Kahan's exp quotient

Percentage Accurate: 53.7% → 100.0%

Time: 10.0s

Alternatives: 14

Speedup: 11.7×

• 25.0% 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: 53.7% 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 55.1%

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

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

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

   2. expm1-define N/A

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

   3. expm1-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 73.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 + x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\\ t_1 := x \cdot t\_0 + -1\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{x}{x \cdot \left(1 + x \cdot -0.5\right)}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{\frac{1}{\frac{1}{x} - t\_0} \cdot \frac{\frac{1}{x}}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+ -0.5 (* x (+ (* x -0.041666666666666664) -0.16666666666666666))))
        (t_1 (+ (* x t_0) -1.0)))
   (if (<= x -1.5)
     (/ x (* x (+ 1.0 (* x -0.5))))
     (if (<= x 1.65e+103)
       (/ (/ 1.0 t_1) (* (/ 1.0 (- (/ 1.0 x) t_0)) (/ (/ 1.0 x) t_1)))
       (* x (* 0.041666666666666664 (* x x)))))))

C

double code(double x) {
	double t_0 = -0.5 + (x * ((x * -0.041666666666666664) + -0.16666666666666666));
	double t_1 = (x * t_0) + -1.0;
	double tmp;
	if (x <= -1.5) {
		tmp = x / (x * (1.0 + (x * -0.5)));
	} else if (x <= 1.65e+103) {
		tmp = (1.0 / t_1) / ((1.0 / ((1.0 / x) - t_0)) * ((1.0 / x) / t_1));
	} else {
		tmp = x * (0.041666666666666664 * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) + (x * ((x * (-0.041666666666666664d0)) + (-0.16666666666666666d0)))
    t_1 = (x * t_0) + (-1.0d0)
    if (x <= (-1.5d0)) then
        tmp = x / (x * (1.0d0 + (x * (-0.5d0))))
    else if (x <= 1.65d+103) then
        tmp = (1.0d0 / t_1) / ((1.0d0 / ((1.0d0 / x) - t_0)) * ((1.0d0 / x) / t_1))
    else
        tmp = x * (0.041666666666666664d0 * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -0.5 + (x * ((x * -0.041666666666666664) + -0.16666666666666666));
	double t_1 = (x * t_0) + -1.0;
	double tmp;
	if (x <= -1.5) {
		tmp = x / (x * (1.0 + (x * -0.5)));
	} else if (x <= 1.65e+103) {
		tmp = (1.0 / t_1) / ((1.0 / ((1.0 / x) - t_0)) * ((1.0 / x) / t_1));
	} else {
		tmp = x * (0.041666666666666664 * (x * x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = -0.5 + (x * ((x * -0.041666666666666664) + -0.16666666666666666))
	t_1 = (x * t_0) + -1.0
	tmp = 0
	if x <= -1.5:
		tmp = x / (x * (1.0 + (x * -0.5)))
	elif x <= 1.65e+103:
		tmp = (1.0 / t_1) / ((1.0 / ((1.0 / x) - t_0)) * ((1.0 / x) / t_1))
	else:
		tmp = x * (0.041666666666666664 * (x * x))
	return tmp

Julia

function code(x)
	t_0 = Float64(-0.5 + Float64(x * Float64(Float64(x * -0.041666666666666664) + -0.16666666666666666)))
	t_1 = Float64(Float64(x * t_0) + -1.0)
	tmp = 0.0
	if (x <= -1.5)
		tmp = Float64(x / Float64(x * Float64(1.0 + Float64(x * -0.5))));
	elseif (x <= 1.65e+103)
		tmp = Float64(Float64(1.0 / t_1) / Float64(Float64(1.0 / Float64(Float64(1.0 / x) - t_0)) * Float64(Float64(1.0 / x) / t_1)));
	else
		tmp = Float64(x * Float64(0.041666666666666664 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -0.5 + (x * ((x * -0.041666666666666664) + -0.16666666666666666));
	t_1 = (x * t_0) + -1.0;
	tmp = 0.0;
	if (x <= -1.5)
		tmp = x / (x * (1.0 + (x * -0.5)));
	elseif (x <= 1.65e+103)
		tmp = (1.0 / t_1) / ((1.0 / ((1.0 / x) - t_0)) * ((1.0 / x) / t_1));
	else
		tmp = x * (0.041666666666666664 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 + N[(x * N[(N[(x * -0.041666666666666664), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -1.5], N[(x / N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+103], N[(N[(1.0 / t$95$1), $MachinePrecision] / N[(N[(1.0 / N[(N[(1.0 / x), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 1.2%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 1.2%

