Kahan's exp quotient

Percentage Accurate: 53.2% → 100.0%

Time: 9.9s

Alternatives: 12

Speedup: 10.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. accelerator-lowering-expm1.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot \left(x \cdot t\_0\right)\\ t_2 := 1 + x \cdot -0.5\\ \mathbf{if}\;x \leq -5000:\\ \;\;\;\;\frac{1}{t\_2}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{1 - 0.000244140625 \cdot \left(t\_1 \cdot t\_1\right)}{1 - t\_1 \cdot -0.015625}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 0.015625 \cdot \left(t\_0 \cdot t\_0\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (* (* x x) (* x t_0)))
        (t_2 (+ 1.0 (* x -0.5))))
   (if (<= x -5000.0)
     (/ 1.0 t_2)
     (if (<= x 2.35e+51)
       (/
        (/ (- 1.0 (* 0.000244140625 (* t_1 t_1))) (- 1.0 (* t_1 -0.015625)))
        t_2)
       (/ (- 1.0 (* 0.015625 (* t_0 t_0))) t_2)))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * (x * t_0);
	double t_2 = 1.0 + (x * -0.5);
	double tmp;
	if (x <= -5000.0) {
		tmp = 1.0 / t_2;
	} else if (x <= 2.35e+51) {
		tmp = ((1.0 - (0.000244140625 * (t_1 * t_1))) / (1.0 - (t_1 * -0.015625))) / t_2;
	} else {
		tmp = (1.0 - (0.015625 * (t_0 * t_0))) / t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = (x * x) * (x * t_0)
    t_2 = 1.0d0 + (x * (-0.5d0))
    if (x <= (-5000.0d0)) then
        tmp = 1.0d0 / t_2
    else if (x <= 2.35d+51) then
        tmp = ((1.0d0 - (0.000244140625d0 * (t_1 * t_1))) / (1.0d0 - (t_1 * (-0.015625d0)))) / t_2
    else
        tmp = (1.0d0 - (0.015625d0 * (t_0 * t_0))) / t_2
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * (x * t_0);
	double t_2 = 1.0 + (x * -0.5);
	double tmp;
	if (x <= -5000.0) {
		tmp = 1.0 / t_2;
	} else if (x <= 2.35e+51) {
		tmp = ((1.0 - (0.000244140625 * (t_1 * t_1))) / (1.0 - (t_1 * -0.015625))) / t_2;
	} else {
		tmp = (1.0 - (0.015625 * (t_0 * t_0))) / t_2;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = (x * x) * (x * t_0)
	t_2 = 1.0 + (x * -0.5)
	tmp = 0
	if x <= -5000.0:
		tmp = 1.0 / t_2
	elif x <= 2.35e+51:
		tmp = ((1.0 - (0.000244140625 * (t_1 * t_1))) / (1.0 - (t_1 * -0.015625))) / t_2
	else:
		tmp = (1.0 - (0.015625 * (t_0 * t_0))) / t_2
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(x * x) * Float64(x * t_0))
	t_2 = Float64(1.0 + Float64(x * -0.5))
	tmp = 0.0
	if (x <= -5000.0)
		tmp = Float64(1.0 / t_2);
	elseif (x <= 2.35e+51)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.000244140625 * Float64(t_1 * t_1))) / Float64(1.0 - Float64(t_1 * -0.015625))) / t_2);
	else
		tmp = Float64(Float64(1.0 - Float64(0.015625 * Float64(t_0 * t_0))) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = (x * x) * (x * t_0);
	t_2 = 1.0 + (x * -0.5);
	tmp = 0.0;
	if (x <= -5000.0)
		tmp = 1.0 / t_2;
	elseif (x <= 2.35e+51)
		tmp = ((1.0 - (0.000244140625 * (t_1 * t_1))) / (1.0 - (t_1 * -0.015625))) / t_2;
	else
		tmp = (1.0 - (0.015625 * (t_0 * t_0))) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5000.0], N[(1.0 / t$95$2), $MachinePrecision], If[LessEqual[x, 2.35e+51], N[(N[(N[(1.0 - N[(0.000244140625 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(t$95$1 * -0.015625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(1.0 - N[(0.015625 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 1.6%

