expax (section 3.5)

Percentage Accurate: 64.9% → 100.0%

Time: 6.0s

Alternatives: 9

Speedup: 6.1×

• 24.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))

C

double code(double a, double x) {
	return exp((a * x)) - 1.0;
}

Fortran

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

Java

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

Python

def code(a, x):
	return math.exp((a * x)) - 1.0

Julia

function code(a, x)
	return Float64(exp(Float64(a * x)) - 1.0)
end

MATLAB

function tmp = code(a, x)
	tmp = exp((a * x)) - 1.0;
end

Wolfram

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

Initial Program: 64.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))

C

double code(double a, double x) {
	return exp((a * x)) - 1.0;
}

Fortran

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

Java

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

Python

def code(a, x):
	return math.exp((a * x)) - 1.0

Julia

function code(a, x)
	return Float64(exp(Float64(a * x)) - 1.0)
end

MATLAB

function tmp = code(a, x)
	tmp = exp((a * x)) - 1.0;
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a x) :precision binary64 (expm1 (* a x)))

C

double code(double a, double x) {
	return expm1((a * x));
}

Java

public static double code(double a, double x) {
	return Math.expm1((a * x));
}

Python

def code(a, x):
	return math.expm1((a * x))

Julia

function code(a, x)
	return expm1(Float64(a * x))
end

Wolfram

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

Derivation

1. Initial program 67.2%

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

   1. expm1-def 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{expm1}\left(a \cdot x\right) \]

Alternative 2: 73.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\\ t_1 := x \cdot t_0\\ t_2 := x \cdot \left(a + t_0\right)\\ \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 10^{+14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot x \leq 5 \cdot 10^{+70}:\\ \;\;\;\;a \cdot \left(x + a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;a \cdot x \leq 5 \cdot 10^{+75}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;a \cdot x \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - t_1 \cdot t_1}{a \cdot x - t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (let* ((t_0 (* a (* x (* a 0.5)))) (t_1 (* x t_0)) (t_2 (* x (+ a t_0))))
   (if (<= (* a x) -500.0)
     -2.0
     (if (<= (* a x) 1e+14)
       t_2
       (if (<= (* a x) 5e+70)
         (* a (+ x (* a (* x (* x 0.5)))))
         (if (<= (* a x) 5e+75)
           (* x (+ a (* x (* a (* a 0.5)))))
           (if (<= (* a x) 2e+126)
             (/ (- (* (* a x) (* a x)) (* t_1 t_1)) (- (* a x) t_1))
             t_2)))))))

C

double code(double a, double x) {
	double t_0 = a * (x * (a * 0.5));
	double t_1 = x * t_0;
	double t_2 = x * (a + t_0);
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0;
	} else if ((a * x) <= 1e+14) {
		tmp = t_2;
	} else if ((a * x) <= 5e+70) {
		tmp = a * (x + (a * (x * (x * 0.5))));
	} else if ((a * x) <= 5e+75) {
		tmp = x * (a + (x * (a * (a * 0.5))));
	} else if ((a * x) <= 2e+126) {
		tmp = (((a * x) * (a * x)) - (t_1 * t_1)) / ((a * x) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = a * (x * (a * 0.5d0))
    t_1 = x * t_0
    t_2 = x * (a + t_0)
    if ((a * x) <= (-500.0d0)) then
        tmp = -2.0d0
    else if ((a * x) <= 1d+14) then
        tmp = t_2
    else if ((a * x) <= 5d+70) then
        tmp = a * (x + (a * (x * (x * 0.5d0))))
    else if ((a * x) <= 5d+75) then
        tmp = x * (a + (x * (a * (a * 0.5d0))))
    else if ((a * x) <= 2d+126) then
        tmp = (((a * x) * (a * x)) - (t_1 * t_1)) / ((a * x) - t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double t_0 = a * (x * (a * 0.5));
	double t_1 = x * t_0;
	double t_2 = x * (a + t_0);
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0;
	} else if ((a * x) <= 1e+14) {
		tmp = t_2;
	} else if ((a * x) <= 5e+70) {
		tmp = a * (x + (a * (x * (x * 0.5))));
	} else if ((a * x) <= 5e+75) {
		tmp = x * (a + (x * (a * (a * 0.5))));
	} else if ((a * x) <= 2e+126) {
		tmp = (((a * x) * (a * x)) - (t_1 * t_1)) / ((a * x) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(a, x):
	t_0 = a * (x * (a * 0.5))
	t_1 = x * t_0
	t_2 = x * (a + t_0)
	tmp = 0
	if (a * x) <= -500.0:
		tmp = -2.0
	elif (a * x) <= 1e+14:
		tmp = t_2
	elif (a * x) <= 5e+70:
		tmp = a * (x + (a * (x * (x * 0.5))))
	elif (a * x) <= 5e+75:
		tmp = x * (a + (x * (a * (a * 0.5))))
	elif (a * x) <= 2e+126:
		tmp = (((a * x) * (a * x)) - (t_1 * t_1)) / ((a * x) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(a, x)
	t_0 = Float64(a * Float64(x * Float64(a * 0.5)))
	t_1 = Float64(x * t_0)
	t_2 = Float64(x * Float64(a + t_0))
	tmp = 0.0
	if (Float64(a * x) <= -500.0)
		tmp = -2.0;
	elseif (Float64(a * x) <= 1e+14)
		tmp = t_2;
	elseif (Float64(a * x) <= 5e+70)
		tmp = Float64(a * Float64(x + Float64(a * Float64(x * Float64(x * 0.5)))));
	elseif (Float64(a * x) <= 5e+75)
		tmp = Float64(x * Float64(a + Float64(x * Float64(a * Float64(a * 0.5)))));
	elseif (Float64(a * x) <= 2e+126)
		tmp = Float64(Float64(Float64(Float64(a * x) * Float64(a * x)) - Float64(t_1 * t_1)) / Float64(Float64(a * x) - t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	t_0 = a * (x * (a * 0.5));
	t_1 = x * t_0;
	t_2 = x * (a + t_0);
	tmp = 0.0;
	if ((a * x) <= -500.0)
		tmp = -2.0;
	elseif ((a * x) <= 1e+14)
		tmp = t_2;
	elseif ((a * x) <= 5e+70)
		tmp = a * (x + (a * (x * (x * 0.5))));
	elseif ((a * x) <= 5e+75)
		tmp = x * (a + (x * (a * (a * 0.5))));
	elseif ((a * x) <= 2e+126)
		tmp = (((a * x) * (a * x)) - (t_1 * t_1)) / ((a * x) - t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := Block[{t$95$0 = N[(a * N[(x * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(a + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * x), $MachinePrecision], -500.0], -2.0, If[LessEqual[N[(a * x), $MachinePrecision], 1e+14], t$95$2, If[LessEqual[N[(a * x), $MachinePrecision], 5e+70], N[(a * N[(x + N[(a * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * x), $MachinePrecision], 5e+75], N[(x * N[(a + N[(x * N[(a * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * x), $MachinePrecision], 2e+126], N[(N[(N[(N[(a * x), $MachinePrecision] * N[(a * x), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(a * x), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 a x) < -500