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

   9. Applied egg-rr 1.2%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

       6. sub-neg N/A

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

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

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

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

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

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

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

       10. +-commutative N/A

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

       11. distribute-neg-in N/A

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

       12. metadata-eval N/A

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

       13. distribute-frac-neg2 N/A

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

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

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

       15. distribute-frac-neg2 N/A

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

       16. distribute-frac-neg N/A

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

   11. Applied egg-rr 1.2%

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

   12. Taylor expanded in x around 0

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

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

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 13.1%

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

   14. Simplified 13.1%

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

   if -1.5 < x < 1.65000000000000004e103

   1. Initial program 22.4%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 86.4%

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

   9. Applied egg-rr 89.5%

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

   if 1.65000000000000004e103 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 100.0%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{x}{x \cdot \left(1 + x \cdot -0.5\right)}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(-0.5 + x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right) + -1}}{\frac{1}{\frac{1}{x} - \left(-0.5 + x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)} \cdot \frac{\frac{1}{x}}{x \cdot \left(-0.5 + x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right)\right) + -1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.6% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.5

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 1.2%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 1.2%

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

   9. Applied egg-rr 1.2%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

       6. sub-neg N/A

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

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

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

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

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

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

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

       10. +-commutative N/A

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

       11. distribute-neg-in N/A

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

       12. metadata-eval N/A

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

       13. distribute-frac-neg2 N/A

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

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

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

       15. distribute-frac-neg2 N/A

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

       16. distribute-frac-neg N/A

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

   11. Applied egg-rr 1.2%

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

   12. Taylor expanded in x around 0

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

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

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 13.1%

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

   14. Simplified 13.1%

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

   if -1.5 < x

   1. Initial program 40.2%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 89.5%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 89.5%

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

   9. Applied egg-rr 89.5%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. sub-neg N/A

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

       4. distribute-lft-in N/A

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

       5. *-rgt-identity N/A

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

       6. distribute-lft-in N/A

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

       7. *-commutative N/A

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

       8. rgt-mult-inverse N/A

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

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

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

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

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

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

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

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

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

       13. associate-*r* N/A

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

   11. Applied egg-rr 89.5%

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

Alternative 4: 71.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.5

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 1.2%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 1.2%

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

   9. Applied egg-rr 1.2%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

       6. sub-neg N/A

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

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

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

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

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

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

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

       10. +-commutative N/A

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

       11. distribute-neg-in N/A

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

       12. metadata-eval N/A

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

       13. distribute-frac-neg2 N/A

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

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

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

       15. distribute-frac-neg2 N/A

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

       16. distribute-frac-neg N/A

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

   11. Applied egg-rr 1.2%

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

   12. Taylor expanded in x around 0

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

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

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 13.1%

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

   14. Simplified 13.1%

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

   if -1.5 < x

   1. Initial program 40.2%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 89.5%

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

4. Final simplification 70.4%

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

Alternative 5: 71.4% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.94999999999999996

   1. Initial program 38.6%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 66.2%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 66.2%

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

   9. Applied egg-rr 66.2%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

       6. sub-neg N/A

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

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

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

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

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

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

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

       10. +-commutative N/A

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

       11. distribute-neg-in N/A

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

       12. metadata-eval N/A

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

       13. distribute-frac-neg2 N/A

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

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

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

       15. distribute-frac-neg2 N/A

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

       16. distribute-frac-neg N/A

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

   11. Applied egg-rr 66.2%

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

   12. Taylor expanded in x around 0

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

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

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 69.9%

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

   14. Simplified 69.9%

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

   if 1.94999999999999996 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 70.8%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 70.8%

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

   9. Applied egg-rr 70.8%

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

   10. Taylor expanded in x around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 70.8%

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

   12. Simplified 70.8%

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

Alternative 6: 69.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{x}{x \cdot \left(1 + x \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\frac{24 + \frac{-96}{x}}{x}}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.0)
   (/ x (* x (+ 1.0 (* x -0.5))))
   (/ x (/ (/ (+ 24.0 (/ -96.0 x)) x) x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = x / (x * (1.0 + (x * -0.5)));
	} else {
		tmp = x / (((24.0 + (-96.0 / x)) / x) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = x / (x * (1.0d0 + (x * (-0.5d0))))
    else
        tmp = x / (((24.0d0 + ((-96.0d0) / x)) / x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = x / (x * (1.0 + (x * -0.5)));
	} else {
		tmp = x / (((24.0 + (-96.0 / x)) / x) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = x / (x * (1.0 + (x * -0.5)))
	else:
		tmp = x / (((24.0 + (-96.0 / x)) / x) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(x / Float64(x * Float64(1.0 + Float64(x * -0.5))));
	else
		tmp = Float64(x / Float64(Float64(Float64(24.0 + Float64(-96.0 / x)) / x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = x / (x * (1.0 + (x * -0.5)));
	else
		tmp = x / (((24.0 + (-96.0 / x)) / x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[(x / N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[(24.0 + N[(-96.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 38.6%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 66.2%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 66.2%

