sign-sub N/A

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

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

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

      4. *-rgt-identity N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 66.2%

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

      1. flip3-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 66.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 72.2%

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

   12. Simplified 72.2%

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

   if 1.80000000000000004 < x < 2.00000000000000007e154

   1. Initial program 100.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. cancel-sign-sub N/A

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

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

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

      4. *-rgt-identity N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 39.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({x}^{4} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right) \cdot {x}^{4}\right), x\right) \]

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. *-lowering-*.f64 39.1%

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

   10. Simplified 39.1%

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

       1. *-commutative N/A

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

       2. flip3-+ N/A

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

       3. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 68.0%

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

   if 2.00000000000000007e154 < x

   1. Initial program 100.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. cancel-sign-sub N/A

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

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

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

      3. *-rgt-identity N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(7.233796296296296 \cdot 10^{-5} \cdot \left(x \cdot \left(x \cdot x\right)\right) + 0.004629629629629629\right)}{0.027777777777777776 + \left(x \cdot 0.041666666666666664\right) \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 - \left(t\_0 \cdot t\_0\right) \cdot 0.001736111111111111}{1 - 0.041666666666666664 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x 1.65)
     (/ (/ x (+ 1.0 (* x -0.5))) x)
     (if (<= x 5.5e+102)
       (/
        (- 1.0 (* (* t_0 t_0) 0.001736111111111111))
        (- 1.0 (* 0.041666666666666664 t_0)))
       (* x (* 0.041666666666666664 (* x x)))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 1.65) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else if (x <= 5.5e+102) {
		tmp = (1.0 - ((t_0 * t_0) * 0.001736111111111111)) / (1.0 - (0.041666666666666664 * t_0));
	} else {
		tmp = x * (0.041666666666666664 * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= 1.65d0) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    else if (x <= 5.5d+102) then
        tmp = (1.0d0 - ((t_0 * t_0) * 0.001736111111111111d0)) / (1.0d0 - (0.041666666666666664d0 * t_0))
    else
        tmp = x * (0.041666666666666664d0 * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 1.65) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else if (x <= 5.5e+102) {
		tmp = (1.0 - ((t_0 * t_0) * 0.001736111111111111)) / (1.0 - (0.041666666666666664 * t_0));
	} else {
		tmp = x * (0.041666666666666664 * (x * x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	tmp = 0
	if x <= 1.65:
		tmp = (x / (1.0 + (x * -0.5))) / x
	elif x <= 5.5e+102:
		tmp = (1.0 - ((t_0 * t_0) * 0.001736111111111111)) / (1.0 - (0.041666666666666664 * t_0))
	else:
		tmp = x * (0.041666666666666664 * (x * x))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= 1.65)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	elseif (x <= 5.5e+102)
		tmp = Float64(Float64(1.0 - Float64(Float64(t_0 * t_0) * 0.001736111111111111)) / Float64(1.0 - Float64(0.041666666666666664 * t_0)));
	else
		tmp = Float64(x * Float64(0.041666666666666664 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= 1.65)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	elseif (x <= 5.5e+102)
		tmp = (1.0 - ((t_0 * t_0) * 0.001736111111111111)) / (1.0 - (0.041666666666666664 * t_0));
	else
		tmp = x * (0.041666666666666664 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.65], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.5e+102], N[(N[(1.0 - N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.001736111111111111), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(0.041666666666666664 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6499999999999999

   1. Initial program 37.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. cancel-sign-sub N/A

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

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

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

      4. *-rgt-identity N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 66.2%

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

      1. flip3-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 66.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 72.2%

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

   12. Simplified 72.2%

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

   if 1.6499999999999999 < x < 5.49999999999999981e102

   1. Initial program 100.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. cancel-sign-sub N/A

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

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

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

      3. *-rgt-identity N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. distribute-lft-neg-out N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

   7. Simplified 5.2%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

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

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

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

      7. sub-neg N/A

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

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

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

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

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

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

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

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

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

      12. metadata-eval N/A

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

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

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

   9. Applied egg-rr 43.9%

      \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + 0.5}{\left(x \cdot \left(\left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right) + -0.25\right) \cdot x}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-24}{{x}^{3}}\right)}\right)\right) \]
   11. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 5.2%

