Kahan's exp quotient

Percentage Accurate: 52.6% → 100.0%

Time: 10.7s

Alternatives: 14

Speedup: 10.5×

• 24.9% 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

\[\begin{array}{l} \\ \frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ x (expm1 x))))

C

double code(double x) {
	return 1.0 / (x / expm1(x));
}

Java

public static double code(double x) {
	return 1.0 / (x / Math.expm1(x));
}

Python

def code(x):
	return 1.0 / (x / math.expm1(x))

Julia

function code(x)
	return Float64(1.0 / Float64(x / expm1(x)))
end

Wolfram

code[x_] := N[(1.0 / N[(x / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.8%

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

   1. clear-num N/A

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

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

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

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

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

   4. expm1-define N/A

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

   5. expm1-lowering-expm1.f64 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}} \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.8%

   \[\frac{e^{x} - 1}{x} \]
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 3: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot x - t\_0 \cdot t\_0}{x - t\_0}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          (* x x)
          (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664)))))))
   (if (<= x 2e-34)
     (/ 1.0 (+ 1.0 (* x -0.5)))
     (if (<= x 2.55e+77)
       (/ (/ (- (* x x) (* t_0 t_0)) (- x t_0)) x)
       (/ (* x (* x (* x (* x 0.041666666666666664)))) x)))))

C

double code(double x) {
	double t_0 = (x * x) * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))));
	double tmp;
	if (x <= 2e-34) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 2.55e+77) {
		tmp = (((x * x) - (t_0 * t_0)) / (x - t_0)) / x;
	} else {
		tmp = (x * (x * (x * (x * 0.041666666666666664)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))))
    if (x <= 2d-34) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else if (x <= 2.55d+77) then
        tmp = (((x * x) - (t_0 * t_0)) / (x - t_0)) / x
    else
        tmp = (x * (x * (x * (x * 0.041666666666666664d0)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))));
	double tmp;
	if (x <= 2e-34) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 2.55e+77) {
		tmp = (((x * x) - (t_0 * t_0)) / (x - t_0)) / x;
	} else {
		tmp = (x * (x * (x * (x * 0.041666666666666664)))) / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))
	tmp = 0
	if x <= 2e-34:
		tmp = 1.0 / (1.0 + (x * -0.5))
	elif x <= 2.55e+77:
		tmp = (((x * x) - (t_0 * t_0)) / (x - t_0)) / x
	else:
		tmp = (x * (x * (x * (x * 0.041666666666666664)))) / x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664)))))
	tmp = 0.0
	if (x <= 2e-34)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	elseif (x <= 2.55e+77)
		tmp = Float64(Float64(Float64(Float64(x * x) - Float64(t_0 * t_0)) / Float64(x - t_0)) / x);
	else
		tmp = Float64(Float64(x * Float64(x * Float64(x * Float64(x * 0.041666666666666664)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))));
	tmp = 0.0;
	if (x <= 2e-34)
		tmp = 1.0 / (1.0 + (x * -0.5));
	elseif (x <= 2.55e+77)
		tmp = (((x * x) - (t_0 * t_0)) / (x - t_0)) / x;
	else
		tmp = (x * (x * (x * (x * 0.041666666666666664)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e-34], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e+77], N[(N[(N[(N[(x * x), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x - t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.99999999999999986e-34

   1. Initial program 39.9%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

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

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

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

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

      4. expm1-define N/A

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

      5. expm1-lowering-expm1.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 69.8%

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

   9. Simplified 69.8%

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

   if 1.99999999999999986e-34 < x < 2.54999999999999985e77

   1. Initial program 75.4%

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

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   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 36.5%