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

   5. Simplified 1.6%

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

   7. Applied egg-rr 0.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 1.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   10. Simplified 1.2%

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

   11. Taylor expanded in x around 0

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

   13. Simplified 18.8%

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

   if -5e3 < x < 2.3500000000000001e51

   1. Initial program 12.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 91.5%

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

   5. Simplified 91.5%

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

   7. Applied egg-rr 91.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 91.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   10. Simplified 91.7%

       \[\leadsto \frac{1 - 0.015625 \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{1 + x \cdot -0.5}} \]
   11. Step-by-step derivation

       1. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{64}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(\frac{1}{64}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{64}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{1 - \left(\mathsf{neg}\left(\frac{1}{64}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right) \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot 1 - \left(\mathsf{neg}\left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{64}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{1 - \left(\mathsf{neg}\left(\frac{1}{64}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot 1 - \left(\mathsf{neg}\left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{1 - \left(\mathsf{neg}\left(\frac{1}{64}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right) \]

       5. sqr-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot 1 - \left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{1 - \left(\mathsf{neg}\left(\frac{1}{64}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right) \]

       6. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot 1 - \left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{1 - \left(\mathsf{neg}\left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 \cdot 1 - \left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), \left(1 - \left(\mathsf{neg}\left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right) \]

   12. Applied egg-rr 97.9%

       \[\leadsto \frac{\color{blue}{\frac{1 - 0.000244140625 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{1 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot -0.015625}}}{1 + x \cdot -0.5} \]

   if 2.3500000000000001e51 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 5.4%

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

   5. Simplified 5.4%

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

   7. Applied egg-rr 9.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \frac{1 - 0.015625 \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{1 + x \cdot -0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot -0.5\\ \mathbf{if}\;x \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{t\_0}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;\frac{t\_0 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.015625\right) \cdot \left(-1 - x \cdot -0.5\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x -0.5))))
   (if (<= x -2e-5)
     (/ 1.0 t_0)
     (if (<= x 2.7e+154)
       (/
        (+
         t_0
         (* (* (* (* x x) (* x (* x (* x x)))) 0.015625) (- -1.0 (* x -0.5))))
        (* t_0 t_0))
       (* x (* x 0.16666666666666666))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * -0.5);
	double tmp;
	if (x <= -2e-5) {
		tmp = 1.0 / t_0;
	} else if (x <= 2.7e+154) {
		tmp = (t_0 + ((((x * x) * (x * (x * (x * x)))) * 0.015625) * (-1.0 - (x * -0.5)))) / (t_0 * 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 = 1.0d0 + (x * (-0.5d0))
    if (x <= (-2d-5)) then
        tmp = 1.0d0 / t_0
    else if (x <= 2.7d+154) then
        tmp = (t_0 + ((((x * x) * (x * (x * (x * x)))) * 0.015625d0) * ((-1.0d0) - (x * (-0.5d0))))) / (t_0 * t_0)
    else
        tmp = x * (x * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (x * -0.5);
	double tmp;
	if (x <= -2e-5) {
		tmp = 1.0 / t_0;
	} else if (x <= 2.7e+154) {
		tmp = (t_0 + ((((x * x) * (x * (x * (x * x)))) * 0.015625) * (-1.0 - (x * -0.5)))) / (t_0 * t_0);
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + (x * -0.5)
	tmp = 0
	if x <= -2e-5:
		tmp = 1.0 / t_0
	elif x <= 2.7e+154:
		tmp = (t_0 + ((((x * x) * (x * (x * (x * x)))) * 0.015625) * (-1.0 - (x * -0.5)))) / (t_0 * t_0)
	else:
		tmp = x * (x * 0.16666666666666666)
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * -0.5))
	tmp = 0.0
	if (x <= -2e-5)
		tmp = Float64(1.0 / t_0);
	elseif (x <= 2.7e+154)
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x)))) * 0.015625) * Float64(-1.0 - Float64(x * -0.5)))) / Float64(t_0 * t_0));
	else
		tmp = Float64(x * Float64(x * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + (x * -0.5);
	tmp = 0.0;
	if (x <= -2e-5)
		tmp = 1.0 / t_0;
	elseif (x <= 2.7e+154)
		tmp = (t_0 + ((((x * x) * (x * (x * (x * x)))) * 0.015625) * (-1.0 - (x * -0.5)))) / (t_0 * t_0);
	else
		tmp = x * (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e-5], N[(1.0 / t$95$0), $MachinePrecision], If[LessEqual[x, 2.7e+154], N[(N[(t$95$0 + N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.015625), $MachinePrecision] * N[(-1.0 - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000016e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 1.6%