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

   12. Simplified 5.2%

       \[\leadsto 1 - \frac{1}{\color{blue}{\frac{-24}{x \cdot \left(x \cdot x\right)}}} \]
   13. Step-by-step derivation

       1. sub-neg N/A

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

       2. flip-+ N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\mathsf{neg}\left(\frac{1}{\frac{-24}{x \cdot \left(x \cdot x\right)}}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{\frac{-24}{x \cdot \left(x \cdot x\right)}}\right)\right)}{\color{blue}{1 - \left(\mathsf{neg}\left(\frac{1}{\frac{-24}{x \cdot \left(x \cdot x\right)}}\right)\right)}} \]

       3. sqr-neg N/A

          \[\leadsto \frac{1 \cdot 1 - \frac{1}{\frac{-24}{x \cdot \left(x \cdot x\right)}} \cdot \frac{1}{\frac{-24}{x \cdot \left(x \cdot x\right)}}}{1 - \left(\mathsf{neg}\left(\frac{1}{\frac{-24}{x \cdot \left(x \cdot x\right)}}\right)\right)} \]

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

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

   14. Applied egg-rr 60.3%

       \[\leadsto \color{blue}{\frac{1 - \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.001736111111111111}{1 - 0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]

   if 5.49999999999999981e102 < x

   1. Initial program 100.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. cancel-sign-sub N/A

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

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

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

      4. *-rgt-identity N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3}} \]
   9. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\frac{x \cdot -0.16666666666666666 + 0.5}{x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001736111111111111\right)\right) + -0.25\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.2)
   (/ (/ x (+ 1.0 (* x -0.5))) x)
   (+
    1.0
    (/
     -1.0
     (/
      (+ (* x -0.16666666666666666) 0.5)
      (* x (+ (* x (* x (* (* x x) 0.001736111111111111))) -0.25)))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.2) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = 1.0 + (-1.0 / (((x * -0.16666666666666666) + 0.5) / (x * ((x * (x * ((x * x) * 0.001736111111111111))) + -0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.2d0)) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    else
        tmp = 1.0d0 + ((-1.0d0) / (((x * (-0.16666666666666666d0)) + 0.5d0) / (x * ((x * (x * ((x * x) * 0.001736111111111111d0))) + (-0.25d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.2) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = 1.0 + (-1.0 / (((x * -0.16666666666666666) + 0.5) / (x * ((x * (x * ((x * x) * 0.001736111111111111))) + -0.25))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.2:
		tmp = (x / (1.0 + (x * -0.5))) / x
	else:
		tmp = 1.0 + (-1.0 / (((x * -0.16666666666666666) + 0.5) / (x * ((x * (x * ((x * x) * 0.001736111111111111))) + -0.25))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.2)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(Float64(Float64(x * -0.16666666666666666) + 0.5) / Float64(x * Float64(Float64(x * Float64(x * Float64(Float64(x * x) * 0.001736111111111111))) + -0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.2)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	else
		tmp = 1.0 + (-1.0 / (((x * -0.16666666666666666) + 0.5) / (x * ((x * (x * ((x * x) * 0.001736111111111111))) + -0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.2], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(N[(N[(x * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] / N[(x * N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001736111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000002

   1. Initial program 100.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. cancel-sign-sub N/A

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

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

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

      4. *-rgt-identity N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 1.2%

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

      1. flip3-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 1.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 18.8%

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

   12. Simplified 18.8%

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

   if -2.2000000000000002 < x

   1. Initial program 37.3%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. cancel-sign-sub N/A

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

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

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

      3. *-rgt-identity N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. distribute-lft-neg-out N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

   7. Simplified 85.6%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

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

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

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

      7. sub-neg N/A

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

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

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

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

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

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

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

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

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

      12. metadata-eval N/A

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

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

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

   9. Applied egg-rr 75.8%

      \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + 0.5}{\left(x \cdot \left(\left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right) + -0.25\right) \cdot x}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{6} \cdot x\right)}, \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)}, \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       2. *-lowering-*.f64 91.5%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)}, \frac{-1}{4}\right), x\right)\right)\right)\right) \]

