Kahan's exp quotient

Percentage Accurate: 52.6% → 100.0%

Time: 10.5s

Alternatives: 16

Speedup: 11.7×

• 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: 52.6% 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 56.6%

   \[\frac{e^{x} - 1}{x} \]
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. Add Preprocessing

Alternative 2: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + x \cdot 0.041666666666666664\\ t_1 := x \cdot t\_0\\ t_2 := t\_0 \cdot t\_0\\ t_3 := \left(\left(x \cdot x\right) \cdot t\_0\right) \cdot \left(x \cdot t\_2\right)\\ \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+51}:\\ \;\;\;\;1 + \frac{\frac{x \cdot \left(0.015625 - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(t\_3 \cdot \left(t\_0 \cdot t\_2\right)\right)\right)}{0.125 - t\_3}}{0.25 + t\_1 \cdot \left(-0.5 + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.004629629629629629 + x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(36 + x \cdot 9\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.16666666666666666 (* x 0.041666666666666664)))
        (t_1 (* x t_0))
        (t_2 (* t_0 t_0))
        (t_3 (* (* (* x x) t_0) (* x t_2))))
   (if (<= x -1.55)
     (/ (/ x (+ 1.0 (* x -0.5))) x)
     (if (<= x 5e+51)
       (+
        1.0
        (/
         (/
          (* x (- 0.015625 (* (* x (* x x)) (* t_3 (* t_0 t_2)))))
          (- 0.125 t_3))
         (+ 0.25 (* t_1 (+ -0.5 t_1)))))
       (*
        (+ 0.004629629629629629 (* x (* (* x x) 7.233796296296296e-5)))
        (* x (* x (+ 36.0 (* x 9.0)))))))))

C

double code(double x) {
	double t_0 = 0.16666666666666666 + (x * 0.041666666666666664);
	double t_1 = x * t_0;
	double t_2 = t_0 * t_0;
	double t_3 = ((x * x) * t_0) * (x * t_2);
	double tmp;
	if (x <= -1.55) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else if (x <= 5e+51) {
		tmp = 1.0 + (((x * (0.015625 - ((x * (x * x)) * (t_3 * (t_0 * t_2))))) / (0.125 - t_3)) / (0.25 + (t_1 * (-0.5 + t_1))));
	} else {
		tmp = (0.004629629629629629 + (x * ((x * x) * 7.233796296296296e-5))) * (x * (x * (36.0 + (x * 9.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 + (x * 0.041666666666666664d0)
    t_1 = x * t_0
    t_2 = t_0 * t_0
    t_3 = ((x * x) * t_0) * (x * t_2)
    if (x <= (-1.55d0)) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    else if (x <= 5d+51) then
        tmp = 1.0d0 + (((x * (0.015625d0 - ((x * (x * x)) * (t_3 * (t_0 * t_2))))) / (0.125d0 - t_3)) / (0.25d0 + (t_1 * ((-0.5d0) + t_1))))
    else
        tmp = (0.004629629629629629d0 + (x * ((x * x) * 7.233796296296296d-5))) * (x * (x * (36.0d0 + (x * 9.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.16666666666666666 + (x * 0.041666666666666664);
	double t_1 = x * t_0;
	double t_2 = t_0 * t_0;
	double t_3 = ((x * x) * t_0) * (x * t_2);
	double tmp;
	if (x <= -1.55) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else if (x <= 5e+51) {
		tmp = 1.0 + (((x * (0.015625 - ((x * (x * x)) * (t_3 * (t_0 * t_2))))) / (0.125 - t_3)) / (0.25 + (t_1 * (-0.5 + t_1))));
	} else {
		tmp = (0.004629629629629629 + (x * ((x * x) * 7.233796296296296e-5))) * (x * (x * (36.0 + (x * 9.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.16666666666666666 + (x * 0.041666666666666664)
	t_1 = x * t_0
	t_2 = t_0 * t_0
	t_3 = ((x * x) * t_0) * (x * t_2)
	tmp = 0
	if x <= -1.55:
		tmp = (x / (1.0 + (x * -0.5))) / x
	elif x <= 5e+51:
		tmp = 1.0 + (((x * (0.015625 - ((x * (x * x)) * (t_3 * (t_0 * t_2))))) / (0.125 - t_3)) / (0.25 + (t_1 * (-0.5 + t_1))))
	else:
		tmp = (0.004629629629629629 + (x * ((x * x) * 7.233796296296296e-5))) * (x * (x * (36.0 + (x * 9.0))))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))
	t_1 = Float64(x * t_0)
	t_2 = Float64(t_0 * t_0)
	t_3 = Float64(Float64(Float64(x * x) * t_0) * Float64(x * t_2))
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	elseif (x <= 5e+51)
		tmp = Float64(1.0 + Float64(Float64(Float64(x * Float64(0.015625 - Float64(Float64(x * Float64(x * x)) * Float64(t_3 * Float64(t_0 * t_2))))) / Float64(0.125 - t_3)) / Float64(0.25 + Float64(t_1 * Float64(-0.5 + t_1)))));
	else
		tmp = Float64(Float64(0.004629629629629629 + Float64(x * Float64(Float64(x * x) * 7.233796296296296e-5))) * Float64(x * Float64(x * Float64(36.0 + Float64(x * 9.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.16666666666666666 + (x * 0.041666666666666664);
	t_1 = x * t_0;
	t_2 = t_0 * t_0;
	t_3 = ((x * x) * t_0) * (x * t_2);
	tmp = 0.0;
	if (x <= -1.55)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	elseif (x <= 5e+51)
		tmp = 1.0 + (((x * (0.015625 - ((x * (x * x)) * (t_3 * (t_0 * t_2))))) / (0.125 - t_3)) / (0.25 + (t_1 * (-0.5 + t_1))));
	else
		tmp = (0.004629629629629629 + (x * ((x * x) * 7.233796296296296e-5))) * (x * (x * (36.0 + (x * 9.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.55], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5e+51], N[(1.0 + N[(N[(N[(x * N[(0.015625 - N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.125 - t$95$3), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(t$95$1 * N[(-0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.004629629629629629 + N[(x * N[(N[(x * x), $MachinePrecision] * 7.233796296296296e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(36.0 + N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 100.0%