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 0.5%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 0.5%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 0.5%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 0.5%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 0.9%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 1.1%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 1.1%

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

      9. associate-*l* 1.1%

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

   6. Simplified 1.1%

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in 0.9%

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

      2. flip-+ 0.6%

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

      3. *-commutative 0.6%

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

      4. *-commutative 0.6%

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

      5. *-commutative 0.6%

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

      6. associate-*l* 0.6%

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

      7. *-commutative 0.6%

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

      8. associate-*l* 0.6%

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

      9. *-commutative 0.6%

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

      10. *-commutative 0.6%

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

      11. associate-*l* 0.5%

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

   8. Applied egg-rr 0.5%

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

   9. Taylor expanded in a around inf 0.4%

      \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{-0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot -0.5}} \]

       2. unpow2 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\left(\color{blue}{\left(a \cdot a\right)} \cdot {x}^{2}\right) \cdot -0.5} \]

       3. unpow2 0.4%

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

   11. Simplified 0.4%

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

   12. Taylor expanded in a around 0 18.8%

       \[\leadsto \color{blue}{-2} \]

   if -500 < (*.f64 a x) < 1e14 or 1.99999999999999985e126 < (*.f64 a x)

   1. Initial program 51.1%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 88.4%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 88.4%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 88.4%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 88.4%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 93.8%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 93.8%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 93.8%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 93.8%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 93.8%

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

      9. associate-*l* 93.8%

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

   6. Simplified 93.8%

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 93.8%

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

      2. distribute-lft-in 93.8%

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

      3. *-commutative 93.8%

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

      4. associate-*l* 97.3%

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

      5. *-commutative 97.3%

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

   8. Applied egg-rr 97.3%

      \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right) + a \cdot x} \]
   9. Step-by-step derivation

      1. *-commutative 97.3%

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

      2. distribute-lft-out 97.3%

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

      3. *-commutative 97.3%

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

   10. Applied egg-rr 97.3%

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

   if 1e14 < (*.f64 a x) < 5.0000000000000002e70

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 1.8%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 1.8%

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot 0.5} + a \cdot x \]

      2. associate-*l* 1.8%

         \[\leadsto \color{blue}{{a}^{2} \cdot \left({x}^{2} \cdot 0.5\right)} + a \cdot x \]

      3. unpow2 1.8%

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left({x}^{2} \cdot 0.5\right) + a \cdot x \]

      4. associate-*l* 21.2%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left({x}^{2} \cdot 0.5\right)\right)} + a \cdot x \]

      5. distribute-lft-out 21.2%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left({x}^{2} \cdot 0.5\right) + x\right)} \]

      6. unpow2 21.2%

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

      7. associate-*l* 21.2%

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

   6. Simplified 21.2%

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

   if 5.0000000000000002e70 < (*.f64 a x) < 5.0000000000000002e75

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 0.0%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 0.0%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 100.0%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 100.0%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 100.0%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 100.0%

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

      9. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   if 5.0000000000000002e75 < (*.f64 a x) < 1.99999999999999985e126

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 13.5%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 13.5%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 13.5%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 13.5%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 5.4%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 5.4%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 5.4%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 5.4%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 5.4%