   12. Simplified 91.5%

       \[\leadsto 1 - \frac{1}{\frac{\color{blue}{x \cdot -0.16666666666666666} + 0.5}{\left(x \cdot \left(\left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right) + -0.25\right) \cdot x}} \]

   13. Taylor expanded in x around inf

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{576} \cdot {x}^{3}\right)}\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{576} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{576} \cdot \left(x \cdot {x}^{2}\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{576} \cdot x\right) \cdot {x}^{2}\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{576} \cdot x\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{576} \cdot x\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(\frac{1}{576} \cdot x\right)\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{576} \cdot x\right)\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{576}\right)\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{576}\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{576}\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{576}\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{576}\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       13. *-lowering-*.f64 91.5%

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]

   15. Simplified 91.5%

       \[\leadsto 1 - \frac{1}{\frac{x \cdot -0.16666666666666666 + 0.5}{\left(x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001736111111111111\right)\right)} + -0.25\right) \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\frac{x \cdot -0.16666666666666666 + 0.5}{x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001736111111111111\right)\right) + -0.25\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.027777777777777776 - \left(x \cdot x\right) \cdot 0.001736111111111111\right)}{x \cdot -0.041666666666666664 + 0.16666666666666666}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8)
   (/ (/ x (+ 1.0 (* x -0.5))) x)
   (/
    (/
     (*
      (* x (* x x))
      (- 0.027777777777777776 (* (* x x) 0.001736111111111111)))
     (+ (* x -0.041666666666666664) 0.16666666666666666))
    x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = (((x * (x * x)) * (0.027777777777777776 - ((x * x) * 0.001736111111111111))) / ((x * -0.041666666666666664) + 0.16666666666666666)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.8d0) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    else
        tmp = (((x * (x * x)) * (0.027777777777777776d0 - ((x * x) * 0.001736111111111111d0))) / ((x * (-0.041666666666666664d0)) + 0.16666666666666666d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = (((x * (x * x)) * (0.027777777777777776 - ((x * x) * 0.001736111111111111))) / ((x * -0.041666666666666664) + 0.16666666666666666)) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = (x / (1.0 + (x * -0.5))) / x
	else:
		tmp = (((x * (x * x)) * (0.027777777777777776 - ((x * x) * 0.001736111111111111))) / ((x * -0.041666666666666664) + 0.16666666666666666)) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	else
		tmp = Float64(Float64(Float64(Float64(x * Float64(x * x)) * Float64(0.027777777777777776 - Float64(Float64(x * x) * 0.001736111111111111))) / Float64(Float64(x * -0.041666666666666664) + 0.16666666666666666)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	else
		tmp = (((x * (x * x)) * (0.027777777777777776 - ((x * x) * 0.001736111111111111))) / ((x * -0.041666666666666664) + 0.16666666666666666)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.027777777777777776 - N[(N[(x * x), $MachinePrecision] * 0.001736111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * -0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 37.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. cancel-sign-sub N/A

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

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

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

      4. *-rgt-identity N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 66.2%

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

      1. flip3-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 66.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 72.2%

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

   12. Simplified 72.2%

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

   if 1.80000000000000004 < x

   1. Initial program 100.0%

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

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

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

      2. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. cancel-sign-sub N/A

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

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

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

      4. *-rgt-identity N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 66.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({x}^{4} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right) \cdot {x}^{4}\right), x\right) \]

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \frac{1}{24}\right)\right)\right), x\right) \]

      21. *-lowering-*.f64 66.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), x\right) \]

   10. Simplified 66.8%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)}}{x} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{6} + x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), x\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right)}{\frac{1}{6} - x \cdot \frac{1}{24}} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), x\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right)}{\frac{1}{6} + \left(\mathsf{neg}\left(x \cdot \frac{1}{24}\right)\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), x\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right)}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) + \left(\mathsf{neg}\left(x \cdot \frac{1}{24}\right)\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), x\right) \]