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

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

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

      2. 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 1.1%

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

   7. Simplified 1.1%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 1.1%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 18.8%

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

   12. Simplified 18.8%

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

   if -1.55000000000000004 < x < 5e51

   1. Initial program 16.5%

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

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
   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 88.2%

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

   7. Simplified 88.2%

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

      1. *-commutative N/A

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

      2. flip3-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 89.6%

      \[\leadsto 1 + \color{blue}{\frac{\left(0.125 + \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot x}{0.25 + \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) + -0.5\right)}} \]

   11. Applied egg-rr 94.0%

       \[\leadsto 1 + \frac{\color{blue}{\frac{\left(0.015625 - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right) \cdot \left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right) \cdot x}{0.125 - \left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}}}{0.25 + \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) + -0.5\right)} \]

   if 5e51 < 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 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*l* N/A

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

      10. lft-mult-inverse N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 87.4%

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

   10. Simplified 87.4%

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

   12. Applied egg-rr 20.0%

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

   13. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(36 + 9 \cdot x\right)\right)}, x\right)\right) \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(36 + 9 \cdot x\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(36, \left(9 \cdot x\right)\right)\right), x\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(36, \left(x \cdot 9\right)\right)\right), x\right)\right) \]

       4. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(36, \mathsf{*.f64}\left(x, 9\right)\right)\right), x\right)\right) \]

   15. Simplified 100.0%

       \[\leadsto \left(0.004629629629629629 + x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(\color{blue}{\left(x \cdot \left(36 + x \cdot 9\right)\right)} \cdot x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+51}:\\ \;\;\;\;1 + \frac{\frac{x \cdot \left(0.015625 - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right) \cdot \left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right)}{0.125 - \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}}{0.25 + \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(-0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.004629629629629629 + x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(36 + x \cdot 9\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) + 0.5\\ t_1 := t\_0 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\\ \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 - t\_1 \cdot t\_1}{\left(1 + t\_1\right) \cdot \left(1 - x \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.004629629629629629 + x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(36 + x \cdot 9\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (* x (+ 0.16666666666666666 (* x 0.041666666666666664))) 0.5))
        (t_1 (* t_0 (* (* x x) t_0))))
   (if (<= x -1.55)
     (/ (/ x (+ 1.0 (* x -0.5))) x)
     (if (<= x 5e+34)
       (/ (- 1.0 (* t_1 t_1)) (* (+ 1.0 t_1) (- 1.0 (* x t_0))))
       (*
        (+ 0.004629629629629629 (* x (* (* x x) 7.233796296296296e-5)))
        (* x (* x (+ 36.0 (* x 9.0)))))))))