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

      9. associate-*l* 5.4%

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

   6. Simplified 5.4%

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in 5.4%

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

      2. flip-+ 91.3%

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

      3. *-commutative 91.3%

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

      4. *-commutative 91.3%

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

      5. *-commutative 91.3%

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

      6. associate-*l* 91.3%

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

      7. *-commutative 91.3%

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

      8. associate-*l* 90.9%

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

      9. *-commutative 90.9%

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

      10. *-commutative 90.9%

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

      11. associate-*l* 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{a \cdot x - x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 10^{+14}:\\ \;\;\;\;x \cdot \left(a + a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;a \cdot x \leq 5 \cdot 10^{+70}:\\ \;\;\;\;a \cdot \left(x + a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;a \cdot x \leq 5 \cdot 10^{+75}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;a \cdot x \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\right)}{a \cdot x - x \cdot \left(a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 3: 72.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\\ t_1 := x \cdot t_0\\ \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 10^{+14}:\\ \;\;\;\;x \cdot \left(a + t_0\right)\\ \mathbf{elif}\;a \cdot x \leq 5 \cdot 10^{+70}:\\ \;\;\;\;a \cdot \left(x + a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;a \cdot x \leq 5 \cdot 10^{+75}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;a \cdot x \leq 10^{+117}:\\ \;\;\;\;\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - t_1 \cdot t_1}{\left(\left(a \cdot a\right) \cdot \left(x \cdot x\right)\right) \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (let* ((t_0 (* a (* x (* a 0.5)))) (t_1 (* x t_0)))
   (if (<= (* a x) -500.0)
     -2.0
     (if (<= (* a x) 1e+14)
       (* x (+ a t_0))
       (if (<= (* a x) 5e+70)
         (* a (+ x (* a (* x (* x 0.5)))))
         (if (<= (* a x) 5e+75)
           (* x (+ a (* x (* a (* a 0.5)))))
           (if (<= (* a x) 1e+117)
             (/
              (- (* (* a x) (* a x)) (* t_1 t_1))
              (* (* (* a a) (* x x)) -0.5))
             (* (* x x) (* 0.5 (* a a))))))))))

C

double code(double a, double x) {
	double t_0 = a * (x * (a * 0.5));
	double t_1 = x * t_0;
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0;
	} else if ((a * x) <= 1e+14) {
		tmp = x * (a + t_0);
	} else if ((a * x) <= 5e+70) {
		tmp = a * (x + (a * (x * (x * 0.5))));
	} else if ((a * x) <= 5e+75) {
		tmp = x * (a + (x * (a * (a * 0.5))));
	} else if ((a * x) <= 1e+117) {
		tmp = (((a * x) * (a * x)) - (t_1 * t_1)) / (((a * a) * (x * x)) * -0.5);
	} else {
		tmp = (x * x) * (0.5 * (a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (x * (a * 0.5d0))
    t_1 = x * t_0
    if ((a * x) <= (-500.0d0)) then
        tmp = -2.0d0
    else if ((a * x) <= 1d+14) then
        tmp = x * (a + t_0)
    else if ((a * x) <= 5d+70) then
        tmp = a * (x + (a * (x * (x * 0.5d0))))
    else if ((a * x) <= 5d+75) then
        tmp = x * (a + (x * (a * (a * 0.5d0))))
    else if ((a * x) <= 1d+117) then
        tmp = (((a * x) * (a * x)) - (t_1 * t_1)) / (((a * a) * (x * x)) * (-0.5d0))
    else
        tmp = (x * x) * (0.5d0 * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double t_0 = a * (x * (a * 0.5));
	double t_1 = x * t_0;
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0;
	} else if ((a * x) <= 1e+14) {
		tmp = x * (a + t_0);
	} else if ((a * x) <= 5e+70) {
		tmp = a * (x + (a * (x * (x * 0.5))));
	} else if ((a * x) <= 5e+75) {
		tmp = x * (a + (x * (a * (a * 0.5))));
	} else if ((a * x) <= 1e+117) {
		tmp = (((a * x) * (a * x)) - (t_1 * t_1)) / (((a * a) * (x * x)) * -0.5);
	} else {
		tmp = (x * x) * (0.5 * (a * a));
	}
	return tmp;
}

Python

def code(a, x):
	t_0 = a * (x * (a * 0.5))
	t_1 = x * t_0
	tmp = 0
	if (a * x) <= -500.0:
		tmp = -2.0
	elif (a * x) <= 1e+14:
		tmp = x * (a + t_0)
	elif (a * x) <= 5e+70:
		tmp = a * (x + (a * (x * (x * 0.5))))
	elif (a * x) <= 5e+75:
		tmp = x * (a + (x * (a * (a * 0.5))))
	elif (a * x) <= 1e+117:
		tmp = (((a * x) * (a * x)) - (t_1 * t_1)) / (((a * a) * (x * x)) * -0.5)
	else:
		tmp = (x * x) * (0.5 * (a * a))
	return tmp

Julia

function code(a, x)
	t_0 = Float64(a * Float64(x * Float64(a * 0.5)))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (Float64(a * x) <= -500.0)
		tmp = -2.0;
	elseif (Float64(a * x) <= 1e+14)
		tmp = Float64(x * Float64(a + t_0));
	elseif (Float64(a * x) <= 5e+70)
		tmp = Float64(a * Float64(x + Float64(a * Float64(x * Float64(x * 0.5)))));
	elseif (Float64(a * x) <= 5e+75)
		tmp = Float64(x * Float64(a + Float64(x * Float64(a * Float64(a * 0.5)))));
	elseif (Float64(a * x) <= 1e+117)
		tmp = Float64(Float64(Float64(Float64(a * x) * Float64(a * x)) - Float64(t_1 * t_1)) / Float64(Float64(Float64(a * a) * Float64(x * x)) * -0.5));
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	t_0 = a * (x * (a * 0.5));
	t_1 = x * t_0;
	tmp = 0.0;
	if ((a * x) <= -500.0)
		tmp = -2.0;
	elseif ((a * x) <= 1e+14)
		tmp = x * (a + t_0);
	elseif ((a * x) <= 5e+70)
		tmp = a * (x + (a * (x * (x * 0.5))));
	elseif ((a * x) <= 5e+75)
		tmp = x * (a + (x * (a * (a * 0.5))));
	elseif ((a * x) <= 1e+117)
		tmp = (((a * x) * (a * x)) - (t_1 * t_1)) / (((a * a) * (x * x)) * -0.5);
	else
		tmp = (x * x) * (0.5 * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := Block[{t$95$0 = N[(a * N[(x * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[N[(a * x), $MachinePrecision], -500.0], -2.0, If[LessEqual[N[(a * x), $MachinePrecision], 1e+14], N[(x * N[(a + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * x), $MachinePrecision], 5e+70], N[(a * N[(x + N[(a * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * x), $MachinePrecision], 5e+75], N[(x * N[(a + N[(x * N[(a * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * x), $MachinePrecision], 1e+117], N[(N[(N[(N[(a * x), $MachinePrecision] * N[(a * x), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a * a), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (*.f64 a x) < -500