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

   5. Simplified 1.6%

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

   7. Applied egg-rr 0.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 1.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   10. Simplified 1.2%

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

   11. Taylor expanded in x around 0

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

   13. Simplified 18.8%

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

   if -2.00000000000000016e-5 < x < 2.70000000000000006e154

   1. Initial program 25.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 78.5%

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

   5. Simplified 78.5%

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

   7. Applied egg-rr 81.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 92.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   10. Simplified 92.9%

       \[\leadsto \frac{1 - 0.015625 \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{1 + x \cdot -0.5}} \]
   11. Step-by-step derivation

       1. div-sub N/A

          \[\leadsto \frac{1}{1 + x \cdot \frac{-1}{2}} - \color{blue}{\frac{\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{1 + x \cdot \frac{-1}{2}}} \]

       2. frac-sub N/A

          \[\leadsto \frac{1 \cdot \left(1 + x \cdot \frac{-1}{2}\right) - \left(1 + x \cdot \frac{-1}{2}\right) \cdot \left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\color{blue}{\left(1 + x \cdot \frac{-1}{2}\right) \cdot \left(1 + x \cdot \frac{-1}{2}\right)}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot \left(1 + x \cdot \frac{-1}{2}\right) - \left(1 + x \cdot \frac{-1}{2}\right) \cdot \left(\frac{1}{64} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), \color{blue}{\left(\left(1 + x \cdot \frac{-1}{2}\right) \cdot \left(1 + x \cdot \frac{-1}{2}\right)\right)}\right) \]

   12. Applied egg-rr 95.8%

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

   if 2.70000000000000006e154 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]
   4. 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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(1 + x \cdot -0.5\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.015625\right) \cdot \left(-1 - x \cdot -0.5\right)}{\left(1 + x \cdot -0.5\right) \cdot \left(1 + x \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot -0.5\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -5000:\\ \;\;\;\;\frac{1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 0.015625 \cdot \left(t\_1 \cdot t\_1\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x -0.5))) (t_1 (* x (* x x))))
   (if (<= x -5000.0) (/ 1.0 t_0) (/ (- 1.0 (* 0.015625 (* t_1 t_1))) t_0))))

C

double code(double x) {
	double t_0 = 1.0 + (x * -0.5);
	double t_1 = x * (x * x);
	double tmp;
	if (x <= -5000.0) {
		tmp = 1.0 / t_0;
	} else {
		tmp = (1.0 - (0.015625 * (t_1 * t_1))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (-0.5d0))
    t_1 = x * (x * x)
    if (x <= (-5000.0d0)) then
        tmp = 1.0d0 / t_0
    else
        tmp = (1.0d0 - (0.015625d0 * (t_1 * t_1))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (x * -0.5);
	double t_1 = x * (x * x);
	double tmp;
	if (x <= -5000.0) {
		tmp = 1.0 / t_0;
	} else {
		tmp = (1.0 - (0.015625 * (t_1 * t_1))) / t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + (x * -0.5)
	t_1 = x * (x * x)
	tmp = 0
	if x <= -5000.0:
		tmp = 1.0 / t_0
	else:
		tmp = (1.0 - (0.015625 * (t_1 * t_1))) / t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * -0.5))
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -5000.0)
		tmp = Float64(1.0 / t_0);
	else
		tmp = Float64(Float64(1.0 - Float64(0.015625 * Float64(t_1 * t_1))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + (x * -0.5);
	t_1 = x * (x * x);
	tmp = 0.0;
	if (x <= -5000.0)
		tmp = 1.0 / t_0;
	else
		tmp = (1.0 - (0.015625 * (t_1 * t_1))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5000.0], N[(1.0 / t$95$0), $MachinePrecision], N[(N[(1.0 - N[(0.015625 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -5e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 1.6%