C

double code(double x) {
	double t_0 = (x * (0.16666666666666666 + (x * 0.041666666666666664))) + 0.5;
	double t_1 = t_0 * ((x * x) * t_0);
	double tmp;
	if (x <= -1.55) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else if (x <= 5e+34) {
		tmp = (1.0 - (t_1 * t_1)) / ((1.0 + t_1) * (1.0 - (x * t_0)));
	} else {
		tmp = (0.004629629629629629 + (x * ((x * x) * 7.233796296296296e-5))) * (x * (x * (36.0 + (x * 9.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))) + 0.5d0
    t_1 = t_0 * ((x * x) * t_0)
    if (x <= (-1.55d0)) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    else if (x <= 5d+34) then
        tmp = (1.0d0 - (t_1 * t_1)) / ((1.0d0 + t_1) * (1.0d0 - (x * t_0)))
    else
        tmp = (0.004629629629629629d0 + (x * ((x * x) * 7.233796296296296d-5))) * (x * (x * (36.0d0 + (x * 9.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * (0.16666666666666666 + (x * 0.041666666666666664))) + 0.5;
	double t_1 = t_0 * ((x * x) * t_0);
	double tmp;
	if (x <= -1.55) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else if (x <= 5e+34) {
		tmp = (1.0 - (t_1 * t_1)) / ((1.0 + t_1) * (1.0 - (x * t_0)));
	} else {
		tmp = (0.004629629629629629 + (x * ((x * x) * 7.233796296296296e-5))) * (x * (x * (36.0 + (x * 9.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * (0.16666666666666666 + (x * 0.041666666666666664))) + 0.5
	t_1 = t_0 * ((x * x) * t_0)
	tmp = 0
	if x <= -1.55:
		tmp = (x / (1.0 + (x * -0.5))) / x
	elif x <= 5e+34:
		tmp = (1.0 - (t_1 * t_1)) / ((1.0 + t_1) * (1.0 - (x * t_0)))
	else:
		tmp = (0.004629629629629629 + (x * ((x * x) * 7.233796296296296e-5))) * (x * (x * (36.0 + (x * 9.0))))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))) + 0.5)
	t_1 = Float64(t_0 * Float64(Float64(x * x) * t_0))
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	elseif (x <= 5e+34)
		tmp = Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(Float64(1.0 + t_1) * Float64(1.0 - Float64(x * t_0))));
	else
		tmp = Float64(Float64(0.004629629629629629 + Float64(x * Float64(Float64(x * x) * 7.233796296296296e-5))) * Float64(x * Float64(x * Float64(36.0 + Float64(x * 9.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * (0.16666666666666666 + (x * 0.041666666666666664))) + 0.5;
	t_1 = t_0 * ((x * x) * t_0);
	tmp = 0.0;
	if (x <= -1.55)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	elseif (x <= 5e+34)
		tmp = (1.0 - (t_1 * t_1)) / ((1.0 + t_1) * (1.0 - (x * t_0)));
	else
		tmp = (0.004629629629629629 + (x * ((x * x) * 7.233796296296296e-5))) * (x * (x * (36.0 + (x * 9.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.55], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5e+34], N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + t$95$1), $MachinePrecision] * N[(1.0 - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.004629629629629629 + N[(x * N[(N[(x * x), $MachinePrecision] * 7.233796296296296e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(36.0 + N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 100.0%

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

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

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

      2. 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 1.1%

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

   7. Simplified 1.1%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 1.1%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 18.8%

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

   12. Simplified 18.8%

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

   if -1.55000000000000004 < x < 4.9999999999999998e34

   1. Initial program 14.6%

      \[\frac{e^{x} - 1}{x} \]
   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 90.1%