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

   9. Applied egg-rr 66.2%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

       6. sub-neg N/A

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

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

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

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

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

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

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

       10. +-commutative N/A

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

       11. distribute-neg-in N/A

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

       12. metadata-eval N/A

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

       13. distribute-frac-neg2 N/A

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

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

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

       15. distribute-frac-neg2 N/A

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

       16. distribute-frac-neg N/A

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

   11. Applied egg-rr 66.2%

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

   12. Taylor expanded in x around 0

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

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

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 69.9%

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

   14. Simplified 69.9%

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

   if 2 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 70.8%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 70.8%

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

   9. Applied egg-rr 70.8%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

       6. sub-neg N/A

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

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

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

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

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

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

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

       10. +-commutative N/A

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

       11. distribute-neg-in N/A

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

       12. metadata-eval N/A

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

       13. distribute-frac-neg2 N/A

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

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

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

       15. distribute-frac-neg2 N/A

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

       16. distribute-frac-neg N/A

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

   11. Applied egg-rr 65.5%

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

   12. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{24 - 96 \cdot \frac{1}{x}}{{x}^{2}}\right)}\right) \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. sub-neg N/A

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-frac N/A

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

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

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

       11. metadata-eval 65.5%

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

   14. Simplified 65.5%

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

Alternative 7: 69.4% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.52

   1. Initial program 38.6%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 66.2%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 66.2%

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

   9. Applied egg-rr 66.2%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

       6. sub-neg N/A

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

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

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

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

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

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

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

       10. +-commutative N/A

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

       11. distribute-neg-in N/A

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

       12. metadata-eval N/A

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

       13. distribute-frac-neg2 N/A

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

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

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

       15. distribute-frac-neg2 N/A

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

       16. distribute-frac-neg N/A

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

   11. Applied egg-rr 66.2%

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

   12. Taylor expanded in x around 0

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

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

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 69.9%

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

   14. Simplified 69.9%

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

   if 1.52 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 70.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)} \]
   9. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

   10. Simplified 65.5%

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

Alternative 8: 69.4% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = x / (x * (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[(x / N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 38.6%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 66.2%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 66.2%

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

   9. Applied egg-rr 66.2%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

       6. sub-neg N/A

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

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

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

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

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

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

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

       10. +-commutative N/A

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

       11. distribute-neg-in N/A

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

       12. metadata-eval N/A

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

       13. distribute-frac-neg2 N/A

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

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

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

       15. distribute-frac-neg2 N/A

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

       16. distribute-frac-neg N/A

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

   11. Applied egg-rr 66.2%

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

   12. Taylor expanded in x around 0

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

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

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 69.9%

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

   14. Simplified 69.9%

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

   if 1.80000000000000004 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 70.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)} \]
   9. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

   10. Simplified 65.5%

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

   11. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

       4. distribute-rgt-in N/A

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

       5. +-commutative N/A

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

       6. associate-*l* N/A

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

       7. fma-define N/A

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

       8. lft-mult-inverse N/A

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

       9. fma-undefine N/A

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

       10. metadata-eval N/A

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

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

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 65.5%

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

   13. Simplified 65.5%

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

Alternative 9: 67.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.2)
   (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))
   (* x (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))

C

double code(double x) {
	double tmp;
	if (x <= 4.2) {
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)));
	} 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 <= 4.2d0) then
        tmp = 1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0)))
    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 <= 4.2) {
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)));
	} else {
		tmp = x * (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.2:
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)))
	else:
		tmp = x * (x * (0.16666666666666666 + (x * 0.041666666666666664)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.2)
		tmp = Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))));
	else
		tmp = Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.2)
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)));
	else
		tmp = x * (x * (0.16666666666666666 + (x * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.2], N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.20000000000000018

   1. Initial program 38.6%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 66.7%

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

   7. Simplified 66.7%

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

   if 4.20000000000000018 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 70.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)} \]
   9. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