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

   7. Simplified 36.5%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. flip-+ N/A

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

      4. *-rgt-identity N/A

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

      5. fmm-def N/A

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

      6. *-commutative N/A

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

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

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

   9. Applied egg-rr 71.9%

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

   if 2.54999999999999985e77 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot x - \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}{x - \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + x \cdot 0.041666666666666664\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 10^{+77}:\\ \;\;\;\;1 + \frac{x \cdot \left(0.125 + t\_1 \cdot \left(t\_0 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)\right)}{0.25 + t\_1 \cdot \left(t\_1 - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.16666666666666666 (* x 0.041666666666666664)))
        (t_1 (* x t_0)))
   (if (<= x -1.52)
     (/ 1.0 (+ 1.0 (* x -0.5)))
     (if (<= x 1e+77)
       (+
        1.0
        (/
         (* x (+ 0.125 (* t_1 (* t_0 (* (* x x) t_0)))))
         (+ 0.25 (* t_1 (- t_1 0.5)))))
       (/ (* x (* x (* x (* x 0.041666666666666664)))) x)))))

C

double code(double x) {
	double t_0 = 0.16666666666666666 + (x * 0.041666666666666664);
	double t_1 = x * t_0;
	double tmp;
	if (x <= -1.52) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 1e+77) {
		tmp = 1.0 + ((x * (0.125 + (t_1 * (t_0 * ((x * x) * t_0))))) / (0.25 + (t_1 * (t_1 - 0.5))));
	} else {
		tmp = (x * (x * (x * (x * 0.041666666666666664)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 + (x * 0.041666666666666664d0)
    t_1 = x * t_0
    if (x <= (-1.52d0)) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else if (x <= 1d+77) then
        tmp = 1.0d0 + ((x * (0.125d0 + (t_1 * (t_0 * ((x * x) * t_0))))) / (0.25d0 + (t_1 * (t_1 - 0.5d0))))
    else
        tmp = (x * (x * (x * (x * 0.041666666666666664d0)))) / x
    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 tmp;
	if (x <= -1.52) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 1e+77) {
		tmp = 1.0 + ((x * (0.125 + (t_1 * (t_0 * ((x * x) * t_0))))) / (0.25 + (t_1 * (t_1 - 0.5))));
	} else {
		tmp = (x * (x * (x * (x * 0.041666666666666664)))) / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.16666666666666666 + (x * 0.041666666666666664)
	t_1 = x * t_0
	tmp = 0
	if x <= -1.52:
		tmp = 1.0 / (1.0 + (x * -0.5))
	elif x <= 1e+77:
		tmp = 1.0 + ((x * (0.125 + (t_1 * (t_0 * ((x * x) * t_0))))) / (0.25 + (t_1 * (t_1 - 0.5))))
	else:
		tmp = (x * (x * (x * (x * 0.041666666666666664)))) / x
	return tmp

Julia

function code(x)
	t_0 = Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= -1.52)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	elseif (x <= 1e+77)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(0.125 + Float64(t_1 * Float64(t_0 * Float64(Float64(x * x) * t_0))))) / Float64(0.25 + Float64(t_1 * Float64(t_1 - 0.5)))));
	else
		tmp = Float64(Float64(x * Float64(x * Float64(x * Float64(x * 0.041666666666666664)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.16666666666666666 + (x * 0.041666666666666664);
	t_1 = x * t_0;
	tmp = 0.0;
	if (x <= -1.52)
		tmp = 1.0 / (1.0 + (x * -0.5));
	elseif (x <= 1e+77)
		tmp = 1.0 + ((x * (0.125 + (t_1 * (t_0 * ((x * x) * t_0))))) / (0.25 + (t_1 * (t_1 - 0.5))));
	else
		tmp = (x * (x * (x * (x * 0.041666666666666664)))) / x;
	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]}, If[LessEqual[x, -1.52], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+77], N[(1.0 + N[(N[(x * N[(0.125 + N[(t$95$1 * N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(t$95$1 * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.52

   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. Step-by-step derivation

      1. clear-num N/A

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

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

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

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

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

      4. expm1-define N/A

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

      5. expm1-lowering-expm1.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 18.8%

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

   9. Simplified 18.8%

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

   if -1.52 < x < 9.99999999999999983e76

   1. Initial program 17.3%

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

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
   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 89.0%

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

   7. Simplified 89.0%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)} \]
   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 94.2%