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

   7. Simplified 90.1%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. clear-num N/A

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

      7. div-inv N/A

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

   9. Applied egg-rr 93.1%

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

   if 4.9999999999999998e34 < 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 83.5%

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

   7. Simplified 83.5%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*l* N/A

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

      10. lft-mult-inverse N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 83.5%

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

   10. Simplified 83.5%

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

   12. Applied egg-rr 19.2%

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

   13. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(36 + 9 \cdot x\right)\right)}, x\right)\right) \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(36 + 9 \cdot x\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(36, \left(9 \cdot x\right)\right)\right), x\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(36, \left(x \cdot 9\right)\right)\right), x\right)\right) \]

       4. *-lowering-*.f64 95.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(36, \mathsf{*.f64}\left(x, 9\right)\right)\right), x\right)\right) \]

   15. Simplified 95.7%

       \[\leadsto \left(0.004629629629629629 + x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(\color{blue}{\left(x \cdot \left(36 + x \cdot 9\right)\right)} \cdot x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 - \left(\left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) + 0.5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) + 0.5\right)\right)\right) \cdot \left(\left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) + 0.5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) + 0.5\right)\right)\right)}{\left(1 + \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) + 0.5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) + 0.5\right)\right)\right) \cdot \left(1 - x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) + 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.004629629629629629 + x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(36 + x \cdot 9\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.004629629629629629 + x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(36 + x \cdot 9\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75)
   (/ (/ x (+ 1.0 (* x -0.5))) x)
   (*
    (+ 0.004629629629629629 (* x (* (* x x) 7.233796296296296e-5)))
    (* x (* x (+ 36.0 (* x 9.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = (0.004629629629629629 + (x * ((x * x) * 7.233796296296296e-5))) * (x * (x * (36.0 + (x * 9.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.75d0) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    else
        tmp = (0.004629629629629629d0 + (x * ((x * x) * 7.233796296296296d-5))) * (x * (x * (36.0d0 + (x * 9.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = (0.004629629629629629 + (x * ((x * x) * 7.233796296296296e-5))) * (x * (x * (36.0 + (x * 9.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.75:
		tmp = (x / (1.0 + (x * -0.5))) / x
	else:
		tmp = (0.004629629629629629 + (x * ((x * x) * 7.233796296296296e-5))) * (x * (x * (36.0 + (x * 9.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	else
		tmp = Float64(Float64(0.004629629629629629 + Float64(x * Float64(Float64(x * x) * 7.233796296296296e-5))) * Float64(x * Float64(x * Float64(36.0 + Float64(x * 9.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.75)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	else
		tmp = (0.004629629629629629 + (x * ((x * x) * 7.233796296296296e-5))) * (x * (x * (36.0 + (x * 9.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.004629629629629629 + N[(x * N[(N[(x * x), $MachinePrecision] * 7.233796296296296e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(36.0 + N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 38.3%

      \[\frac{e^{x} - 1}{x} \]
   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 65.2%

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

   7. Simplified 65.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 65.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 71.3%

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

   12. Simplified 71.3%

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

   if 1.75 < 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 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*l* N/A

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

      10. lft-mult-inverse N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 69.8%

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

   10. Simplified 69.8%

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

   12. Applied egg-rr 16.6%

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

   13. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(36 + 9 \cdot x\right)\right)}, x\right)\right) \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(36 + 9 \cdot x\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(36, \left(9 \cdot x\right)\right)\right), x\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(36, \left(x \cdot 9\right)\right)\right), x\right)\right) \]

       4. *-lowering-*.f64 80.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{13824}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(36, \mathsf{*.f64}\left(x, 9\right)\right)\right), x\right)\right) \]

   15. Simplified 80.1%

       \[\leadsto \left(0.004629629629629629 + x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(\color{blue}{\left(x \cdot \left(36 + x \cdot 9\right)\right)} \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.004629629629629629 + x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(36 + x \cdot 9\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.004629629629629629 + x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(x \cdot \frac{x}{0.027777777777777776}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8)
   (/ (/ x (+ 1.0 (* x -0.5))) x)
   (*
    (+ 0.004629629629629629 (* x (* (* x x) 7.233796296296296e-5)))
    (* x (/ x 0.027777777777777776)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.004629629629629629 + N[(x * N[(N[(x * x), $MachinePrecision] * 7.233796296296296e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x / 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 38.3%

      \[\frac{e^{x} - 1}{x} \]
   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 65.2%