   10. Simplified 65.5%

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

   11. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

       4. distribute-rgt-in N/A

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

       5. +-commutative N/A

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

       6. associate-*l* N/A

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

       7. fma-define N/A

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

       8. lft-mult-inverse N/A

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

       9. fma-undefine N/A

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

       10. metadata-eval N/A

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

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

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 65.5%

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

   13. Simplified 65.5%

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

Alternative 10: 66.9% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x * (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 2.0], 1.0, N[(x * N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 38.6%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 66.2%

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

   if 2 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 70.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)} \]
   9. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

   10. Simplified 65.5%

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

   11. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

       4. distribute-rgt-in N/A

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

       5. +-commutative N/A

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

       6. associate-*l* N/A

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

       7. fma-define N/A

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

       8. lft-mult-inverse N/A

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

       9. fma-undefine N/A

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

       10. metadata-eval N/A

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

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

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 65.5%

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

   13. Simplified 65.5%

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

Alternative 11: 66.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.85:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.85) 1.0 (* x (* 0.041666666666666664 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.85) {
		tmp = 1.0;
	} else {
		tmp = x * (0.041666666666666664 * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.85d0) then
        tmp = 1.0d0
    else
        tmp = x * (0.041666666666666664d0 * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.85) {
		tmp = 1.0;
	} else {
		tmp = x * (0.041666666666666664 * (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.85:
		tmp = 1.0
	else:
		tmp = x * (0.041666666666666664 * (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.85)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(0.041666666666666664 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.85)
		tmp = 1.0;
	else
		tmp = x * (0.041666666666666664 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.85], 1.0, N[(x * N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.85000000000000009

   1. Initial program 38.6%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 66.2%

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

   if 2.85000000000000009 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 70.8%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 70.8%

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

   9. Applied egg-rr 70.8%

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

   10. Taylor expanded in x around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 65.5%

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

   12. Simplified 65.5%

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

Alternative 12: 63.2% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.5d0) then
        tmp = 1.0d0
    else
        tmp = (x * x) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.5)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * x) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 38.6%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 66.2%

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

   if 2.5 < x

   1. Initial program 100.0%

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

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

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

      2. expm1-define N/A

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

      3. expm1-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 55.7%

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

   7. Simplified 55.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 55.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   10. Simplified 55.7%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.9% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ 1 - -0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (* -0.041666666666666664 (* x (* x x)))))

C

double code(double x) {
	return 1.0 - (-0.041666666666666664 * (x * (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - ((-0.041666666666666664d0) * (x * (x * x)))
end function

Java

public static double code(double x) {
	return 1.0 - (-0.041666666666666664 * (x * (x * x)));
}

Python

def code(x):
	return 1.0 - (-0.041666666666666664 * (x * (x * x)))

Julia

function code(x)
	return Float64(1.0 - Float64(-0.041666666666666664 * Float64(x * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (-0.041666666666666664 * (x * (x * x)));
end

Wolfram

code[x_] := N[(1.0 - N[(-0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

   \[\frac{e^{x} - 1}{x} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

   2. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(x\right)\right), x\right) \]

   3. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
6. Step-by-step derivation

   1. remove-double-neg N/A

      \[\leadsto 1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) \]

   2. distribute-lft-neg-out N/A

      \[\leadsto 1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto 1 - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right) \]

   5. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)}\right)\right) \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \left(-1 \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)}\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(\color{blue}{\frac{1}{2}} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

   14. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{-1}{2}\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot \color{blue}{-1}\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

   18. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right) \]

7. Simplified 66.0%

   \[\leadsto \color{blue}{1 - x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + -0.16666666666666666\right) + -0.5\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{-1}{24} \cdot {x}^{3}\right)}\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{3} \cdot \color{blue}{\frac{-1}{24}}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{\frac{-1}{24}}\right)\right) \]

   3. cube-mult N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \frac{-1}{24}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), \frac{-1}{24}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), \frac{-1}{24}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \frac{-1}{24}\right)\right) \]

   7. *-lowering-*.f64 65.0%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{-1}{24}\right)\right) \]

10. Simplified 65.0%

    \[\leadsto 1 - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.041666666666666664} \]

11. Final simplification 65.0%

    \[\leadsto 1 - -0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right) \]
12. Add Preprocessing

Alternative 14: 50.5% 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 55.1%

   \[\frac{e^{x} - 1}{x} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x} - 1\right), \color{blue}{x}\right) \]

   2. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(x\right)\right), x\right) \]

   3. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(x\right), x\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
6. Step-by-step derivation

7. Simplified 49.2%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Developer Target 1: 53.3% 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 2024164 
(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))