       5. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right)}{\mathsf{neg}\left(\left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), x\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right)}{\mathsf{neg}\left(\left(\frac{-1}{6} + \frac{1}{24} \cdot x\right)\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), x\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right)}{\mathsf{neg}\left(\left(\frac{-1}{6} + \left(\mathsf{neg}\left(\frac{-1}{24}\right)\right) \cdot x\right)\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), x\right) \]

       8. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right)}{\mathsf{neg}\left(\left(\frac{-1}{6} - \frac{-1}{24} \cdot x\right)\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), x\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right)}{\mathsf{neg}\left(\left(\frac{-1}{6} - x \cdot \frac{-1}{24}\right)\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), x\right) \]

       10. associate-*l/ N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\mathsf{neg}\left(\left(\frac{-1}{6} - x \cdot \frac{-1}{24}\right)\right)}\right), x\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{-1}{6} - x \cdot \frac{-1}{24}\right)\right)\right)\right), x\right) \]

   12. Applied egg-rr 75.4%

       \[\leadsto \frac{\color{blue}{\frac{\left(0.027777777777777776 - \left(x \cdot x\right) \cdot 0.001736111111111111\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{0.16666666666666666 + x \cdot -0.041666666666666664}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.027777777777777776 - \left(x \cdot x\right) \cdot 0.001736111111111111\right)}{x \cdot -0.041666666666666664 + 0.16666666666666666}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.010416666666666666 - \frac{-0.11458333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8)
   (/ (/ x (+ 1.0 (* x -0.5))) x)
   (*
    (* (* x x) (* x x))
    (- 0.010416666666666666 (/ -0.11458333333333333 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = ((x * x) * (x * x)) * (0.010416666666666666 - (-0.11458333333333333 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.8d0) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    else
        tmp = ((x * x) * (x * x)) * (0.010416666666666666d0 - ((-0.11458333333333333d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = ((x * x) * (x * x)) * (0.010416666666666666 - (-0.11458333333333333 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = (x / (1.0 + (x * -0.5))) / x
	else:
		tmp = ((x * x) * (x * x)) * (0.010416666666666666 - (-0.11458333333333333 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(0.010416666666666666 - Float64(-0.11458333333333333 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	else
		tmp = ((x * x) * (x * x)) * (0.010416666666666666 - (-0.11458333333333333 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.010416666666666666 - N[(-0.11458333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 37.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

      2. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right), x\right) \]

      2. cancel-sign-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right), x\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(\mathsf{neg}\left(x \cdot 1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(x \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(x \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), x\right) \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot -1\right)\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), x\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right), \left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right), x\right) \]

   7. Simplified 66.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 - x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + -0.5\right)\right)}}{x} \]
   8. Step-by-step derivation

      1. flip3-- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{{1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)\right)}\right), x\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)\right)}{{1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)}^{3}}}\right), x\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)\right)}{{1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)}^{3}}}\right), x\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)\right)}{{1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)}^{3}}\right)\right), x\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{\frac{{1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)\right)}}\right)\right), x\right) \]

   9. Applied egg-rr 66.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{1}{1 - x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + -0.5\right)}}}}{x} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right), x\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot x\right)\right)\right), x\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{-1}{2}\right)\right)\right), x\right) \]

       3. *-lowering-*.f64 72.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right), x\right) \]

   12. Simplified 72.2%

       \[\leadsto \frac{\frac{x}{\color{blue}{1 + x \cdot -0.5}}}{x} \]

   if 1.80000000000000004 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

      2. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. cancel-sign-sub N/A

         \[\leadsto 1 - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\left(\mathsf{neg}\left(x \cdot 1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\left(x \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(\color{blue}{\frac{1}{2}} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      11. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot \color{blue}{-1}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right) \]