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 0.5%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 0.5%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 0.5%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 0.5%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 0.9%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 1.1%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 1.1%

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

      9. associate-*l* 1.1%

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

   6. Simplified 1.1%

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in 0.9%

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

      2. flip-+ 0.6%

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

      3. *-commutative 0.6%

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

      4. *-commutative 0.6%

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

      5. *-commutative 0.6%

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

      6. associate-*l* 0.6%

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

      7. *-commutative 0.6%

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

      8. associate-*l* 0.6%

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

      9. *-commutative 0.6%

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

      10. *-commutative 0.6%

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

      11. associate-*l* 0.5%

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

   8. Applied egg-rr 0.5%

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

   9. Taylor expanded in a around inf 0.4%

      \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{-0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot -0.5}} \]

       2. unpow2 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\left(\color{blue}{\left(a \cdot a\right)} \cdot {x}^{2}\right) \cdot -0.5} \]

       3. unpow2 0.4%

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

   11. Simplified 0.4%

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

   12. Taylor expanded in a around 0 18.8%

       \[\leadsto \color{blue}{-2} \]

   if -500 < (*.f64 a x) < 1e14

   1. Initial program 35.8%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 85.5%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 85.5%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 85.5%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 85.5%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 92.5%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 92.5%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 92.5%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 92.5%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 92.5%

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

      9. associate-*l* 92.5%

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

   6. Simplified 92.5%

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 92.5%

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

      2. distribute-lft-in 92.5%

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

      3. *-commutative 92.5%

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

      4. associate-*l* 97.2%

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

      5. *-commutative 97.2%

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

   8. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right) + a \cdot x} \]
   9. Step-by-step derivation

      1. *-commutative 97.2%

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

      2. distribute-lft-out 97.2%

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

      3. *-commutative 97.2%

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

   10. Applied egg-rr 97.2%

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

   if 1e14 < (*.f64 a x) < 5.0000000000000002e70

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 1.8%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 1.8%

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot 0.5} + a \cdot x \]

      2. associate-*l* 1.8%

         \[\leadsto \color{blue}{{a}^{2} \cdot \left({x}^{2} \cdot 0.5\right)} + a \cdot x \]

      3. unpow2 1.8%

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left({x}^{2} \cdot 0.5\right) + a \cdot x \]

      4. associate-*l* 21.2%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left({x}^{2} \cdot 0.5\right)\right)} + a \cdot x \]

      5. distribute-lft-out 21.2%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left({x}^{2} \cdot 0.5\right) + x\right)} \]

      6. unpow2 21.2%

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

      7. associate-*l* 21.2%

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

   6. Simplified 21.2%

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

   if 5.0000000000000002e70 < (*.f64 a x) < 5.0000000000000002e75

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 0.0%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 0.0%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 100.0%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 100.0%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 100.0%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 100.0%

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

      9. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   if 5.0000000000000002e75 < (*.f64 a x) < 1.00000000000000005e117

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 4.9%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 4.9%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 4.9%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 4.9%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 5.3%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 5.3%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 5.3%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 5.3%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 5.3%

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

      9. associate-*l* 5.3%

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

   6. Simplified 5.3%

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in 5.3%

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

      2. flip-+ 90.4%

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

      3. *-commutative 90.4%

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

      4. *-commutative 90.4%

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

      5. *-commutative 90.4%

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

      6. associate-*l* 90.4%

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

      7. *-commutative 90.4%

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

      8. associate-*l* 90.0%

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

      9. *-commutative 90.0%

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

      10. *-commutative 90.0%

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

      11. associate-*l* 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in a around inf 90.0%

      \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{-0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 90.0%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot -0.5}} \]

       2. unpow2 90.0%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\left(\color{blue}{\left(a \cdot a\right)} \cdot {x}^{2}\right) \cdot -0.5} \]

       3. unpow2 90.0%

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

   11. Simplified 90.0%

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

   if 1.00000000000000005e117 < (*.f64 a x)

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 97.6%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 97.6%

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot 0.5} + a \cdot x \]

      2. associate-*l* 97.6%

         \[\leadsto \color{blue}{{a}^{2} \cdot \left({x}^{2} \cdot 0.5\right)} + a \cdot x \]

      3. unpow2 97.6%

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left({x}^{2} \cdot 0.5\right) + a \cdot x \]

      4. associate-*l* 95.5%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left({x}^{2} \cdot 0.5\right)\right)} + a \cdot x \]

      5. distribute-lft-out 95.5%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left({x}^{2} \cdot 0.5\right) + x\right)} \]

      6. unpow2 95.5%

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

      7. associate-*l* 95.5%

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

   6. Simplified 95.5%

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

   7. Taylor expanded in a around inf 97.6%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 97.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {x}^{2}\right) \]

      2. unpow2 97.6%

         \[\leadsto 0.5 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

      3. associate-*r* 95.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(x \cdot x\right)\right)\right)} \]

   9. Simplified 95.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(a \cdot \left(a \cdot \left(x \cdot x\right)\right)\right)} \]

   10. Taylor expanded in a around 0 97.6%

       \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 97.6%

          \[\leadsto 0.5 \cdot \left({a}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       2. associate-*r* 97.6%

          \[\leadsto \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot \left(x \cdot x\right)} \]

       3. unpow2 97.6%

          \[\leadsto \left(0.5 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(x \cdot x\right) \]