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

   if 9.99999999999999983e76 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 10^{+77}:\\ \;\;\;\;1 + \frac{x \cdot \left(0.125 + \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(\left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)}{0.25 + \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - \left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_0\right)\right)}{1 - x \cdot t\_0}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))
   (if (<= x -1.52)
     (/ 1.0 (+ 1.0 (* x -0.5)))
     (if (<= x 1.65e+103)
       (/ (/ (* x (- 1.0 (* (* x x) (* t_0 t_0)))) (- 1.0 (* x t_0))) x)
       (* x (* (* x x) 0.041666666666666664))))))

C

double code(double x) {
	double t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	double tmp;
	if (x <= -1.52) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 1.65e+103) {
		tmp = ((x * (1.0 - ((x * x) * (t_0 * t_0)))) / (1.0 - (x * t_0))) / x;
	} else {
		tmp = x * ((x * x) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0)))
    if (x <= (-1.52d0)) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else if (x <= 1.65d+103) then
        tmp = ((x * (1.0d0 - ((x * x) * (t_0 * t_0)))) / (1.0d0 - (x * t_0))) / x
    else
        tmp = x * ((x * x) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	double tmp;
	if (x <= -1.52) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 1.65e+103) {
		tmp = ((x * (1.0 - ((x * x) * (t_0 * t_0)))) / (1.0 - (x * t_0))) / x;
	} else {
		tmp = x * ((x * x) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))
	tmp = 0
	if x <= -1.52:
		tmp = 1.0 / (1.0 + (x * -0.5))
	elif x <= 1.65e+103:
		tmp = ((x * (1.0 - ((x * x) * (t_0 * t_0)))) / (1.0 - (x * t_0))) / x
	else:
		tmp = x * ((x * x) * 0.041666666666666664)
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))
	tmp = 0.0
	if (x <= -1.52)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	elseif (x <= 1.65e+103)
		tmp = Float64(Float64(Float64(x * Float64(1.0 - Float64(Float64(x * x) * Float64(t_0 * t_0)))) / Float64(1.0 - Float64(x * t_0))) / x);
	else
		tmp = Float64(x * Float64(Float64(x * x) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	tmp = 0.0;
	if (x <= -1.52)
		tmp = 1.0 / (1.0 + (x * -0.5));
	elseif (x <= 1.65e+103)
		tmp = ((x * (1.0 - ((x * x) * (t_0 * t_0)))) / (1.0 - (x * t_0))) / x;
	else
		tmp = x * ((x * x) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.52], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+103], N[(N[(N[(x * N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.52

   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. Step-by-step derivation

      1. clear-num N/A

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

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

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

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

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

      4. expm1-define N/A

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

      5. expm1-lowering-expm1.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 18.8%

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

   9. Simplified 18.8%

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

   if -1.52 < x < 1.65000000000000004e103

   1. Initial program 19.5%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 89.3%

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

   7. Simplified 89.3%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 94.3%

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

   if 1.65000000000000004e103 < x

   1. Initial program 100.0%

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

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

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

      2. 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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - \left(x \cdot x\right) \cdot \left(\left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)}{1 - x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + x \cdot 0.041666666666666664\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{x + \frac{\left(x \cdot x\right) \cdot \left(0.25 - t\_0 \cdot \left(x \cdot t\_1\right)\right)}{0.5 - t\_1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.16666666666666666 (* x 0.041666666666666664)))
        (t_1 (* x t_0)))
   (if (<= x -1.52)
     (/ 1.0 (+ 1.0 (* x -0.5)))
     (if (<= x 2e+148)
       (/ (+ x (/ (* (* x x) (- 0.25 (* t_0 (* x t_1)))) (- 0.5 t_1))) x)
       (* (* x x) 0.16666666666666666)))))