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

   5. Simplified 1.6%

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

   7. Applied egg-rr 0.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 1.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   10. Simplified 1.2%

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

   11. Taylor expanded in x around 0

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

   13. Simplified 18.8%

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

   if -5e3 < x

   1. Initial program 37.3%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 67.2%

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

   5. Simplified 67.2%

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

   7. Applied egg-rr 68.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 94.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   10. Simplified 94.0%

       \[\leadsto \frac{1 - 0.015625 \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{1 + x \cdot -0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.7% accurate, 4.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (/ 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 <= -1.55) {
		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 <= (-1.55d0)) 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 <= -1.55) {
		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 <= -1.55:
		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 <= -1.55)
		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 <= -1.55)
		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, -1.55], 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 < -1.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 1.6%

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

   5. Simplified 1.6%

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

   7. Applied egg-rr 0.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 1.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   10. Simplified 1.2%

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

   11. Taylor expanded in x around 0

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

   13. Simplified 18.8%

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

   if -1.55000000000000004 < x

   1. Initial program 37.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)}, x\right) \]
   4. 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 88.5%

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

   5. Simplified 88.5%

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

Alternative 6: 73.4% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.94999999999999996

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 66.3%

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

   5. Simplified 66.3%

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

   7. Applied egg-rr 65.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 66.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   10. Simplified 66.2%

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

   11. Taylor expanded in x around 0

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

   13. Simplified 72.2%

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

   if 1.94999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)}, x\right) \]
   4. 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 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 66.8%

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

   8. Simplified 66.8%

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

4. Final simplification 70.8%

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

Alternative 7: 71.3% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 66.3%

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

   5. Simplified 66.3%

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

   7. Applied egg-rr 65.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 66.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   10. Simplified 66.2%

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

   11. Taylor expanded in x around 0

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

   13. Simplified 72.2%

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

   if 1.80000000000000004 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)}, x\right) \]
   4. 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 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)} \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*l* N/A

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

      10. lft-mult-inverse N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 58.4%

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

   8. Simplified 58.4%

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

Alternative 8: 71.3% 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 37.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 66.3%

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

   5. Simplified 66.3%

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

   7. Applied egg-rr 65.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 66.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   10. Simplified 66.2%

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

   11. Taylor expanded in x around 0

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

   13. Simplified 72.2%

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

   if 1.94999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)}, x\right) \]
   4. 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 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 58.4%

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

   8. Simplified 58.4%

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

4. Final simplification 68.6%

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

Alternative 9: 67.4% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999991

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.2%

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

   if 2.89999999999999991 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)}, x\right) \]
   4. 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 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 58.4%

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

   8. Simplified 58.4%

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

4. Final simplification 64.9%

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

Alternative 10: 63.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.2%

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

   if 2.39999999999999991 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]
   4. 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 48.1%

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

   5. Simplified 48.1%

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

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 48.1%

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

   8. Simplified 48.1%

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

Alternative 11: 51.1% accurate, 105.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.7%

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

Alternative 12: 3.3% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 3.4%

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

   1. metadata-eval N/A

      \[\leadsto \frac{0}{x} \]

   2. div0 3.4%

      \[\leadsto 0 \]

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 52.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :alt
  (! :herbie-platform default (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x)))

  (/ (- (exp x) 1.0) x))