   12. Simplified 97.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 10^{+14}:\\ \;\;\;\;x \cdot \left(a + a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;a \cdot x \leq 5 \cdot 10^{+70}:\\ \;\;\;\;a \cdot \left(x + a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;a \cdot x \leq 5 \cdot 10^{+75}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;a \cdot x \leq 10^{+117}:\\ \;\;\;\;\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\right)}{\left(\left(a \cdot a\right) \cdot \left(x \cdot x\right)\right) \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 4: 71.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 10^{+14} \lor \neg \left(a \cdot x \leq 5 \cdot 10^{+70}\right):\\ \;\;\;\;x \cdot \left(a + a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -500.0)
   -2.0
   (if (or (<= (* a x) 1e+14) (not (<= (* a x) 5e+70)))
     (* x (+ a (* a (* x (* a 0.5)))))
     (* a (+ x (* a (* x (* x 0.5))))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0;
	} else if (((a * x) <= 1e+14) || !((a * x) <= 5e+70)) {
		tmp = x * (a + (a * (x * (a * 0.5))));
	} else {
		tmp = a * (x + (a * (x * (x * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-500.0d0)) then
        tmp = -2.0d0
    else if (((a * x) <= 1d+14) .or. (.not. ((a * x) <= 5d+70))) then
        tmp = x * (a + (a * (x * (a * 0.5d0))))
    else
        tmp = a * (x + (a * (x * (x * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0;
	} else if (((a * x) <= 1e+14) || !((a * x) <= 5e+70)) {
		tmp = x * (a + (a * (x * (a * 0.5))));
	} else {
		tmp = a * (x + (a * (x * (x * 0.5))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -500.0:
		tmp = -2.0
	elif ((a * x) <= 1e+14) or not ((a * x) <= 5e+70):
		tmp = x * (a + (a * (x * (a * 0.5))))
	else:
		tmp = a * (x + (a * (x * (x * 0.5))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -500.0)
		tmp = -2.0;
	elseif ((Float64(a * x) <= 1e+14) || !(Float64(a * x) <= 5e+70))
		tmp = Float64(x * Float64(a + Float64(a * Float64(x * Float64(a * 0.5)))));
	else
		tmp = Float64(a * Float64(x + Float64(a * Float64(x * Float64(x * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -500.0)
		tmp = -2.0;
	elseif (((a * x) <= 1e+14) || ~(((a * x) <= 5e+70)))
		tmp = x * (a + (a * (x * (a * 0.5))));
	else
		tmp = a * (x + (a * (x * (x * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -500.0], -2.0, If[Or[LessEqual[N[(a * x), $MachinePrecision], 1e+14], N[Not[LessEqual[N[(a * x), $MachinePrecision], 5e+70]], $MachinePrecision]], N[(x * N[(a + N[(a * N[(x * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x + N[(a * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a x) < -500

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 0.5%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 0.5%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 0.5%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 0.5%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 0.9%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 1.1%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 1.1%

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

      9. associate-*l* 1.1%

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

   6. Simplified 1.1%

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in 0.9%

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

      2. flip-+ 0.6%

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

      3. *-commutative 0.6%

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

      4. *-commutative 0.6%

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

      5. *-commutative 0.6%

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

      6. associate-*l* 0.6%

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

      7. *-commutative 0.6%

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

      8. associate-*l* 0.6%

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

      9. *-commutative 0.6%

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

      10. *-commutative 0.6%

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

      11. associate-*l* 0.5%

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

   8. Applied egg-rr 0.5%

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

   9. Taylor expanded in a around inf 0.4%

      \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{-0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot -0.5}} \]

       2. unpow2 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\left(\color{blue}{\left(a \cdot a\right)} \cdot {x}^{2}\right) \cdot -0.5} \]

       3. unpow2 0.4%

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

   11. Simplified 0.4%

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

   12. Taylor expanded in a around 0 18.8%

       \[\leadsto \color{blue}{-2} \]

   if -500 < (*.f64 a x) < 1e14 or 5.0000000000000002e70 < (*.f64 a x)

   1. Initial program 54.3%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 83.4%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 83.4%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 83.4%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 83.4%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 88.5%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 88.5%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 88.5%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 88.5%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 88.5%

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

      9. associate-*l* 88.5%

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

   6. Simplified 88.5%

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 88.5%

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

      2. distribute-lft-in 88.5%

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

      3. *-commutative 88.5%

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

      4. associate-*l* 91.8%

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

      5. *-commutative 91.8%

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

   8. Applied egg-rr 91.8%

      \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right) + a \cdot x} \]
   9. Step-by-step derivation

      1. *-commutative 91.8%

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

      2. distribute-lft-out 91.9%

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

      3. *-commutative 91.9%

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

   10. Applied egg-rr 91.9%

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

   if 1e14 < (*.f64 a x) < 5.0000000000000002e70

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 1.8%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 1.8%

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot 0.5} + a \cdot x \]

      2. associate-*l* 1.8%

         \[\leadsto \color{blue}{{a}^{2} \cdot \left({x}^{2} \cdot 0.5\right)} + a \cdot x \]

      3. unpow2 1.8%

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left({x}^{2} \cdot 0.5\right) + a \cdot x \]

      4. associate-*l* 21.2%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left({x}^{2} \cdot 0.5\right)\right)} + a \cdot x \]

      5. distribute-lft-out 21.2%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left({x}^{2} \cdot 0.5\right) + x\right)} \]