C

double code(double x) {
	double t_0 = 0.16666666666666666 + (x * 0.041666666666666664);
	double t_1 = x * t_0;
	double tmp;
	if (x <= -1.52) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 2e+148) {
		tmp = (x + (((x * x) * (0.25 - (t_0 * (x * t_1)))) / (0.5 - t_1))) / x;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 + (x * 0.041666666666666664d0)
    t_1 = x * t_0
    if (x <= (-1.52d0)) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else if (x <= 2d+148) then
        tmp = (x + (((x * x) * (0.25d0 - (t_0 * (x * t_1)))) / (0.5d0 - t_1))) / x
    else
        tmp = (x * x) * 0.16666666666666666d0
    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 tmp;
	if (x <= -1.52) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 2e+148) {
		tmp = (x + (((x * x) * (0.25 - (t_0 * (x * t_1)))) / (0.5 - t_1))) / x;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.16666666666666666 + (x * 0.041666666666666664)
	t_1 = x * t_0
	tmp = 0
	if x <= -1.52:
		tmp = 1.0 / (1.0 + (x * -0.5))
	elif x <= 2e+148:
		tmp = (x + (((x * x) * (0.25 - (t_0 * (x * t_1)))) / (0.5 - t_1))) / x
	else:
		tmp = (x * x) * 0.16666666666666666
	return tmp

Julia

function code(x)
	t_0 = Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= -1.52)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	elseif (x <= 2e+148)
		tmp = Float64(Float64(x + Float64(Float64(Float64(x * x) * Float64(0.25 - Float64(t_0 * Float64(x * t_1)))) / Float64(0.5 - t_1))) / x);
	else
		tmp = Float64(Float64(x * x) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.16666666666666666 + (x * 0.041666666666666664);
	t_1 = x * t_0;
	tmp = 0.0;
	if (x <= -1.52)
		tmp = 1.0 / (1.0 + (x * -0.5));
	elseif (x <= 2e+148)
		tmp = (x + (((x * x) * (0.25 - (t_0 * (x * t_1)))) / (0.5 - t_1))) / x;
	else
		tmp = (x * x) * 0.16666666666666666;
	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]}, If[LessEqual[x, -1.52], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+148], N[(N[(x + N[(N[(N[(x * x), $MachinePrecision] * N[(0.25 - N[(t$95$0 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.52

   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. Step-by-step derivation

      1. clear-num N/A

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

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

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

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

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

      4. expm1-define N/A

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

      5. expm1-lowering-expm1.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 18.8%

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

   9. Simplified 18.8%

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

   if -1.52 < x < 2.0000000000000001e148

   1. Initial program 23.1%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 89.8%

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

   7. Simplified 89.8%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 89.8%

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

   9. Applied egg-rr 89.8%

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

       1. flip-+ N/A

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

       2. associate-*l/ N/A

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

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

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

   11. Applied egg-rr 94.0%

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

   if 2.0000000000000001e148 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. *-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 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{x + \frac{\left(x \cdot x\right) \cdot \left(0.25 - \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)}{0.5 - x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + x \cdot 0.041666666666666664\\ \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+148}:\\ \;\;\;\;1 + \frac{x \cdot \left(0.25 - t\_0 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)}{0.5 - x \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.16666666666666666 (* x 0.041666666666666664))))
   (if (<= x -1.52)
     (/ 1.0 (+ 1.0 (* x -0.5)))
     (if (<= x 2e+148)
       (+ 1.0 (/ (* x (- 0.25 (* t_0 (* (* x x) t_0)))) (- 0.5 (* x t_0))))
       (* (* x x) 0.16666666666666666)))))

C

double code(double x) {
	double t_0 = 0.16666666666666666 + (x * 0.041666666666666664);
	double tmp;
	if (x <= -1.52) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 2e+148) {
		tmp = 1.0 + ((x * (0.25 - (t_0 * ((x * x) * t_0)))) / (0.5 - (x * t_0)));
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 + (x * 0.041666666666666664d0)
    if (x <= (-1.52d0)) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else if (x <= 2d+148) then
        tmp = 1.0d0 + ((x * (0.25d0 - (t_0 * ((x * x) * t_0)))) / (0.5d0 - (x * t_0)))
    else
        tmp = (x * x) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.16666666666666666 + (x * 0.041666666666666664);
	double tmp;
	if (x <= -1.52) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else if (x <= 2e+148) {
		tmp = 1.0 + ((x * (0.25 - (t_0 * ((x * x) * t_0)))) / (0.5 - (x * t_0)));
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.16666666666666666 + (x * 0.041666666666666664)
	tmp = 0
	if x <= -1.52:
		tmp = 1.0 / (1.0 + (x * -0.5))
	elif x <= 2e+148:
		tmp = 1.0 + ((x * (0.25 - (t_0 * ((x * x) * t_0)))) / (0.5 - (x * t_0)))
	else:
		tmp = (x * x) * 0.16666666666666666
	return tmp