   7. Simplified 58.4%

      \[\leadsto \color{blue}{1 - x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + -0.5\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right) \cdot \color{blue}{x}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}}{x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}} \cdot x\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x}{\color{blue}{x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}}}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}}{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}}{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}\right), \color{blue}{\left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x\right)}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\color{blue}{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right)} \cdot x\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\color{blue}{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right)} \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\left(\color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right)} - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \frac{-1}{24}\right), \frac{-1}{6}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right)} - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(\color{blue}{x} \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right), \left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \color{blue}{\frac{-1}{2} \cdot \frac{-1}{2}}\right) \cdot x\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right), \color{blue}{x}\right)\right)\right)\right) \]

   9. Applied egg-rr 29.9%

      \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + 0.5}{\left(x \cdot \left(\left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right) + -0.25\right) \cdot x}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{6} \cdot x\right)}, \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)}, \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       2. *-lowering-*.f64 75.5%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)}, \frac{-1}{4}\right), x\right)\right)\right)\right) \]

   12. Simplified 75.5%

       \[\leadsto 1 - \frac{1}{\frac{\color{blue}{x \cdot -0.16666666666666666} + 0.5}{\left(x \cdot \left(\left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right) + -0.25\right) \cdot x}} \]

   13. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{96} + \frac{11}{96} \cdot \frac{1}{x}\right)} \]
   14. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{96} + \frac{11}{96} \cdot \frac{1}{x}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{96} + \frac{11}{96} \cdot \frac{1}{x}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{96}} + \frac{11}{96} \cdot \frac{1}{x}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right), \left(\color{blue}{\frac{1}{96}} + \frac{11}{96} \cdot \frac{1}{x}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right), \left(\frac{1}{96} + \frac{11}{96} \cdot \frac{1}{x}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right), \left(\frac{1}{96} + \frac{11}{96} \cdot \frac{1}{x}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right), \left(\frac{1}{96} + \frac{11}{96} \cdot \frac{1}{x}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{96} + \frac{11}{96} \cdot \frac{1}{x}\right)\right) \]

       9. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{96} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{11}{96} \cdot \frac{1}{x}\right)\right)\right)\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{96} - \color{blue}{\left(\mathsf{neg}\left(\frac{11}{96} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

       11. --lowering--.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{96}, \color{blue}{\left(\mathsf{neg}\left(\frac{11}{96} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

       12. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{96}, \left(\mathsf{neg}\left(\frac{\frac{11}{96} \cdot 1}{x}\right)\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{96}, \left(\mathsf{neg}\left(\frac{\frac{11}{96}}{x}\right)\right)\right)\right) \]

       14. distribute-neg-frac N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{96}, \left(\frac{\mathsf{neg}\left(\frac{11}{96}\right)}{\color{blue}{x}}\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{96}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{11}{96}\right)\right), \color{blue}{x}\right)\right)\right) \]

       16. metadata-eval 67.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{96}, \mathsf{/.f64}\left(\frac{-11}{96}, x\right)\right)\right) \]

   15. Simplified 67.2%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.010416666666666666 - \frac{-0.11458333333333333}{x}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 73.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.010416666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.95)
   (/ (/ x (+ 1.0 (* x -0.5))) x)
   (* (* (* x x) (* x x)) 0.010416666666666666)))

C

double code(double x) {
	double tmp;
	if (x <= 1.95) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = ((x * x) * (x * x)) * 0.010416666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.95d0) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    else
        tmp = ((x * x) * (x * x)) * 0.010416666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.95) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = ((x * x) * (x * x)) * 0.010416666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.95:
		tmp = (x / (1.0 + (x * -0.5))) / x
	else:
		tmp = ((x * x) * (x * x)) * 0.010416666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.95)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * 0.010416666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.95)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	else
		tmp = ((x * x) * (x * x)) * 0.010416666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.95], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.010416666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.94999999999999996

   1. Initial program 37.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

      2. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right), x\right) \]

      2. cancel-sign-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right), x\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(\mathsf{neg}\left(x \cdot 1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(x \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(x \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), x\right) \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot -1\right)\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), x\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right), \left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right), x\right) \]

   7. Simplified 66.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 - x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + -0.5\right)\right)}}{x} \]
   8. Step-by-step derivation