      6. unpow2 21.2%

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

      7. associate-*l* 21.2%

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

   6. Simplified 21.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 10^{+14} \lor \neg \left(a \cdot x \leq 5 \cdot 10^{+70}\right):\\ \;\;\;\;x \cdot \left(a + a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 5: 71.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 0.0001:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -500.0)
   -2.0
   (if (<= (* a x) 0.0001) (* a x) (* a (+ x (* a (* x (* x 0.5))))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0;
	} else if ((a * x) <= 0.0001) {
		tmp = a * x;
	} else {
		tmp = a * (x + (a * (x * (x * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-500.0d0)) then
        tmp = -2.0d0
    else if ((a * x) <= 0.0001d0) then
        tmp = a * x
    else
        tmp = a * (x + (a * (x * (x * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0;
	} else if ((a * x) <= 0.0001) {
		tmp = a * x;
	} else {
		tmp = a * (x + (a * (x * (x * 0.5))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -500.0:
		tmp = -2.0
	elif (a * x) <= 0.0001:
		tmp = a * x
	else:
		tmp = a * (x + (a * (x * (x * 0.5))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -500.0)
		tmp = -2.0;
	elseif (Float64(a * x) <= 0.0001)
		tmp = Float64(a * x);
	else
		tmp = Float64(a * Float64(x + Float64(a * Float64(x * Float64(x * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -500.0)
		tmp = -2.0;
	elseif ((a * x) <= 0.0001)
		tmp = a * x;
	else
		tmp = a * (x + (a * (x * (x * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -500.0], -2.0, If[LessEqual[N[(a * x), $MachinePrecision], 0.0001], N[(a * x), $MachinePrecision], N[(a * N[(x + N[(a * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a x) < -500

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 0.5%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 0.5%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 0.5%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 0.5%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 0.9%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 1.1%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 1.1%

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

      9. associate-*l* 1.1%

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

   6. Simplified 1.1%

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in 0.9%

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

      2. flip-+ 0.6%

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

      3. *-commutative 0.6%

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

      4. *-commutative 0.6%

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

      5. *-commutative 0.6%

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

      6. associate-*l* 0.6%

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

      7. *-commutative 0.6%

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

      8. associate-*l* 0.6%

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

      9. *-commutative 0.6%

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

      10. *-commutative 0.6%

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

      11. associate-*l* 0.5%

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

   8. Applied egg-rr 0.5%

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

   9. Taylor expanded in a around inf 0.4%

      \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{-0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot -0.5}} \]

       2. unpow2 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\left(\color{blue}{\left(a \cdot a\right)} \cdot {x}^{2}\right) \cdot -0.5} \]

       3. unpow2 0.4%

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

   11. Simplified 0.4%

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

   12. Taylor expanded in a around 0 18.8%

       \[\leadsto \color{blue}{-2} \]

   if -500 < (*.f64 a x) < 1.00000000000000005e-4

   1. Initial program 34.3%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 98.6%

      \[\leadsto \color{blue}{a \cdot x} \]

   if 1.00000000000000005e-4 < (*.f64 a x)

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 62.3%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 62.3%

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot 0.5} + a \cdot x \]

      2. associate-*l* 62.3%

         \[\leadsto \color{blue}{{a}^{2} \cdot \left({x}^{2} \cdot 0.5\right)} + a \cdot x \]

      3. unpow2 62.3%

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left({x}^{2} \cdot 0.5\right) + a \cdot x \]

      4. associate-*l* 68.7%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left({x}^{2} \cdot 0.5\right)\right)} + a \cdot x \]

      5. distribute-lft-out 68.7%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left({x}^{2} \cdot 0.5\right) + x\right)} \]

      6. unpow2 68.7%

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

      7. associate-*l* 68.7%

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

   6. Simplified 68.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 0.0001:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 6: 70.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 5 \cdot 10^{-21}:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -500.0)
   -2.0
   (if (<= (* a x) 5e-21) (* a x) (* x (+ a (* x (* a (* a 0.5))))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0;
	} else if ((a * x) <= 5e-21) {
		tmp = a * x;
	} else {
		tmp = x * (a + (x * (a * (a * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-500.0d0)) then
        tmp = -2.0d0
    else if ((a * x) <= 5d-21) then
        tmp = a * x
    else
        tmp = x * (a + (x * (a * (a * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0;
	} else if ((a * x) <= 5e-21) {
		tmp = a * x;
	} else {
		tmp = x * (a + (x * (a * (a * 0.5))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -500.0:
		tmp = -2.0
	elif (a * x) <= 5e-21:
		tmp = a * x
	else:
		tmp = x * (a + (x * (a * (a * 0.5))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -500.0)
		tmp = -2.0;
	elseif (Float64(a * x) <= 5e-21)
		tmp = Float64(a * x);
	else
		tmp = Float64(x * Float64(a + Float64(x * Float64(a * Float64(a * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -500.0)
		tmp = -2.0;
	elseif ((a * x) <= 5e-21)
		tmp = a * x;
	else
		tmp = x * (a + (x * (a * (a * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -500.0], -2.0, If[LessEqual[N[(a * x), $MachinePrecision], 5e-21], N[(a * x), $MachinePrecision], N[(x * N[(a + N[(x * N[(a * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a x) < -500

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 0.5%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 0.5%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 0.5%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 0.5%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 0.9%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 1.1%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 1.1%

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

      9. associate-*l* 1.1%

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

   6. Simplified 1.1%

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in 0.9%

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

      2. flip-+ 0.6%

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

      3. *-commutative 0.6%

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

      4. *-commutative 0.6%

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

      5. *-commutative 0.6%

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

      6. associate-*l* 0.6%

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

      7. *-commutative 0.6%

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

      8. associate-*l* 0.6%

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

      9. *-commutative 0.6%

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

      10. *-commutative 0.6%

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

      11. associate-*l* 0.5%

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

   8. Applied egg-rr 0.5%

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

   9. Taylor expanded in a around inf 0.4%

      \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{-0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot -0.5}} \]