Julia

function code(x)
	t_0 = Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))
	tmp = 0.0
	if (x <= -1.52)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	elseif (x <= 2e+148)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(0.25 - Float64(t_0 * Float64(Float64(x * x) * t_0)))) / Float64(0.5 - Float64(x * t_0))));
	else
		tmp = Float64(Float64(x * x) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.16666666666666666 + (x * 0.041666666666666664);
	tmp = 0.0;
	if (x <= -1.52)
		tmp = 1.0 / (1.0 + (x * -0.5));
	elseif (x <= 2e+148)
		tmp = 1.0 + ((x * (0.25 - (t_0 * ((x * x) * t_0)))) / (0.5 - (x * t_0)));
	else
		tmp = (x * x) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.52], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+148], N[(1.0 + N[(N[(x * N[(0.25 - N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.52

   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. Step-by-step derivation

      1. clear-num N/A

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

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

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

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

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

      4. expm1-define N/A

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

      5. expm1-lowering-expm1.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 18.8%

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

   9. Simplified 18.8%

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

   if -1.52 < x < 2.0000000000000001e148

   1. Initial program 23.1%

      \[\frac{e^{x} - 1}{x} \]
   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 87.4%

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

   7. Simplified 87.4%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)} \]
   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. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 93.4%

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

   if 2.0000000000000001e148 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. *-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 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+148}:\\ \;\;\;\;1 + \frac{x \cdot \left(0.25 - \left(0.16666666666666666 + x \cdot 0.041666666666666664\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}{0.5 - x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.52)
   (/ 1.0 (+ 1.0 (* x -0.5)))
   (/
    (+
     x
     (*
      (* x x)
      (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.52) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = (x + ((x * x) * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.52d0)) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = (x + ((x * x) * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0)))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.52) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = (x + ((x * x) * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.52:
		tmp = 1.0 / (1.0 + (x * -0.5))
	else:
		tmp = (x + ((x * x) * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.52)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64(Float64(x + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664)))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.52)
		tmp = 1.0 / (1.0 + (x * -0.5));
	else
		tmp = (x + ((x * x) * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.52], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.52

   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. Step-by-step derivation

      1. clear-num N/A

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

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

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

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

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

      4. expm1-define N/A

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

      5. expm1-lowering-expm1.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 18.8%

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

   9. Simplified 18.8%

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

   if -1.52 < x

   1. Initial program 35.4%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 91.4%

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

   7. Simplified 91.4%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 91.4%

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

   9. Applied egg-rr 91.4%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.52)
   (/ 1.0 (+ 1.0 (* x -0.5)))
   (/
    (*
     x
     (+
      1.0
      (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664)))))))
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.52) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.52d0)) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = (x * (1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.52) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.52:
		tmp = 1.0 / (1.0 + (x * -0.5))
	else:
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.52)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64(Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.52)
		tmp = 1.0 / (1.0 + (x * -0.5));
	else
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.52], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.52

   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. Step-by-step derivation

      1. clear-num N/A

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

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

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

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

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

      4. expm1-define N/A

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

      5. expm1-lowering-expm1.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 18.8%

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

   9. Simplified 18.8%

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

   if -1.52 < x

   1. Initial program 35.4%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 91.4%

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

   7. Simplified 91.4%

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

Alternative 10: 74.1% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.95d0) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = (x * (x * (x * (x * 0.041666666666666664d0)))) / x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.95], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.94999999999999996

   1. Initial program 39.6%

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

      1. clear-num N/A

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

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

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

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

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

      4. expm1-define N/A

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

      5. expm1-lowering-expm1.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 70.5%