      1. flip3-- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{{1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)\right)}\right), x\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)\right)}{{1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)}^{3}}}\right), x\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)\right)}{{1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)}^{3}}}\right), x\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)\right)}{{1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)}^{3}}\right)\right), x\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{\frac{{1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right)\right)\right)}}\right)\right), x\right) \]

   9. Applied egg-rr 66.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{1}{1 - x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + -0.5\right)}}}}{x} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right), x\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot x\right)\right)\right), x\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{-1}{2}\right)\right)\right), x\right) \]

       3. *-lowering-*.f64 72.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right), x\right) \]

   12. Simplified 72.2%

       \[\leadsto \frac{\frac{x}{\color{blue}{1 + x \cdot -0.5}}}{x} \]

   if 1.94999999999999996 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

      2. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. cancel-sign-sub N/A

         \[\leadsto 1 - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\left(\mathsf{neg}\left(x \cdot 1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\left(x \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(\color{blue}{\frac{1}{2}} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      11. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot \color{blue}{-1}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right) \]

   7. Simplified 58.4%

      \[\leadsto \color{blue}{1 - x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + -0.5\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right) \cdot \color{blue}{x}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}}{x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}} \cdot x\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x}{\color{blue}{x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}}}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}}{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}}{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}\right), \color{blue}{\left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x\right)}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\color{blue}{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right)} \cdot x\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\color{blue}{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right)} \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\left(\color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right)} - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \frac{-1}{24}\right), \frac{-1}{6}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right)} - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(\color{blue}{x} \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right), \left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \color{blue}{\frac{-1}{2} \cdot \frac{-1}{2}}\right) \cdot x\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right), \color{blue}{x}\right)\right)\right)\right) \]

   9. Applied egg-rr 29.9%

      \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + 0.5}{\left(x \cdot \left(\left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right) + -0.25\right) \cdot x}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{6} \cdot x\right)}, \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)}, \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       2. *-lowering-*.f64 75.5%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)}, \frac{-1}{4}\right), x\right)\right)\right)\right) \]

   12. Simplified 75.5%

       \[\leadsto 1 - \frac{1}{\frac{\color{blue}{x \cdot -0.16666666666666666} + 0.5}{\left(x \cdot \left(\left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right) + -0.25\right) \cdot x}} \]

   13. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{96} \cdot {x}^{4}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\frac{1}{96}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\frac{1}{96}}\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \frac{1}{96}\right) \]

       4. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \frac{1}{96}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right), \frac{1}{96}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right), \frac{1}{96}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right), \frac{1}{96}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right), \frac{1}{96}\right) \]

       9. *-lowering-*.f64 67.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{96}\right) \]

   15. Simplified 67.2%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.010416666666666666} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.010416666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.1) 1.0 (* (* (* x x) (* x x)) 0.010416666666666666)))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = 1.0;
	} else {
		tmp = ((x * x) * (x * x)) * 0.010416666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.1d0) then
        tmp = 1.0d0
    else
        tmp = ((x * x) * (x * x)) * 0.010416666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = 1.0;
	} else {
		tmp = ((x * x) * (x * x)) * 0.010416666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.1:
		tmp = 1.0
	else:
		tmp = ((x * x) * (x * x)) * 0.010416666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.1)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * 0.010416666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.1)
		tmp = 1.0;
	else
		tmp = ((x * x) * (x * x)) * 0.010416666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.1], 1.0, N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.010416666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 37.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

      2. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 67.2%

      \[\leadsto \color{blue}{1} \]

   if 3.10000000000000009 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

      2. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. cancel-sign-sub N/A

         \[\leadsto 1 - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\left(\mathsf{neg}\left(x \cdot 1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\left(x \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(\color{blue}{\frac{1}{2}} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      11. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot \color{blue}{-1}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right) \]

