Kahan's exp quotient

Percentage Accurate: 53.6% → 100.0%

Time: 9.9s

Alternatives: 13

Speedup: 10.5×

• 24.8% 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: 53.6% 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 52.7%

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

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

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

   2. expm1-define N/A

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

   3. expm1-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: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ t_1 := x \cdot t\_0\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\frac{x \cdot \left(1 - \left(x \cdot x\right) \cdot \left(t\_2 \cdot t\_2\right)\right)}{1 + x \cdot t\_2}}{1 - t\_1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.4166666666666667 + x \cdot 0.3333333333333333\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664)))))
        (t_1 (* x t_0))
        (t_2 (* t_0 t_1)))
   (if (<= x -1.5)
     (/ 1.0 (+ 1.0 (* x -0.5)))
     (if (<= x 6.8e+51)
       (/
        (/
         (/ (* x (- 1.0 (* (* x x) (* t_2 t_2)))) (+ 1.0 (* x t_2)))
         (- 1.0 t_1))
        x)
       (/
        (*
         x
         (+
          1.0
          (* x (+ 0.5 (* x (+ 0.4166666666666667 (* x 0.3333333333333333)))))))
        x)))))

C

double code(double x) {
	double t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	double t_1 = x * t_0;
	double t_2 = t_0 * t_1;
	double tmp;
	if (x <= -1.5) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 6.8e+51) {
		tmp = (((x * (1.0 - ((x * x) * (t_2 * t_2)))) / (1.0 + (x * t_2))) / (1.0 - t_1)) / x;
	} else {
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.4166666666666667 + (x * 0.3333333333333333))))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0)))
    t_1 = x * t_0
    t_2 = t_0 * t_1
    if (x <= (-1.5d0)) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else if (x <= 6.8d+51) then
        tmp = (((x * (1.0d0 - ((x * x) * (t_2 * t_2)))) / (1.0d0 + (x * t_2))) / (1.0d0 - t_1)) / x
    else
        tmp = (x * (1.0d0 + (x * (0.5d0 + (x * (0.4166666666666667d0 + (x * 0.3333333333333333d0))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	double t_1 = x * t_0;
	double t_2 = t_0 * t_1;
	double tmp;
	if (x <= -1.5) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 6.8e+51) {
		tmp = (((x * (1.0 - ((x * x) * (t_2 * t_2)))) / (1.0 + (x * t_2))) / (1.0 - t_1)) / x;
	} else {
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.4166666666666667 + (x * 0.3333333333333333))))))) / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))
	t_1 = x * t_0
	t_2 = t_0 * t_1
	tmp = 0
	if x <= -1.5:
		tmp = 1.0 / (1.0 + (x * -0.5))
	elif x <= 6.8e+51:
		tmp = (((x * (1.0 - ((x * x) * (t_2 * t_2)))) / (1.0 + (x * t_2))) / (1.0 - t_1)) / x
	else:
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.4166666666666667 + (x * 0.3333333333333333))))))) / x
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))
	t_1 = Float64(x * t_0)
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (x <= -1.5)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	elseif (x <= 6.8e+51)
		tmp = Float64(Float64(Float64(Float64(x * Float64(1.0 - Float64(Float64(x * x) * Float64(t_2 * t_2)))) / Float64(1.0 + Float64(x * t_2))) / Float64(1.0 - t_1)) / x);
	else
		tmp = Float64(Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.4166666666666667 + Float64(x * 0.3333333333333333))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	t_1 = x * t_0;
	t_2 = t_0 * t_1;
	tmp = 0.0;
	if (x <= -1.5)
		tmp = 1.0 / (1.0 + (x * -0.5));
	elseif (x <= 6.8e+51)
		tmp = (((x * (1.0 - ((x * x) * (t_2 * t_2)))) / (1.0 + (x * t_2))) / (1.0 - t_1)) / x;
	else
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.4166666666666667 + (x * 0.3333333333333333))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[x, -1.5], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e+51], N[(N[(N[(N[(x * N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.4166666666666667 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 1.2%