       2. unpow2 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\left(\color{blue}{\left(a \cdot a\right)} \cdot {x}^{2}\right) \cdot -0.5} \]

       3. unpow2 0.4%

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

   11. Simplified 0.4%

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

   12. Taylor expanded in a around 0 18.8%

       \[\leadsto \color{blue}{-2} \]

   if -500 < (*.f64 a x) < 4.99999999999999973e-21

   1. Initial program 34.2%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 99.3%

      \[\leadsto \color{blue}{a \cdot x} \]

   if 4.99999999999999973e-21 < (*.f64 a x)

   1. Initial program 96.6%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 60.1%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 60.1%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 60.1%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 60.1%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 67.4%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 67.4%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 67.4%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 67.4%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 67.4%

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

      9. associate-*l* 67.4%

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

   6. Simplified 67.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 5 \cdot 10^{-21}:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 7: 71.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 10^{+14}:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(a \cdot \left(a \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -500.0)
   -2.0
   (if (<= (* a x) 1e+14) (* a x) (* 0.5 (* a (* a (* x x)))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0;
	} else if ((a * x) <= 1e+14) {
		tmp = a * x;
	} else {
		tmp = 0.5 * (a * (a * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-500.0d0)) then
        tmp = -2.0d0
    else if ((a * x) <= 1d+14) then
        tmp = a * x
    else
        tmp = 0.5d0 * (a * (a * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0;
	} else if ((a * x) <= 1e+14) {
		tmp = a * x;
	} else {
		tmp = 0.5 * (a * (a * (x * x)));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -500.0:
		tmp = -2.0
	elif (a * x) <= 1e+14:
		tmp = a * x
	else:
		tmp = 0.5 * (a * (a * (x * x)))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -500.0)
		tmp = -2.0;
	elseif (Float64(a * x) <= 1e+14)
		tmp = Float64(a * x);
	else
		tmp = Float64(0.5 * Float64(a * Float64(a * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -500.0)
		tmp = -2.0;
	elseif ((a * x) <= 1e+14)
		tmp = a * x;
	else
		tmp = 0.5 * (a * (a * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -500.0], -2.0, If[LessEqual[N[(a * x), $MachinePrecision], 1e+14], N[(a * x), $MachinePrecision], N[(0.5 * N[(a * N[(a * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a x) < -500

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 0.5%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 0.5%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 0.5%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 0.5%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 0.9%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 1.1%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 1.1%

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

      9. associate-*l* 1.1%

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

   6. Simplified 1.1%

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in 0.9%

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

      2. flip-+ 0.6%

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

      3. *-commutative 0.6%

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

      4. *-commutative 0.6%

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

      5. *-commutative 0.6%

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

      6. associate-*l* 0.6%

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

      7. *-commutative 0.6%

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

      8. associate-*l* 0.6%

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

      9. *-commutative 0.6%

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

      10. *-commutative 0.6%

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

      11. associate-*l* 0.5%

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

   8. Applied egg-rr 0.5%

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

   9. Taylor expanded in a around inf 0.4%

      \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{-0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot -0.5}} \]

       2. unpow2 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\left(\color{blue}{\left(a \cdot a\right)} \cdot {x}^{2}\right) \cdot -0.5} \]

       3. unpow2 0.4%

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

   11. Simplified 0.4%

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

   12. Taylor expanded in a around 0 18.8%

       \[\leadsto \color{blue}{-2} \]

   if -500 < (*.f64 a x) < 1e14

   1. Initial program 35.8%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 96.4%

      \[\leadsto \color{blue}{a \cdot x} \]

   if 1e14 < (*.f64 a x)

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 65.1%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 65.1%

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot 0.5} + a \cdot x \]

      2. associate-*l* 65.1%

         \[\leadsto \color{blue}{{a}^{2} \cdot \left({x}^{2} \cdot 0.5\right)} + a \cdot x \]

      3. unpow2 65.1%

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left({x}^{2} \cdot 0.5\right) + a \cdot x \]

      4. associate-*l* 68.7%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left({x}^{2} \cdot 0.5\right)\right)} + a \cdot x \]

      5. distribute-lft-out 68.7%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left({x}^{2} \cdot 0.5\right) + x\right)} \]

      6. unpow2 68.7%

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

      7. associate-*l* 68.7%

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

   6. Simplified 68.7%

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

   7. Taylor expanded in a around inf 65.1%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 65.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {x}^{2}\right) \]

      2. unpow2 65.1%

         \[\leadsto 0.5 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

      3. associate-*r* 68.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(x \cdot x\right)\right)\right)} \]

   9. Simplified 68.5%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 10^{+14}:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(a \cdot \left(a \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]

Alternative 8: 61.5% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (if (<= (* a x) -2.0) -2.0 (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2.0) {
		tmp = -2.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-2.0d0)) then
        tmp = -2.0d0
    else
        tmp = a * x
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2.0) {
		tmp = -2.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -2.0:
		tmp = -2.0
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -2.0)
		tmp = -2.0;
	else
		tmp = Float64(a * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -2.0)
		tmp = -2.0;
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -2.0], -2.0, N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -2

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 0.5%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 0.5%

         \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*r* 0.5%

         \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

      3. unpow2 0.5%

         \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* 0.9%

         \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

      5. distribute-rgt-out 1.1%

         \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

      6. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

      7. *-commutative 1.1%

         \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

      8. unpow2 1.1%

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

      9. associate-*l* 1.1%

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

   6. Simplified 1.1%

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in 0.9%

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

      2. flip-+ 0.6%

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

      3. *-commutative 0.6%

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

      4. *-commutative 0.6%

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

      5. *-commutative 0.6%

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

      6. associate-*l* 0.6%

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

      7. *-commutative 0.6%

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

      8. associate-*l* 0.6%

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

      9. *-commutative 0.6%

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

      10. *-commutative 0.6%

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

      11. associate-*l* 0.5%

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

   8. Applied egg-rr 0.5%

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

   9. Taylor expanded in a around inf 0.4%

      \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{-0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot -0.5}} \]