   7. Simplified 58.4%

      \[\leadsto \color{blue}{1 - x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + -0.5\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \frac{-1}{2}\right) \cdot \color{blue}{x}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}}{x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}} \cdot x\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x}{\color{blue}{x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}}}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}}{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}}{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) - \frac{-1}{2}\right), \color{blue}{\left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x\right)}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\color{blue}{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right)} \cdot x\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\color{blue}{\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right)} \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\left(\color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right)} - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \frac{-1}{24}\right), \frac{-1}{6}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right)} - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), \left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(\color{blue}{x} \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right) \cdot x\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right), \left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \color{blue}{\frac{-1}{2} \cdot \frac{-1}{2}}\right) \cdot x\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\left(\left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{24} + \frac{-1}{6}\right)\right) - \frac{-1}{2} \cdot \frac{-1}{2}\right), \color{blue}{x}\right)\right)\right)\right) \]

   9. Applied egg-rr 29.9%

      \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + 0.5}{\left(x \cdot \left(\left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right) + -0.25\right) \cdot x}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{6} \cdot x\right)}, \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right), \frac{-1}{4}\right), x\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)}, \frac{-1}{4}\right), x\right)\right)\right)\right) \]

       2. *-lowering-*.f64 75.5%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)}, \frac{-1}{4}\right), x\right)\right)\right)\right) \]

   12. Simplified 75.5%

       \[\leadsto 1 - \frac{1}{\frac{\color{blue}{x \cdot -0.16666666666666666} + 0.5}{\left(x \cdot \left(\left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right) + -0.25\right) \cdot x}} \]

   13. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{96} \cdot {x}^{4}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\frac{1}{96}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\frac{1}{96}}\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \frac{1}{96}\right) \]

       4. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \frac{1}{96}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right), \frac{1}{96}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right), \frac{1}{96}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right), \frac{1}{96}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right), \frac{1}{96}\right) \]

       9. *-lowering-*.f64 67.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{96}\right) \]

   15. Simplified 67.2%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.010416666666666666} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.041666666666666664 + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.0)
   1.0
   (* (* x x) (+ (* x 0.041666666666666664) 0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * ((x * 0.041666666666666664) + 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * x) * ((x * 0.041666666666666664d0) + 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * ((x * 0.041666666666666664) + 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 1.0
	else:
		tmp = (x * x) * ((x * 0.041666666666666664) + 0.16666666666666666)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(x * 0.041666666666666664) + 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = 1.0;
	else
		tmp = (x * x) * ((x * 0.041666666666666664) + 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], 1.0, N[(N[(x * x), $MachinePrecision] * N[(N[(x * 0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 37.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

      2. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 67.2%

      \[\leadsto \color{blue}{1} \]

   if 2 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

      2. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right), x\right) \]

      2. cancel-sign-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right), x\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(\mathsf{neg}\left(x \cdot 1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(x \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(x \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), x\right) \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot -1\right)\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), x\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right), \left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right), x\right) \]

   7. Simplified 66.8%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 - x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + -0.5\right)\right)}}{x} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{6} \cdot \frac{1}{x}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({x}^{2} \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right) \]

      3. associate-*l* N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{x} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{x} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(\frac{1}{6} \cdot \frac{1}{x} + \color{blue}{\frac{1}{24}}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot x + \color{blue}{\frac{1}{24} \cdot x}\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{6} \cdot \left(\frac{1}{x} \cdot x\right) + \color{blue}{\frac{1}{24}} \cdot x\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{6} \cdot 1 + \frac{1}{24} \cdot x\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{6} + \color{blue}{\frac{1}{24}} \cdot x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{24} \cdot x\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

      14. *-lowering-*.f64 58.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right) \]

   10. Simplified 58.4%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.041666666666666664 + 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.9) 1.0 (* x (* 0.041666666666666664 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = 1.0;
	} else {
		tmp = x * (0.041666666666666664 * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.9d0) then
        tmp = 1.0d0
    else
        tmp = x * (0.041666666666666664d0 * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = 1.0;
	} else {
		tmp = x * (0.041666666666666664 * (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.9:
		tmp = 1.0
	else:
		tmp = x * (0.041666666666666664 * (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.9)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(0.041666666666666664 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.9)
		tmp = 1.0;
	else
		tmp = x * (0.041666666666666664 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.9], 1.0, N[(x * N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999991

