expax (section 3.5)

Percentage Accurate: 64.9% → 100.0%

Time: 4.5s

Alternatives: 5

Speedup: 8.0×

• 23.8% 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 65.1%

   \[e^{a \cdot x} - 1 \]
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. Final simplification 100.0%

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

Alternative 2: 65.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{-158}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + a \cdot \left(\left(a \cdot x\right) \cdot \left(x \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= x -3.25e-158)
   (* x (+ a (* x (* a (* a 0.5)))))
   (+ (* a x) (* a (* (* a x) (* x 0.5))))))

C

double code(double a, double x) {
	double tmp;
	if (x <= -3.25e-158) {
		tmp = x * (a + (x * (a * (a * 0.5))));
	} else {
		tmp = (a * x) + (a * ((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 (x <= (-3.25d-158)) then
        tmp = x * (a + (x * (a * (a * 0.5d0))))
    else
        tmp = (a * x) + (a * ((a * x) * (x * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if (x <= -3.25e-158) {
		tmp = x * (a + (x * (a * (a * 0.5))));
	} else {
		tmp = (a * x) + (a * ((a * x) * (x * 0.5)));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if x <= -3.25e-158:
		tmp = x * (a + (x * (a * (a * 0.5))))
	else:
		tmp = (a * x) + (a * ((a * x) * (x * 0.5)))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (x <= -3.25e-158)
		tmp = Float64(x * Float64(a + Float64(x * Float64(a * Float64(a * 0.5)))));
	else
		tmp = Float64(Float64(a * x) + Float64(a * Float64(Float64(a * x) * Float64(x * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if (x <= -3.25e-158)
		tmp = x * (a + (x * (a * (a * 0.5))));
	else
		tmp = (a * x) + (a * ((a * x) * (x * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[x, -3.25e-158], N[(x * N[(a + N[(x * N[(a * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * x), $MachinePrecision] + N[(a * N[(N[(a * x), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.24999999999999986e-158

   1. Initial program 72.2%

      \[e^{a \cdot x} - 1 \]
   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 53.3%

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

      1. +-commutative 53.3%

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

      2. associate-*r* 53.3%

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

      3. unpow2 53.3%

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

      4. associate-*r* 51.0%

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

      5. distribute-rgt-out 51.1%

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

      6. *-commutative 51.1%

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

      7. *-commutative 51.1%

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

      8. unpow2 51.1%

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

      9. associate-*l* 51.1%

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

   6. Simplified 51.1%

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

   if -3.24999999999999986e-158 < x

   1. Initial program 60.3%

      \[e^{a \cdot x} - 1 \]
   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 64.7%

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

      1. *-commutative 64.7%

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

      2. associate-*l* 64.7%

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

      3. unpow2 64.7%

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

      4. associate-*l* 67.7%

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

      5. distribute-lft-out 67.7%

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

      6. unpow2 67.7%

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

      7. associate-*l* 67.7%

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

   6. Simplified 67.7%

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

      1. distribute-lft-in 67.7%

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

      2. associate-*r* 72.4%

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

      3. *-commutative 72.4%

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

   8. Applied egg-rr 72.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{-158}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + a \cdot \left(\left(a \cdot x\right) \cdot \left(x \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 3: 67.7% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq 2 \cdot 10^{+44}:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x \cdot x\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) 2e+44) (* a x) (* 0.5 (* (* x x) (* a a)))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= 2e+44) {
		tmp = a * x;
	} else {
		tmp = 0.5 * ((x * x) * (a * a));
	}
	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) <= 2d+44) then
        tmp = a * x
    else
        tmp = 0.5d0 * ((x * x) * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= 2e+44) {
		tmp = a * x;
	} else {
		tmp = 0.5 * ((x * x) * (a * a));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= 2e+44:
		tmp = a * x
	else:
		tmp = 0.5 * ((x * x) * (a * a))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= 2e+44)
		tmp = Float64(a * x);
	else
		tmp = Float64(0.5 * Float64(Float64(x * x) * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= 2e+44)
		tmp = a * x;
	else
		tmp = 0.5 * ((x * x) * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], 2e+44], N[(a * x), $MachinePrecision], N[(0.5 * N[(N[(x * x), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < 2.0000000000000002e44

