expax (section 3.5)

Percentage Accurate: 53.3% → 100.0%

Time: 9.9s

Alternatives: 6

Speedup: 11.7×

Specification

\[710 > a \cdot x\]

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: 53.3% 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 54.2%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 N/A

      \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 72.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1}{x \cdot \left(\frac{a}{a} + -1.5\right) + \frac{1}{a}} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (* x (/ 1.0 (+ (* x (+ (/ a a) -1.5)) (/ 1.0 a)))))

C

double code(double a, double x) {
	return x * (1.0 / ((x * ((a / a) + -1.5)) + (1.0 / a)));
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = x * (1.0d0 / ((x * ((a / a) + (-1.5d0))) + (1.0d0 / a)))
end function

Java

public static double code(double a, double x) {
	return x * (1.0 / ((x * ((a / a) + -1.5)) + (1.0 / a)));
}

Python

def code(a, x):
	return x * (1.0 / ((x * ((a / a) + -1.5)) + (1.0 / a)))

Julia

function code(a, x)
	return Float64(x * Float64(1.0 / Float64(Float64(x * Float64(Float64(a / a) + -1.5)) + Float64(1.0 / a))))
end

MATLAB

function tmp = code(a, x)
	tmp = x * (1.0 / ((x * ((a / a) + -1.5)) + (1.0 / a)));
end

Wolfram

code[a_, x_] := N[(x * N[(1.0 / N[(N[(x * N[(N[(a / a), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 N/A

      \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

   1. sub-neg N/A

      \[\leadsto e^{a \cdot x} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

   2. flip3-+ N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + {\left(\mathsf{neg}\left(1\right)\right)}^{3}}{\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + {-1}^{3}}{e^{a \cdot x} \cdot e^{\color{blue}{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + -1}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + \left(\mathsf{neg}\left(1\right)\right)}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   6. sub-neg N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   8. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}}} \]

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

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

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \color{blue}{\left({\left(e^{a \cdot x}\right)}^{3} - {1}^{3}\right)}\right)\right) \]

6. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + e^{a \cdot \left(x \cdot 2\right)}\right) - e^{a \cdot x} \cdot -1}{\mathsf{expm1}\left(a \cdot \left(x \cdot 3\right)\right)}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a} - \frac{3}{2}\right) + \frac{1}{a}}{x}\right)}\right) \]
8. Step-by-step derivation

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

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

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

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

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

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

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a}\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{2 \cdot a - -1 \cdot a}{a}\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(2 \cdot a - -1 \cdot a\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

   8. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(a \cdot \left(2 - -1\right)\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(a \cdot 3\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \frac{-3}{2}\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

   12. /-lowering-/.f64 71.2%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \frac{-3}{2}\right)\right), \mathsf{/.f64}\left(1, a\right)\right), x\right)\right) \]

9. Simplified 71.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \left(0.3333333333333333 \cdot \frac{a \cdot 3}{a} + -1.5\right) + \frac{1}{a}}{x}}} \]
10. Step-by-step derivation

    1. associate-/r/ N/A

       \[\leadsto \frac{1}{x \cdot \left(\frac{1}{3} \cdot \frac{a \cdot 3}{a} + \frac{-3}{2}\right) + \frac{1}{a}} \cdot \color{blue}{x} \]

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

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

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

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

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

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

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

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

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

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

    7. associate-*r/ N/A

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

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

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

    9. *-commutative N/A

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

    10. associate-*r* N/A

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

    11. metadata-eval N/A

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

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, a\right), a\right), \frac{-3}{2}\right)\right), \left(\frac{1}{a}\right)\right)\right), x\right) \]

    13. /-lowering-/.f64 71.9%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, a\right), a\right), \frac{-3}{2}\right)\right), \mathsf{/.f64}\left(1, a\right)\right)\right), x\right) \]

11. Applied egg-rr 71.9%

    \[\leadsto \color{blue}{\frac{1}{x \cdot \left(\frac{1 \cdot a}{a} + -1.5\right) + \frac{1}{a}} \cdot x} \]

12. Final simplification 71.9%

    \[\leadsto x \cdot \frac{1}{x \cdot \left(\frac{a}{a} + -1.5\right) + \frac{1}{a}} \]
13. Add Preprocessing

Alternative 3: 72.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500:\\ \;\;\;\;-2 + \frac{-4}{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -500.0) (+ -2.0 (/ -4.0 (* a x))) (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -500.0) {
		tmp = -2.0 + (-4.0 / (a * x));
	} 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) <= (-500.0d0)) then
        tmp = (-2.0d0) + ((-4.0d0) / (a * x))
    else
        tmp = a * 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 + (-4.0 / (a * x));
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -500.0:
		tmp = -2.0 + (-4.0 / (a * x))
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -500.0)
		tmp = Float64(-2.0 + Float64(-4.0 / Float64(a * x)));
	else
		tmp = Float64(a * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -500.0)
		tmp = -2.0 + (-4.0 / (a * x));
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -500.0], N[(-2.0 + N[(-4.0 / N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. expm1-define N/A