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

   7. Simplified 1.2%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 1.2%

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

   9. Applied egg-rr 1.2%

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

       1. clear-num N/A

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

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

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

       3. *-commutative N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

   11. Applied egg-rr 1.2%

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

   12. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 18.8%

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

   14. Simplified 18.8%

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

   if -1.5 < x < 6.79999999999999969e51

   1. Initial program 14.7%

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

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

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

      2. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 91.5%

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

   7. Simplified 91.5%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 92.9%

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

       1. flip-- N/A

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

       2. associate-*l/ N/A

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

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

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

   11. Applied egg-rr 96.9%

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

   if 6.79999999999999969e51 < 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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 92.7%

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

   7. Simplified 92.7%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 21.2%

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

   10. Taylor expanded in x around 0

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

   12. Simplified 1.5%

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

   13. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{5}{12} + \frac{1}{3} \cdot x\right)\right)\right)\right)}, x\right) \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{5}{12} + \frac{1}{3} \cdot x\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{5}{12} + \frac{1}{3} \cdot x\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + x \cdot \left(\frac{5}{12} + \frac{1}{3} \cdot x\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{5}{12} + \frac{1}{3} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{5}{12} + \frac{1}{3} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{5}{12}, \left(\frac{1}{3} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{5}{12}, \left(x \cdot \frac{1}{3}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       8. *-lowering-*.f64 92.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{5}{12}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right)\right)\right)\right)\right), x\right) \]

   15. Simplified 92.7%

       \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.4166666666666667 + x \cdot 0.3333333333333333\right)\right)\right)}}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\frac{x \cdot \left(1 - \left(x \cdot x\right) \cdot \left(\left(\left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right) \cdot \left(\left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right)\right)}{1 + x \cdot \left(\left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)}}{1 - x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.4166666666666667 + x \cdot 0.3333333333333333\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{x \cdot x - t\_0 \cdot t\_0}{x - t\_0}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          (* x x)
          (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664)))))))
   (if (<= x 2e-154)
     (/ 1.0 (+ 1.0 (* x -0.5)))
     (if (<= x 2e+76)
       (/ (/ (- (* x x) (* t_0 t_0)) (- x t_0)) x)
       (/ (* 0.041666666666666664 (* x (* x (* x x)))) x)))))

C

double code(double x) {
	double t_0 = (x * x) * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))));
	double tmp;
	if (x <= 2e-154) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 2e+76) {
		tmp = (((x * x) - (t_0 * t_0)) / (x - t_0)) / x;
	} else {
		tmp = (0.041666666666666664 * (x * (x * (x * 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) * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))))
    if (x <= 2d-154) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else if (x <= 2d+76) then
        tmp = (((x * x) - (t_0 * t_0)) / (x - t_0)) / x
    else
        tmp = (0.041666666666666664d0 * (x * (x * (x * x)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))));
	double tmp;
	if (x <= 2e-154) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 2e+76) {
		tmp = (((x * x) - (t_0 * t_0)) / (x - t_0)) / x;
	} else {
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))
	tmp = 0
	if x <= 2e-154:
		tmp = 1.0 / (1.0 + (x * -0.5))
	elif x <= 2e+76:
		tmp = (((x * x) - (t_0 * t_0)) / (x - t_0)) / x
	else:
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664)))))
	tmp = 0.0
	if (x <= 2e-154)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	elseif (x <= 2e+76)
		tmp = Float64(Float64(Float64(Float64(x * x) - Float64(t_0 * t_0)) / Float64(x - t_0)) / x);
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(x * Float64(x * Float64(x * x)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))));
	tmp = 0.0;
	if (x <= 2e-154)
		tmp = 1.0 / (1.0 + (x * -0.5));
	elseif (x <= 2e+76)
		tmp = (((x * x) - (t_0 * t_0)) / (x - t_0)) / x;
	else
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e-154], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+76], N[(N[(N[(N[(x * x), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x - t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.9999999999999999e-154

   1. Initial program 42.5%

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

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

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

      2. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 61.0%

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

   7. Simplified 61.0%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 61.0%

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

   9. Applied egg-rr 61.0%

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

       1. clear-num N/A

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

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

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

       3. *-commutative N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

   11. Applied egg-rr 61.0%

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

   12. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 67.8%

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

   14. Simplified 67.8%

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

   if 1.9999999999999999e-154 < x < 2.0000000000000001e76

   1. Initial program 39.5%

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

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

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

      2. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 69.3%

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

   7. Simplified 69.3%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. flip-+ N/A

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

      4. *-rgt-identity N/A

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

      5. fmm-def N/A

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

      6. *-commutative N/A

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

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

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

   9. Applied egg-rr 82.9%

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

   if 2.0000000000000001e76 < 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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{x \cdot x - \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}{x - \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - x \cdot \left(t\_0 \cdot t\_1\right)\right)}{1 - t\_1}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664)))))
        (t_1 (* x t_0)))
   (if (<= x -1.5)
     (/ 1.0 (+ 1.0 (* x -0.5)))
     (if (<= x 1.6e+103)
       (/ (/ (* x (- 1.0 (* x (* t_0 t_1)))) (- 1.0 t_1)) x)
       (* x (* (* x x) 0.041666666666666664))))))