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

   7. Simplified 65.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 65.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 71.3%

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

   12. Simplified 71.3%

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

   if 1.80000000000000004 < x

   1. Initial program 100.0%

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

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

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

      2. 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 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*l* N/A

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

      10. lft-mult-inverse N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 69.8%

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

   10. Simplified 69.8%

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

   12. Applied egg-rr 16.6%

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

   13. Taylor expanded in x around 0

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

   15. Simplified 78.7%

       \[\leadsto \left(0.004629629629629629 + x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(\frac{x}{\color{blue}{0.027777777777777776}} \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

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

Alternative 6: 73.5% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 38.3%

      \[\frac{e^{x} - 1}{x} \]
   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 65.2%

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

   7. Simplified 65.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 65.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 71.3%

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

   12. Simplified 71.3%

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

   if 1.80000000000000004 < x

   1. Initial program 100.0%

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

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

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

      2. 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 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*l* N/A

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

      10. lft-mult-inverse N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 69.8%

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

   10. Simplified 69.8%

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

4. Final simplification 70.8%

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

Alternative 7: 73.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6)
   (/ (/ x (+ 1.0 (* x -0.5))) x)
   (/ (* x (+ 1.0 (* x (* x (* x 0.041666666666666664))))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = (x * (1.0 + (x * (x * (x * 0.041666666666666664))))) / x;
	}
	return tmp;
}

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = (x / (1.0 + (x * -0.5))) / x
	else:
		tmp = (x * (1.0 + (x * (x * (x * 0.041666666666666664))))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	else
		tmp = Float64(Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(x * 0.041666666666666664))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.6)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	else
		tmp = (x * (1.0 + (x * (x * (x * 0.041666666666666664))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * N[(1.0 + N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 38.3%

      \[\frac{e^{x} - 1}{x} \]
   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 65.2%

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

   7. Simplified 65.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 65.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 71.3%

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

   12. Simplified 71.3%

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

   if 1.6000000000000001 < 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 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 69.8%

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

   10. Simplified 69.8%

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

Alternative 8: 73.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.95)
   (/ (/ x (+ 1.0 (* x -0.5))) x)
   (/ (* 0.041666666666666664 (* x (* x (* x x)))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.95) {
		tmp = (x / (1.0 + (x * -0.5))) / 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) :: tmp
    if (x <= 1.95d0) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    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 <= 1.95) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.95:
		tmp = (x / (1.0 + (x * -0.5))) / x
	else:
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.95)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	else
		tmp = Float64(Float64(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 <= 1.95)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	else
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.95], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(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 < 1.94999999999999996

   1. Initial program 38.3%

      \[\frac{e^{x} - 1}{x} \]
   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 65.2%

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

   7. Simplified 65.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 65.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 71.3%

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

   12. Simplified 71.3%

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

   if 1.94999999999999996 < x

   1. Initial program 100.0%

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

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

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

      2. 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 69.8%

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

   7. Simplified 69.8%

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

   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 69.8%

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

   10. Simplified 69.8%

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

Alternative 9: 71.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x \cdot x}{24 + \frac{-96}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.0)
   (/ (/ x (+ 1.0 (* x -0.5))) x)
   (* x (/ (* x x) (+ 24.0 (/ -96.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = x * ((x * x) / (24.0 + (-96.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.0d0) then
        tmp = (x / (1.0d0 + (x * (-0.5d0)))) / x
    else
        tmp = x * ((x * x) / (24.0d0 + ((-96.0d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = x * ((x * x) / (24.0 + (-96.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.0:
		tmp = (x / (1.0 + (x * -0.5))) / x
	else:
		tmp = x * ((x * x) / (24.0 + (-96.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	else
		tmp = Float64(x * Float64(Float64(x * x) / Float64(24.0 + Float64(-96.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.0)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	else
		tmp = x * ((x * x) / (24.0 + (-96.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.0], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(N[(x * x), $MachinePrecision] / N[(24.0 + N[(-96.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 38.3%

      \[\frac{e^{x} - 1}{x} \]
   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 65.2%