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

      2. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

      3. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

      1. sub-neg N/A

         \[\leadsto e^{a \cdot x} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. flip3-+ N/A

         \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + {\left(\mathsf{neg}\left(1\right)\right)}^{3}}{\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + {-1}^{3}}{e^{a \cdot x} \cdot e^{\color{blue}{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + -1}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + \left(\mathsf{neg}\left(1\right)\right)}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}}} \]

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

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

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \color{blue}{\left({\left(e^{a \cdot x}\right)}^{3} - {1}^{3}\right)}\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + e^{a \cdot \left(x \cdot 2\right)}\right) - e^{a \cdot x} \cdot -1}{\mathsf{expm1}\left(a \cdot \left(x \cdot 3\right)\right)}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a} - \frac{3}{2}\right) + \frac{1}{a}}{x}\right)}\right) \]
   8. Step-by-step derivation

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

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

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

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

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

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

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a}\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{2 \cdot a - -1 \cdot a}{a}\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(2 \cdot a - -1 \cdot a\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(a \cdot \left(2 - -1\right)\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(a \cdot 3\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \frac{-3}{2}\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

      12. /-lowering-/.f64 18.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \frac{-3}{2}\right)\right), \mathsf{/.f64}\left(1, a\right)\right), x\right)\right) \]

   9. Simplified 18.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \left(0.3333333333333333 \cdot \frac{a \cdot 3}{a} + -1.5\right) + \frac{1}{a}}{x}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-1 \cdot \left(2 + 4 \cdot \frac{1}{a \cdot x}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\left(2 + 4 \cdot \frac{1}{a \cdot x}\right)\right) \]

       2. distribute-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(2\right)\right) + \color{blue}{\left(\mathsf{neg}\left(4 \cdot \frac{1}{a \cdot x}\right)\right)} \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{\left(\mathsf{neg}\left(4 \cdot \frac{1}{a \cdot x}\right)\right)}\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\mathsf{neg}\left(\color{blue}{4 \cdot \frac{1}{a \cdot x}}\right)\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(-1 \cdot \color{blue}{\left(4 \cdot \frac{1}{a \cdot x}\right)}\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(-1 \cdot \frac{4 \cdot 1}{\color{blue}{a \cdot x}}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(-1 \cdot \frac{4}{\color{blue}{a} \cdot x}\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{-1 \cdot 4}{\color{blue}{a \cdot x}}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{-4}{\color{blue}{a} \cdot x}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\mathsf{neg}\left(4\right)}{\color{blue}{a} \cdot x}\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(4\right)\right), \color{blue}{\left(a \cdot x\right)}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(-4, \left(\color{blue}{a} \cdot x\right)\right)\right) \]

       13. *-lowering-*.f64 18.8%

           \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(a, \color{blue}{x}\right)\right)\right) \]

   12. Simplified 18.8%

       \[\leadsto \color{blue}{-2 + \frac{-4}{a \cdot x}} \]

   if -500 < (*.f64 a x)

   1. Initial program 31.0%

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

      1. expm1-define N/A

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

      2. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

      3. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 98.6%

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

   7. Simplified 98.6%

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

Alternative 4: 72.2% accurate, 10.5× 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-define N/A

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

      2. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

      3. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

      1. sub-neg N/A

         \[\leadsto e^{a \cdot x} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. flip3-+ N/A

         \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + {\left(\mathsf{neg}\left(1\right)\right)}^{3}}{\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + {-1}^{3}}{e^{a \cdot x} \cdot e^{\color{blue}{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + -1}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + \left(\mathsf{neg}\left(1\right)\right)}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}}} \]

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

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

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \color{blue}{\left({\left(e^{a \cdot x}\right)}^{3} - {1}^{3}\right)}\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + e^{a \cdot \left(x \cdot 2\right)}\right) - e^{a \cdot x} \cdot -1}{\mathsf{expm1}\left(a \cdot \left(x \cdot 3\right)\right)}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a} - \frac{3}{2}\right) + \frac{1}{a}}{x}\right)}\right) \]
   8. Step-by-step derivation

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

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

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

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

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

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

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a}\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{2 \cdot a - -1 \cdot a}{a}\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(2 \cdot a - -1 \cdot a\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(a \cdot \left(2 - -1\right)\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(a \cdot 3\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \frac{-3}{2}\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

      12. /-lowering-/.f64 18.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \frac{-3}{2}\right)\right), \mathsf{/.f64}\left(1, a\right)\right), x\right)\right) \]

   9. Simplified 18.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \left(0.3333333333333333 \cdot \frac{a \cdot 3}{a} + -1.5\right) + \frac{1}{a}}{x}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-2} \]
   11. Step-by-step derivation

   12. Simplified 18.8%

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

   if -2 < (*.f64 a x)