C

double code(double x) {
	double t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	double t_1 = x * t_0;
	double tmp;
	if (x <= -1.5) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 1.6e+103) {
		tmp = ((x * (1.0 - (x * (t_0 * t_1)))) / (1.0 - t_1)) / x;
	} else {
		tmp = x * ((x * x) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0)))
    t_1 = x * t_0
    if (x <= (-1.5d0)) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else if (x <= 1.6d+103) then
        tmp = ((x * (1.0d0 - (x * (t_0 * t_1)))) / (1.0d0 - t_1)) / x
    else
        tmp = x * ((x * x) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	double t_1 = x * t_0;
	double tmp;
	if (x <= -1.5) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 1.6e+103) {
		tmp = ((x * (1.0 - (x * (t_0 * t_1)))) / (1.0 - t_1)) / x;
	} else {
		tmp = x * ((x * x) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))
	t_1 = x * t_0
	tmp = 0
	if x <= -1.5:
		tmp = 1.0 / (1.0 + (x * -0.5))
	elif x <= 1.6e+103:
		tmp = ((x * (1.0 - (x * (t_0 * t_1)))) / (1.0 - t_1)) / x
	else:
		tmp = x * ((x * x) * 0.041666666666666664)
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= -1.5)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	elseif (x <= 1.6e+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(Float64(x * x) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	t_1 = x * t_0;
	tmp = 0.0;
	if (x <= -1.5)
		tmp = 1.0 / (1.0 + (x * -0.5));
	elseif (x <= 1.6e+103)
		tmp = ((x * (1.0 - (x * (t_0 * t_1)))) / (1.0 - t_1)) / x;
	else
		tmp = x * ((x * x) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[x, -1.5], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e+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[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 1.2%

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

   7. Simplified 1.2%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 1.2%

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

   9. Applied egg-rr 1.2%

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

       1. clear-num N/A

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

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

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

       3. *-commutative N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

   11. Applied egg-rr 1.2%

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

   12. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 18.8%

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

   14. Simplified 18.8%

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

   if -1.5 < x < 1.59999999999999996e103

   1. Initial program 20.9%

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

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

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

      2. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 89.7%

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

   7. Simplified 89.7%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 93.4%

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

   if 1.59999999999999996e103 < 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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   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 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - x \cdot \left(\left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right)}{1 - x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - x \cdot \left(t\_0 \cdot \left(x \cdot t\_0\right)\right)\right)}{1 - x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))
   (if (<= x -2.2)
     (/ 1.0 (+ 1.0 (* x -0.5)))
     (if (<= x 3.3e+154)
       (/
        (/
         (* x (- 1.0 (* x (* t_0 (* x t_0)))))
         (- 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))
        x)
       (* x (* x 0.16666666666666666))))))

C

double code(double x) {
	double t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	double tmp;
	if (x <= -2.2) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 3.3e+154) {
		tmp = ((x * (1.0 - (x * (t_0 * (x * t_0))))) / (1.0 - (x * (0.5 + (x * 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 = 0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0)))
    if (x <= (-2.2d0)) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else if (x <= 3.3d+154) then
        tmp = ((x * (1.0d0 - (x * (t_0 * (x * t_0))))) / (1.0d0 - (x * (0.5d0 + (x * 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 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	double tmp;
	if (x <= -2.2) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 3.3e+154) {
		tmp = ((x * (1.0 - (x * (t_0 * (x * t_0))))) / (1.0 - (x * (0.5 + (x * 0.16666666666666666))))) / x;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))
	tmp = 0
	if x <= -2.2:
		tmp = 1.0 / (1.0 + (x * -0.5))
	elif x <= 3.3e+154:
		tmp = ((x * (1.0 - (x * (t_0 * (x * t_0))))) / (1.0 - (x * (0.5 + (x * 0.16666666666666666))))) / x
	else:
		tmp = x * (x * 0.16666666666666666)
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))
	tmp = 0.0
	if (x <= -2.2)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	elseif (x <= 3.3e+154)
		tmp = Float64(Float64(Float64(x * Float64(1.0 - Float64(x * Float64(t_0 * Float64(x * t_0))))) / Float64(1.0 - Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))))) / x);
	else
		tmp = Float64(x * Float64(x * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	tmp = 0.0;
	if (x <= -2.2)
		tmp = 1.0 / (1.0 + (x * -0.5));
	elseif (x <= 3.3e+154)
		tmp = ((x * (1.0 - (x * (t_0 * (x * t_0))))) / (1.0 - (x * (0.5 + (x * 0.16666666666666666))))) / x;
	else
		tmp = x * (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.2], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e+154], N[(N[(N[(x * N[(1.0 - N[(x * N[(t$95$0 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 1.2%