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

   7. Simplified 65.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 65.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 71.3%

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

   12. Simplified 71.3%

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

   if 4 < 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 69.8%

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

   7. Simplified 69.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 69.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \color{blue}{\left(\frac{24 - 96 \cdot \frac{1}{x}}{{x}^{3}}\right)}\right), x\right) \]
   11. Step-by-step derivation

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(24, \left(96 \cdot \frac{1}{x}\right)\right), \left({x}^{3}\right)\right)\right), x\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(24, \left(\frac{96 \cdot 1}{x}\right)\right), \left({x}^{3}\right)\right)\right), x\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(24, \left(\frac{96}{x}\right)\right), \left({x}^{3}\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(24, \mathsf{/.f64}\left(96, x\right)\right), \left({x}^{3}\right)\right)\right), x\right) \]

       6. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(24, \mathsf{/.f64}\left(96, x\right)\right), \left(x \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(24, \mathsf{/.f64}\left(96, x\right)\right), \left(x \cdot {x}^{2}\right)\right)\right), x\right) \]

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 69.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(24, \mathsf{/.f64}\left(96, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), x\right) \]

   12. Simplified 69.8%

       \[\leadsto \frac{\frac{x}{\color{blue}{\frac{24 - \frac{96}{x}}{x \cdot \left(x \cdot x\right)}}}}{x} \]
   13. Step-by-step derivation

       1. div-inv N/A

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

       2. associate-/r/ N/A

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

       3. associate-*l* N/A

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

       4. cube-unmult N/A

          \[\leadsto \frac{x}{24 - \frac{96}{x}} \cdot \left({x}^{3} \cdot \frac{\color{blue}{1}}{x}\right) \]

       5. inv-pow N/A

          \[\leadsto \frac{x}{24 - \frac{96}{x}} \cdot \left({x}^{3} \cdot {x}^{\color{blue}{-1}}\right) \]

       6. pow-prod-up N/A

          \[\leadsto \frac{x}{24 - \frac{96}{x}} \cdot {x}^{\color{blue}{\left(3 + -1\right)}} \]

       7. metadata-eval N/A

          \[\leadsto \frac{x}{24 - \frac{96}{x}} \cdot {x}^{2} \]

       8. pow2 N/A

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

       9. clear-num N/A

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

       10. associate-/r/ N/A

           \[\leadsto \frac{1}{\color{blue}{\frac{\frac{24 - \frac{96}{x}}{x}}{x \cdot x}}} \]

       11. associate-/r* N/A

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

       12. clear-num N/A

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

       13. associate-/l* N/A

           \[\leadsto x \cdot \color{blue}{\frac{x \cdot x}{24 - \frac{96}{x}}} \]

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

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

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

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

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

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

       17. sub-neg N/A

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

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

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

       19. distribute-neg-frac N/A

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

       20. /-lowering-/.f64 N/A

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

       21. metadata-eval 62.6%

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

   14. Applied egg-rr 62.6%

       \[\leadsto \color{blue}{x \cdot \frac{x \cdot x}{24 + \frac{-96}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 71.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\frac{\frac{x}{1 + x \cdot -0.5}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55)
   (/ (/ x (+ 1.0 (* x -0.5))) x)
   (* x (+ (* x (+ 0.16666666666666666 (* x 0.041666666666666664))) 0.5))))

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = (x / (1.0 + (x * -0.5))) / x
	else:
		tmp = x * ((x * (0.16666666666666666 + (x * 0.041666666666666664))) + 0.5)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.55)
		tmp = (x / (1.0 + (x * -0.5))) / x;
	else
		tmp = x * ((x * (0.16666666666666666 + (x * 0.041666666666666664))) + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 38.3%

      \[\frac{e^{x} - 1}{x} \]
   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 65.2%

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

   7. Simplified 65.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 65.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 71.3%

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

   12. Simplified 71.3%

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

   if 1.55000000000000004 < 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 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 62.6%