       2. unpow2 0.4%

          \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\left(\color{blue}{\left(a \cdot a\right)} \cdot {x}^{2}\right) \cdot -0.5} \]

       3. unpow2 0.4%

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

   11. Simplified 0.4%

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

   12. Taylor expanded in a around 0 18.8%

       \[\leadsto \color{blue}{-2} \]

   if -2 < (*.f64 a x)

   1. Initial program 56.9%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

   4. Taylor expanded in a around 0 75.7%

      \[\leadsto \color{blue}{a \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \]

Alternative 9: 6.8% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (a x) :precision binary64 -2.0)

C

double code(double a, double x) {
	return -2.0;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = -2.0d0
end function

Java

public static double code(double a, double x) {
	return -2.0;
}

Python

def code(a, x):
	return -2.0

Julia

function code(a, x)
	return -2.0
end

MATLAB

function tmp = code(a, x)
	tmp = -2.0;
end

Wolfram

code[a_, x_] := -2.0

Derivation

1. Initial program 67.2%

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

   1. expm1-def 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

4. Taylor expanded in a around 0 60.2%

   \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
5. Step-by-step derivation

   1. +-commutative 60.2%

      \[\leadsto \color{blue}{a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)} \]

   2. associate-*r* 60.2%

      \[\leadsto a \cdot x + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {x}^{2}} \]

   3. unpow2 60.2%

      \[\leadsto a \cdot x + \left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. associate-*r* 65.5%

      \[\leadsto a \cdot x + \color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot x\right) \cdot x} \]

   5. distribute-rgt-out 65.6%

      \[\leadsto \color{blue}{x \cdot \left(a + \left(0.5 \cdot {a}^{2}\right) \cdot x\right)} \]

   6. *-commutative 65.6%

      \[\leadsto x \cdot \left(a + \color{blue}{x \cdot \left(0.5 \cdot {a}^{2}\right)}\right) \]

   7. *-commutative 65.6%

      \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left({a}^{2} \cdot 0.5\right)}\right) \]

   8. unpow2 65.6%

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

   9. associate-*l* 65.6%

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

6. Simplified 65.6%

   \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} \]
7. Step-by-step derivation

   1. distribute-lft-in 65.5%

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

   2. flip-+ 22.6%

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

   3. *-commutative 22.6%

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

   4. *-commutative 22.6%

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

   5. *-commutative 22.6%

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

   6. associate-*l* 22.6%

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

   7. *-commutative 22.6%

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

   8. associate-*l* 22.6%

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

   9. *-commutative 22.6%

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

   10. *-commutative 22.6%

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

   11. associate-*l* 25.4%

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

8. Applied egg-rr 25.4%

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

9. Taylor expanded in a around inf 4.1%

   \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{-0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)}} \]
10. Step-by-step derivation

    1. *-commutative 4.1%

       \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\color{blue}{\left({a}^{2} \cdot {x}^{2}\right) \cdot -0.5}} \]

    2. unpow2 4.1%

       \[\leadsto \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right) \cdot \left(x \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot x\right)\right)\right)}{\left(\color{blue}{\left(a \cdot a\right)} \cdot {x}^{2}\right) \cdot -0.5} \]

    3. unpow2 4.1%

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

11. Simplified 4.1%

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

12. Taylor expanded in a around 0 6.4%

    \[\leadsto \color{blue}{-2} \]

13. Final simplification 6.4%

    \[\leadsto -2 \]

Developer target: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (< (fabs (* a x)) 0.1)
   (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0))))
   (- (exp (* a x)) 1.0)))

C

double code(double a, double x) {
	double tmp;
	if (fabs((a * x)) < 0.1) {
		tmp = (a * x) * (1.0 + (((a * x) / 2.0) + (pow((a * x), 2.0) / 6.0)));
	} else {
		tmp = exp((a * x)) - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs((a * x)) < 0.1d0) then
        tmp = (a * x) * (1.0d0 + (((a * x) / 2.0d0) + (((a * x) ** 2.0d0) / 6.0d0)))
    else
        tmp = exp((a * x)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if (Math.abs((a * x)) < 0.1) {
		tmp = (a * x) * (1.0 + (((a * x) / 2.0) + (Math.pow((a * x), 2.0) / 6.0)));
	} else {
		tmp = Math.exp((a * x)) - 1.0;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if math.fabs((a * x)) < 0.1:
		tmp = (a * x) * (1.0 + (((a * x) / 2.0) + (math.pow((a * x), 2.0) / 6.0)))
	else:
		tmp = math.exp((a * x)) - 1.0
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (abs(Float64(a * x)) < 0.1)
		tmp = Float64(Float64(a * x) * Float64(1.0 + Float64(Float64(Float64(a * x) / 2.0) + Float64((Float64(a * x) ^ 2.0) / 6.0))));
	else
		tmp = Float64(exp(Float64(a * x)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if (abs((a * x)) < 0.1)
		tmp = (a * x) * (1.0 + (((a * x) / 2.0) + (((a * x) ^ 2.0) / 6.0)));
	else
		tmp = exp((a * x)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[Less[N[Abs[N[(a * x), $MachinePrecision]], $MachinePrecision], 0.1], N[(N[(a * x), $MachinePrecision] * N[(1.0 + N[(N[(N[(a * x), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Power[N[(a * x), $MachinePrecision], 2.0], $MachinePrecision] / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]

Reproduce

herbie shell --seed 2023200 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))