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

   7. Simplified 1.2%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 1.2%

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

   9. Applied egg-rr 1.2%

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

       1. clear-num N/A

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

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

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

       3. *-commutative N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

   11. Applied egg-rr 1.2%

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

   12. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 18.8%

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

   14. Simplified 18.8%

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

   if -2.2000000000000002 < x < 3.3e154

   1. Initial program 25.3%

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

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

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

      2. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 90.3%

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

   7. Simplified 90.3%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 88.2%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 93.6%

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

   12. Simplified 93.6%

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

   if 3.3e154 < 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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-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 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - x \cdot \left(\left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right)}{1 - x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.5)
   (/ 1.0 (+ 1.0 (* x -0.5)))
   (/
    (*
     x
     (+
      1.0
      (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664)))))))
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.5) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.5d0)) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = (x * (1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.5) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.5:
		tmp = 1.0 / (1.0 + (x * -0.5))
	else:
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.5)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64(Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.5)
		tmp = 1.0 / (1.0 + (x * -0.5));
	else
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.5], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5

   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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 1.2%

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

   7. Simplified 1.2%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 1.2%

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

   9. Applied egg-rr 1.2%

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

       1. clear-num N/A

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

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

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

       3. *-commutative N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

   11. Applied egg-rr 1.2%

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

   12. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 18.8%

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

   14. Simplified 18.8%

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

   if -1.5 < x

   1. Initial program 37.6%

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

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

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

      2. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 91.9%

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

   7. Simplified 91.9%

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

Alternative 7: 73.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.92:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.92)
   (/ 1.0 (+ 1.0 (* x -0.5)))
   (/ (* 0.041666666666666664 (* x (* x (* x x)))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.92) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.92d0) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = (0.041666666666666664d0 * (x * (x * (x * x)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.92) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.92:
		tmp = 1.0 / (1.0 + (x * -0.5))
	else:
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.92)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(x * Float64(x * Float64(x * x)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.92)
		tmp = 1.0 / (1.0 + (x * -0.5));
	else
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.92], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9199999999999999

   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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 67.5%

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

   7. Simplified 67.5%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 67.5%

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

   9. Applied egg-rr 67.5%

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

       1. clear-num N/A

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

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

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

       3. *-commutative N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

   11. Applied egg-rr 67.5%

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

   12. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 72.8%

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

   14. Simplified 72.8%

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

   if 1.9199999999999999 < 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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 77.2%

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

   10. Simplified 77.2%

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

Alternative 8: 71.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.45)
   (/ 1.0 (+ 1.0 (* x -0.5)))
   (+ 1.0 (* x (* x (+ 0.16666666666666666 (* x 0.041666666666666664)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = 1.0 + (x * (x * (0.16666666666666666 + (x * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.45d0) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = 1.0d0 + (x * (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = 1.0 + (x * (x * (0.16666666666666666 + (x * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45:
		tmp = 1.0 / (1.0 + (x * -0.5))
	else:
		tmp = 1.0 + (x * (x * (0.16666666666666666 + (x * 0.041666666666666664))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.45)
		tmp = 1.0 / (1.0 + (x * -0.5));
	else
		tmp = 1.0 + (x * (x * (0.16666666666666666 + (x * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.45], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 67.5%

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

   7. Simplified 67.5%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 67.5%

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

   9. Applied egg-rr 67.5%

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

       1. clear-num N/A

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

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

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

       3. *-commutative N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

   11. Applied egg-rr 67.5%

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

   12. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 72.8%

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

   14. Simplified 72.8%

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

   if 1.44999999999999996 < 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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 67.1%