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

4. Final simplification 68.7%

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

Alternative 11: 71.7% accurate, 7.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = (x / (1.0 + (x * -0.5))) / x;
	} 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 = (x / (1.0d0 + (x * (-0.5d0)))) / x
    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 = (x / (1.0 + (x * -0.5))) / x;
	} else {
		tmp = (x * x) * (0.16666666666666666 + (x * 0.041666666666666664));
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(Float64(x / Float64(1.0 + Float64(x * -0.5))) / x);
	else
		tmp = Float64(Float64(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 = (x / (1.0 + (x * -0.5))) / x;
	else
		tmp = (x * x) * (0.16666666666666666 + (x * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[(N[(x / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(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 38.3%

      \[\frac{e^{x} - 1}{x} \]
   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 65.2%

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

   7. Simplified 65.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

   9. Applied egg-rr 65.2%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 71.3%

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

   12. Simplified 71.3%

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

   if 1.80000000000000004 < x

   1. Initial program 100.0%

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

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

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

      2. 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 69.8%

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

   7. Simplified 69.8%

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

   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 62.6%

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

   10. Simplified 62.6%

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

4. Final simplification 68.7%

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

Alternative 12: 67.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.0)
   1.0
   (* (* x x) (+ 0.16666666666666666 (* x 0.041666666666666664)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 1.0;
	} 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 <= 2.0d0) then
        tmp = 1.0d0
    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 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * (0.16666666666666666 + (x * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 1.0
	else:
		tmp = (x * x) * (0.16666666666666666 + (x * 0.041666666666666664))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = 1.0;
	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 <= 2.0)
		tmp = 1.0;
	else
		tmp = (x * x) * (0.16666666666666666 + (x * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], 1.0, 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 < 2

   1. Initial program 38.3%

      \[\frac{e^{x} - 1}{x} \]
   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 66.2%

      \[\leadsto \color{blue}{1} \]

   if 2 < x

   1. Initial program 100.0%

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

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

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

      2. 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 69.8%

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

   7. Simplified 69.8%

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

   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 62.6%

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

   10. Simplified 62.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664\\ \end{array} \end{array} \]

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(Float64(x * 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[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999991

   1. Initial program 38.3%

      \[\frac{e^{x} - 1}{x} \]
   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 66.2%

      \[\leadsto \color{blue}{1} \]

   if 2.89999999999999991 < x

   1. Initial program 100.0%

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

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

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

      2. 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 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in x around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 62.6%

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

   10. Simplified 62.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.9% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) 1.0 (* (* x x) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = 1.0d0
    else
        tmp = (x * x) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = 1.0
	else:
		tmp = (x * x) * 0.16666666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * x) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = 1.0;
	else
		tmp = (x * x) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], 1.0, N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 38.3%

      \[\frac{e^{x} - 1}{x} \]
   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 66.2%

      \[\leadsto \color{blue}{1} \]

   if 2.39999999999999991 < x

   1. Initial program 100.0%

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

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

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

      2. 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. *-commutative N/A

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

      4. remove-double-neg N/A

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

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

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

      6. unsub-neg N/A

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval 48.9%

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

   7. Simplified 48.9%

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

   8. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 48.9%

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

   10. Simplified 48.9%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.8% accurate, 11.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (* x (* x (* x 0.041666666666666664)))))

C

double code(double x) {
	return 1.0 + (x * (x * (x * 0.041666666666666664)));
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 + (x * (x * (x * 0.041666666666666664)));
}

Python

def code(x):
	return 1.0 + (x * (x * (x * 0.041666666666666664)))

Julia

function code(x)
	return Float64(1.0 + Float64(x * Float64(x * Float64(x * 0.041666666666666664))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (x * (x * (x * 0.041666666666666664)));
end

Wolfram

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

Derivation

1. Initial program 56.6%

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

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
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 64.5%

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

7. Simplified 64.5%

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

8. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 64.0%

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

10. Simplified 64.0%

    \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)} \]
11. Add Preprocessing

Alternative 16: 51.6% 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 56.6%

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

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

7. Simplified 47.5%

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

Developer Target 1: 52.0% 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 2024149 
(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))