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

   9. Simplified 70.5%

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

   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 74.5%

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

   7. Simplified 74.5%

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

   8. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 74.5%

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

   10. Simplified 74.5%

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

Alternative 11: 72.4% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.95d0) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = x * ((x * x) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.95], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.94999999999999996

   1. Initial program 39.6%

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

      1. clear-num N/A

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

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

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

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

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

      4. expm1-define N/A

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

      5. expm1-lowering-expm1.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 70.5%

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

   9. Simplified 70.5%

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

   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

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 67.8%

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

   7. Simplified 67.8%

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

   8. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 67.8%

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

   10. Simplified 67.8%

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

4. Final simplification 69.9%

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

Alternative 12: 68.6% accurate, 8.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.9) 1.0 (* x (* (* x x) 0.041666666666666664))))

C

double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = 1.0;
	} else {
		tmp = x * ((x * x) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.9d0) then
        tmp = 1.0d0
    else
        tmp = x * ((x * x) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = 1.0;
	} else {
		tmp = x * ((x * x) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.9:
		tmp = 1.0
	else:
		tmp = x * ((x * x) * 0.041666666666666664)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.9)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(Float64(x * x) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.9)
		tmp = 1.0;
	else
		tmp = x * ((x * x) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.9], 1.0, N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999991

   1. Initial program 39.6%

      \[\frac{e^{x} - 1}{x} \]
   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 65.0%

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

   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

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 67.8%

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

   7. Simplified 67.8%

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

   8. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 67.8%

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

   10. Simplified 67.8%

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

4. Final simplification 65.6%

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

Alternative 13: 64.6% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.5) 1.0 (* (* x x) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if (x <= 2.5) {
		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 <= 2.5d0) 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 <= 2.5) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.5:
		tmp = 1.0
	else:
		tmp = (x * x) * 0.16666666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.5)
		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 <= 2.5)
		tmp = 1.0;
	else
		tmp = (x * x) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.5], 1.0, N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 39.6%

      \[\frac{e^{x} - 1}{x} \]
   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 65.0%

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

   if 2.5 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 55.8%

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

   7. Simplified 55.8%

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

   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 55.8%

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

   10. Simplified 55.8%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 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 52.8%

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

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

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

   2. expm1-define N/A

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

   3. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 51.5%

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

Developer Target 1: 52.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - 1\\ \mathbf{if}\;x < 1 \land x > -1:\\ \;\;\;\;\frac{t\_0}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) 1.0)))
   (if (and (< x 1.0) (> x -1.0)) (/ t_0 (log (exp x))) (/ t_0 x))))

C

double code(double x) {
	double t_0 = exp(x) - 1.0;
	double tmp;
	if ((x < 1.0) && (x > -1.0)) {
		tmp = t_0 / log(exp(x));
	} else {
		tmp = t_0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(x) - 1.0d0
    if ((x < 1.0d0) .and. (x > (-1.0d0))) then
        tmp = t_0 / log(exp(x))
    else
        tmp = t_0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(x) - 1.0;
	double tmp;
	if ((x < 1.0) && (x > -1.0)) {
		tmp = t_0 / Math.log(Math.exp(x));
	} else {
		tmp = t_0 / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(x) - 1.0
	tmp = 0
	if (x < 1.0) and (x > -1.0):
		tmp = t_0 / math.log(math.exp(x))
	else:
		tmp = t_0 / x
	return tmp

Julia

function code(x)
	t_0 = Float64(exp(x) - 1.0)
	tmp = 0.0
	if ((x < 1.0) && (x > -1.0))
		tmp = Float64(t_0 / log(exp(x)));
	else
		tmp = Float64(t_0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(x) - 1.0;
	tmp = 0.0;
	if ((x < 1.0) && (x > -1.0))
		tmp = t_0 / log(exp(x));
	else
		tmp = t_0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[And[Less[x, 1.0], Greater[x, -1.0]], N[(t$95$0 / N[Log[N[Exp[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024163 
(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))