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

   10. Simplified 67.1%

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

Alternative 9: 71.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8)
   (/ 1.0 (+ 1.0 (* x -0.5)))
   (* (* x x) (+ 0.16666666666666666 (* x 0.041666666666666664)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = (x * x) * (0.16666666666666666 + (x * 0.041666666666666664));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.8d0) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = (x * x) * (0.16666666666666666d0 + (x * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = (x * x) * (0.16666666666666666 + (x * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = 1.0 / (1.0 + (x * -0.5))
	else:
		tmp = (x * x) * (0.16666666666666666 + (x * 0.041666666666666664))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = 1.0 / (1.0 + (x * -0.5));
	else
		tmp = (x * x) * (0.16666666666666666 + (x * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 67.5%

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

   7. Simplified 67.5%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 67.5%

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

   9. Applied egg-rr 67.5%

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

       1. clear-num N/A

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

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

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

       3. *-commutative N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

   11. Applied egg-rr 67.5%

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

   12. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 72.8%

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

   14. Simplified 72.8%

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

   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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 77.2%

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

   7. Simplified 77.2%

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

   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. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 67.1%

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

   10. Simplified 67.1%

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

4. Final simplification 71.4%

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

Alternative 10: 71.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.92:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.92)
   (/ 1.0 (+ 1.0 (* x -0.5)))
   (* x (* (* x x) 0.041666666666666664))))

C

double code(double x) {
	double tmp;
	if (x <= 1.92) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = x * ((x * x) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.92d0) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = x * ((x * x) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.92) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = x * ((x * x) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.92:
		tmp = 1.0 / (1.0 + (x * -0.5))
	else:
		tmp = x * ((x * x) * 0.041666666666666664)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.92)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64(x * Float64(Float64(x * x) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.92)
		tmp = 1.0 / (1.0 + (x * -0.5));
	else
		tmp = x * ((x * x) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.92], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9199999999999999

   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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 67.5%

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

   7. Simplified 67.5%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 67.5%

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

   9. Applied egg-rr 67.5%

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

       1. clear-num N/A

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

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

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

       3. *-commutative N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

   11. Applied egg-rr 67.5%

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

   12. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 72.8%

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

   14. Simplified 72.8%

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

   if 1.9199999999999999 < 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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 77.2%

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

   7. Simplified 77.2%

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

   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 67.1%

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

   10. Simplified 67.1%

       \[\leadsto \color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.92:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.9) 1.0 (* x (* (* x x) 0.041666666666666664))))

C

double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = 1.0;
	} else {
		tmp = x * ((x * x) * 0.041666666666666664);
	}
	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 * ((x * x) * 0.041666666666666664d0)
    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 * ((x * x) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.9:
		tmp = 1.0
	else:
		tmp = x * ((x * x) * 0.041666666666666664)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.9)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(Float64(x * x) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.9)
		tmp = 1.0;
	else
		tmp = x * ((x * x) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.9], 1.0, N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999991

   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. expm1-define N/A

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

      3. expm1-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 67.5%

      \[\leadsto \color{blue}{1} \]

   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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 77.2%

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

   7. Simplified 77.2%

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

   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 67.1%

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

   10. Simplified 67.1%

       \[\leadsto \color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.3% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.45:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.45) 1.0 (* x (* x 0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 2.45) {
		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 <= 2.45d0) 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 <= 2.45) {
		tmp = 1.0;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.45:
		tmp = 1.0
	else:
		tmp = x * (x * 0.16666666666666666)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.45)
		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 <= 2.45)
		tmp = 1.0;
	else
		tmp = x * (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.45], 1.0, N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4500000000000002

   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. expm1-define N/A

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

      3. expm1-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 67.5%

      \[\leadsto \color{blue}{1} \]

   if 2.4500000000000002 < 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. expm1-define N/A

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

      3. expm1-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. +-lowering-+.f64 N/A

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 53.4%

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

   7. Simplified 53.4%

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

   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 53.4%

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

   10. Simplified 53.4%

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

Alternative 13: 50.7% 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 52.7%

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

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

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

   2. expm1-define N/A

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

   3. expm1-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 51.7%

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

Developer Target 1: 53.1% 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 2024161 
(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))