   1. Initial program 31.0%

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

      1. expm1-define N/A

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

      2. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

      3. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 98.6%

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

   7. Simplified 98.6%

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

Alternative 5: 72.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{a \cdot x} + -0.5} \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (/ 1.0 (+ (/ 1.0 (* a x)) -0.5)))

C

double code(double a, double x) {
	return 1.0 / ((1.0 / (a * x)) + -0.5);
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = 1.0d0 / ((1.0d0 / (a * x)) + (-0.5d0))
end function

Java

public static double code(double a, double x) {
	return 1.0 / ((1.0 / (a * x)) + -0.5);
}

Python

def code(a, x):
	return 1.0 / ((1.0 / (a * x)) + -0.5)

Julia

function code(a, x)
	return Float64(1.0 / Float64(Float64(1.0 / Float64(a * x)) + -0.5))
end

MATLAB

function tmp = code(a, x)
	tmp = 1.0 / ((1.0 / (a * x)) + -0.5);
end

Wolfram

code[a_, x_] := N[(1.0 / N[(N[(1.0 / N[(a * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 N/A

      \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

   1. sub-neg N/A

      \[\leadsto e^{a \cdot x} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

   2. flip3-+ N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + {\left(\mathsf{neg}\left(1\right)\right)}^{3}}{\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + {-1}^{3}}{e^{a \cdot x} \cdot e^{\color{blue}{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + -1}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + \left(\mathsf{neg}\left(1\right)\right)}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   6. sub-neg N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   8. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}}} \]

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

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

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \color{blue}{\left({\left(e^{a \cdot x}\right)}^{3} - {1}^{3}\right)}\right)\right) \]

6. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + e^{a \cdot \left(x \cdot 2\right)}\right) - e^{a \cdot x} \cdot -1}{\mathsf{expm1}\left(a \cdot \left(x \cdot 3\right)\right)}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a} - \frac{3}{2}\right) + \frac{1}{a}}{x}\right)}\right) \]
8. Step-by-step derivation

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

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

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

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

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

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

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a}\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{2 \cdot a - -1 \cdot a}{a}\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(2 \cdot a - -1 \cdot a\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

   8. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(a \cdot \left(2 - -1\right)\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(a \cdot 3\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \frac{-3}{2}\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

   12. /-lowering-/.f64 71.2%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \frac{-3}{2}\right)\right), \mathsf{/.f64}\left(1, a\right)\right), x\right)\right) \]

9. Simplified 71.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \left(0.3333333333333333 \cdot \frac{a \cdot 3}{a} + -1.5\right) + \frac{1}{a}}{x}}} \]

10. Taylor expanded in x around inf

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

    1. sub-neg N/A

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

    2. metadata-eval N/A

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

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

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

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

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

    5. *-lowering-*.f64 71.3%

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

12. Simplified 71.3%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{a \cdot x} + -0.5}} \]
13. Add Preprocessing

Alternative 6: 8.6% 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 54.2%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 N/A

      \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

   1. sub-neg N/A

      \[\leadsto e^{a \cdot x} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

   2. flip3-+ N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + {\left(\mathsf{neg}\left(1\right)\right)}^{3}}{\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + {-1}^{3}}{e^{a \cdot x} \cdot e^{\color{blue}{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + -1}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} + \left(\mathsf{neg}\left(1\right)\right)}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   6. sub-neg N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   8. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)}{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}}} \]

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

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

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) - e^{a \cdot x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \color{blue}{\left({\left(e^{a \cdot x}\right)}^{3} - {1}^{3}\right)}\right)\right) \]

6. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + e^{a \cdot \left(x \cdot 2\right)}\right) - e^{a \cdot x} \cdot -1}{\mathsf{expm1}\left(a \cdot \left(x \cdot 3\right)\right)}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a} - \frac{3}{2}\right) + \frac{1}{a}}{x}\right)}\right) \]
8. Step-by-step derivation

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

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

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

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

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

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

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot \frac{2 \cdot a - -1 \cdot a}{a}\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{2 \cdot a - -1 \cdot a}{a}\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(2 \cdot a - -1 \cdot a\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

   8. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(a \cdot \left(2 - -1\right)\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(a \cdot 3\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \frac{-3}{2}\right)\right), \left(\frac{1}{a}\right)\right), x\right)\right) \]

   12. /-lowering-/.f64 71.2%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 3\right), a\right)\right), \frac{-3}{2}\right)\right), \mathsf{/.f64}\left(1, a\right)\right), x\right)\right) \]

9. Simplified 71.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \left(0.3333333333333333 \cdot \frac{a \cdot 3}{a} + -1.5\right) + \frac{1}{a}}{x}}} \]

10. Taylor expanded in x around inf

    \[\leadsto \color{blue}{-2} \]
11. Step-by-step derivation

12. Simplified 8.7%

    \[\leadsto \color{blue}{-2} \]
13. Add Preprocessing

Developer Target 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]

Reproduce

herbie shell --seed 2024161 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :pre (> 710.0 (* a x))

  :alt
  (! :herbie-platform default (expm1 (* a x)))

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