   1. Initial program 37.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

      2. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 67.2%

      \[\leadsto \color{blue}{1} \]

   if 2.89999999999999991 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

      2. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right), x\right) \]

      2. cancel-sign-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right), x\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(\mathsf{neg}\left(x \cdot 1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\left(x \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(x \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), x\right) \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot -1\right)\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), x\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right), \left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right), x\right) \]

   7. Simplified 66.8%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 - x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + -0.5\right)\right)}}{x} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3}} \]
   9. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{1}{24} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{24} \cdot \left(x \cdot {x}^{\color{blue}{2}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{24} \cdot x\right) \cdot \color{blue}{{x}^{2}} \]

      4. *-commutative N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot x\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot x\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot x\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot x\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{\color{blue}{2}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      13. *-lowering-*.f64 58.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 58.4%

       \[\leadsto \color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 63.4% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) 1.0 (* x (* x 0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = 1.0d0
    else
        tmp = x * (x * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = 1.0
	else:
		tmp = x * (x * 0.16666666666666666)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = 1.0;
	else
		tmp = x * (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], 1.0, N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 37.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

      2. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 67.2%

      \[\leadsto \color{blue}{1} \]

   if 2.39999999999999991 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

      2. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]
   6. Step-by-step derivation

      1. cancel-sign-sub N/A

         \[\leadsto 1 - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\left(\mathsf{neg}\left(x \cdot 1\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\left(x \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot x\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot x\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6} \cdot x\right)\right)}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot x}\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot -1 + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot x}\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\frac{1}{6} \cdot x}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot -1\right), \left(\frac{1}{6} \cdot x\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\frac{-1}{2}, \left(\color{blue}{\frac{1}{6}} \cdot x\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 48.1%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   7. Simplified 48.1%

      \[\leadsto \color{blue}{1 - x \cdot \left(-0.5 - x \cdot 0.16666666666666666\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{1}{6} \cdot \left(x \cdot \color{blue}{x}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{6} \cdot x\right) \cdot \color{blue}{x} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{6} \cdot x\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

      6. *-lowering-*.f64 48.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right) \]

   10. Simplified 48.1%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 51.1% accurate, 105.0× 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 53.2%

   \[\frac{e^{x} - 1}{x} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

   2. accelerator-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
6. Step-by-step derivation

7. Simplified 50.7%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Developer Target 1: 52.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - 1\\ \mathbf{if}\;x < 1 \land x > -1:\\ \;\;\;\;\frac{t\_0}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) 1.0)))
   (if (and (< x 1.0) (> x -1.0)) (/ t_0 (log (exp x))) (/ t_0 x))))

C

double code(double x) {
	double t_0 = exp(x) - 1.0;
	double tmp;
	if ((x < 1.0) && (x > -1.0)) {
		tmp = t_0 / log(exp(x));
	} else {
		tmp = t_0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(x) - 1.0d0
    if ((x < 1.0d0) .and. (x > (-1.0d0))) then
        tmp = t_0 / log(exp(x))
    else
        tmp = t_0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(x) - 1.0;
	double tmp;
	if ((x < 1.0) && (x > -1.0)) {
		tmp = t_0 / Math.log(Math.exp(x));
	} else {
		tmp = t_0 / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(x) - 1.0
	tmp = 0
	if (x < 1.0) and (x > -1.0):
		tmp = t_0 / math.log(math.exp(x))
	else:
		tmp = t_0 / x
	return tmp

Julia

function code(x)
	t_0 = Float64(exp(x) - 1.0)
	tmp = 0.0
	if ((x < 1.0) && (x > -1.0))
		tmp = Float64(t_0 / log(exp(x)));
	else
		tmp = Float64(t_0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(x) - 1.0;
	tmp = 0.0;
	if ((x < 1.0) && (x > -1.0))
		tmp = t_0 / log(exp(x));
	else
		tmp = t_0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[And[Less[x, 1.0], Greater[x, -1.0]], N[(t$95$0 / N[Log[N[Exp[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :alt
  (! :herbie-platform default (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x)))

  (/ (- (exp x) 1.0) x))