   1. Initial program 55.3%

      \[e^{a \cdot x} - 1 \]
   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 61.9%

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

   if 2.0000000000000002e44 < (*.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 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-*l* 84.3%

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

      3. unpow2 84.3%

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

      4. associate-*l* 74.8%

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

      5. distribute-lft-out 74.8%

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

      6. unpow2 74.8%

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

      7. associate-*l* 74.8%

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

   6. Simplified 74.8%

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

   7. Taylor expanded in a around inf 84.3%

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

      1. unpow2 84.3%

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

      2. unpow2 84.3%

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

      3. *-commutative 84.3%

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

   9. Simplified 84.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq 2 \cdot 10^{+44}:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x \cdot x\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 4: 65.5% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.7 \cdot 10^{-259}:\\ \;\;\;\;x \cdot \left(a + a \cdot \left(a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= a 4.7e-259)
   (* x (+ a (* a (* a (* x 0.5)))))
   (* a (+ x (* a (* x (* x 0.5)))))))

C

double code(double a, double x) {
	double tmp;
	if (a <= 4.7e-259) {
		tmp = x * (a + (a * (a * (x * 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 <= 4.7d-259) then
        tmp = x * (a + (a * (a * (x * 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 <= 4.7e-259) {
		tmp = x * (a + (a * (a * (x * 0.5))));
	} else {
		tmp = a * (x + (a * (x * (x * 0.5))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if a <= 4.7e-259:
		tmp = x * (a + (a * (a * (x * 0.5))))
	else:
		tmp = a * (x + (a * (x * (x * 0.5))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (a <= 4.7e-259)
		tmp = Float64(x * Float64(a + Float64(a * Float64(a * Float64(x * 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 <= 4.7e-259)
		tmp = x * (a + (a * (a * (x * 0.5))));
	else
		tmp = a * (x + (a * (x * (x * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[a, 4.7e-259], N[(x * N[(a + N[(a * N[(a * N[(x * 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 2 regimes

2. if a < 4.69999999999999998e-259

   1. Initial program 58.9%

      \[e^{a \cdot x} - 1 \]
   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 62.7%

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

      1. +-commutative 62.7%

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

      2. associate-*r* 62.7%

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

      3. unpow2 62.7%

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

      4. associate-*r* 66.0%

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

      5. distribute-rgt-out 66.0%

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

      6. *-commutative 66.0%

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

      7. *-commutative 66.0%

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

      8. unpow2 66.0%

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

      9. associate-*l* 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in x around 0 66.0%

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

      1. unpow2 66.0%

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

      2. associate-*r* 66.0%

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

      3. *-commutative 66.0%

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

      4. associate-*l* 66.0%

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

      5. associate-*r* 68.3%

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

      6. associate-*l* 68.3%

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

   9. Simplified 68.3%

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

   if 4.69999999999999998e-259 < a

   1. Initial program 72.4%

      \[e^{a \cdot x} - 1 \]
   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 57.0%

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

      1. *-commutative 57.0%

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

      2. associate-*l* 57.0%

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

      3. unpow2 57.0%

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

      4. associate-*l* 55.8%

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

      5. distribute-lft-out 55.9%

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

      6. unpow2 55.9%

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

      7. associate-*l* 55.9%

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

   6. Simplified 55.9%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.7 \cdot 10^{-259}:\\ \;\;\;\;x \cdot \left(a + a \cdot \left(a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 5: 57.8% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ a \cdot x \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = a * x
end function

Java

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

Python

def code(a, x):
	return a * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.1%

   \[e^{a \cdot x} - 1 \]
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 56.5%

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

5. Final simplification 56.5%

   \[\leadsto a \cdot x \]

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 2023256 
(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))