expax (section 3.5)

Percentage Accurate: 65.0% → 100.0%

Time: 5.3s

Alternatives: 5

Speedup: 8.0×

• 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: 65.0% 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 73.6%

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

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

Alternative 2: 67.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq 100000000:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(a \cdot a\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) 100000000.0) (* a x) (* 0.5 (* (* a a) (* x x)))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= 100000000.0) {
		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) <= 100000000.0d0) 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) <= 100000000.0) {
		tmp = a * x;
	} else {
		tmp = 0.5 * ((a * a) * (x * x));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= 100000000.0:
		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) <= 100000000.0)
		tmp = Float64(a * x);
	else
		tmp = Float64(0.5 * Float64(Float64(a * a) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= 100000000.0)
		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], 100000000.0], N[(a * x), $MachinePrecision], N[(0.5 * N[(N[(a * a), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < 1e8

   1. Initial program 63.4%

      \[e^{a \cdot x} - 1 \]
   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 62.0%

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

   if 1e8 < (*.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 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-*l* 78.3%

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

      3. unpow2 78.3%

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

      4. associate-*l* 74.7%

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

      5. distribute-lft-out 74.7%

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

      6. unpow2 74.7%

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

      7. associate-*l* 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in a around inf 78.3%

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

      1. unpow2 78.3%

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

      2. unpow2 78.3%

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

      3. *-commutative 78.3%

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

   9. Simplified 78.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq 100000000:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(a \cdot a\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 3: 67.4% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq 0.0005:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot 0.5\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) 0.0005) (* a x) (* x (* (* x 0.5) (* a a)))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= 0.0005) {
		tmp = a * x;
	} 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) :: tmp
    if ((a * x) <= 0.0005d0) then
        tmp = a * x
    else
        tmp = x * ((x * 0.5d0) * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= 0.0005) {
		tmp = a * x;
	} else {
		tmp = x * ((x * 0.5) * (a * a));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= 0.0005:
		tmp = a * x
	else:
		tmp = x * ((x * 0.5) * (a * a))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= 0.0005)
		tmp = Float64(a * x);
	else
		tmp = Float64(x * Float64(Float64(x * 0.5) * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= 0.0005)
		tmp = a * x;
	else
		tmp = x * ((x * 0.5) * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], 0.0005], N[(a * x), $MachinePrecision], N[(x * N[(N[(x * 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < 5.0000000000000001e-4

   1. Initial program 62.6%

      \[e^{a \cdot x} - 1 \]
   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 63.2%

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

   if 5.0000000000000001e-4 < (*.f64 a x)

   1. Initial program 99.9%

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

      1. expm1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 74.4%

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

      1. +-commutative 74.4%

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

      2. associate-*r* 74.4%

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

      3. unpow2 74.4%

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

      4. associate-*r* 73.7%

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

      5. distribute-rgt-out 73.7%

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

      6. *-commutative 73.7%

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

      7. *-commutative 73.7%

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

      8. unpow2 73.7%

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

      9. associate-*l* 73.7%

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

   6. Simplified 73.7%

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

      1. expm1-log1p-u 35.6%

         \[\leadsto x \cdot \left(a + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\right)}\right) \]

      2. expm1-udef 35.6%

         \[\leadsto x \cdot \left(a + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} - 1\right)}\right) \]

      3. associate-*r* 33.0%

         \[\leadsto x \cdot \left(a + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot a\right) \cdot \left(a \cdot 0.5\right)}\right)} - 1\right)\right) \]

      4. *-commutative 33.0%

         \[\leadsto x \cdot \left(a + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(a \cdot x\right)} \cdot \left(a \cdot 0.5\right)\right)} - 1\right)\right) \]

      5. associate-*l* 33.0%

         \[\leadsto x \cdot \left(a + \left(e^{\mathsf{log1p}\left(\color{blue}{a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)}\right)} - 1\right)\right) \]

   8. Applied egg-rr 33.0%

      \[\leadsto x \cdot \left(a + \color{blue}{\left(e^{\mathsf{log1p}\left(a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)} - 1\right)}\right) \]
   9. Step-by-step derivation

      1. expm1-def 33.1%

         \[\leadsto x \cdot \left(a + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\right)}\right) \]

      2. expm1-log1p 68.6%

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

      3. associate-*r* 68.6%

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

      4. *-commutative 68.6%

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

      5. associate-*r* 68.6%

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

   10. Simplified 68.6%

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

       1. *-un-lft-identity 68.6%

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

       2. *-commutative 68.6%

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

       3. associate-*r* 68.6%

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

       4. *-commutative 68.6%

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

       5. associate-*r* 68.6%

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

       6. distribute-rgt-out 68.6%

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

       7. associate-*r* 68.6%

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

       8. *-commutative 68.6%

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

       9. associate-*r* 68.6%

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

   12. Applied egg-rr 68.6%

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

   13. Taylor expanded in a around inf 73.6%

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

       1. unpow2 73.6%

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

       2. *-commutative 73.6%

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

       3. associate-*r* 73.6%

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

       4. *-commutative 73.6%

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

   15. Simplified 73.6%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq 0.0005:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot 0.5\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 4: 65.5% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-278}:\\ \;\;\;\;x \cdot \left(a \cdot \left(1 + a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= a 3.5e-278)
   (* x (* a (+ 1.0 (* a (* x 0.5)))))
   (* a (+ x (* a (* x (* x 0.5)))))))

C

double code(double a, double x) {
	double tmp;
	if (a <= 3.5e-278) {
		tmp = x * (a * (1.0 + (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 <= 3.5d-278) then
        tmp = x * (a * (1.0d0 + (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 <= 3.5e-278) {
		tmp = x * (a * (1.0 + (a * (x * 0.5))));
	} else {
		tmp = a * (x + (a * (x * (x * 0.5))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if a <= 3.5e-278:
		tmp = x * (a * (1.0 + (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 <= 3.5e-278)
		tmp = Float64(x * Float64(a * Float64(1.0 + 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 <= 3.5e-278)
		tmp = x * (a * (1.0 + (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, 3.5e-278], N[(x * N[(a * N[(1.0 + 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 < 3.4999999999999997e-278

   1. Initial program 74.6%

      \[e^{a \cdot x} - 1 \]
   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 60.2%

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x} \]
   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* 61.2%

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

      5. distribute-rgt-out 61.2%

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

      6. *-commutative 61.2%

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

      7. *-commutative 61.2%

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

      8. unpow2 61.2%

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

      9. associate-*l* 61.2%

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

   6. Simplified 61.2%

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

      1. expm1-log1p-u 40.5%

         \[\leadsto x \cdot \left(a + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\right)}\right) \]

      2. expm1-udef 40.6%

         \[\leadsto x \cdot \left(a + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)} - 1\right)}\right) \]

      3. associate-*r* 40.6%

         \[\leadsto x \cdot \left(a + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot a\right) \cdot \left(a \cdot 0.5\right)}\right)} - 1\right)\right) \]

      4. *-commutative 40.6%

         \[\leadsto x \cdot \left(a + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(a \cdot x\right)} \cdot \left(a \cdot 0.5\right)\right)} - 1\right)\right) \]

      5. associate-*l* 40.6%

         \[\leadsto x \cdot \left(a + \left(e^{\mathsf{log1p}\left(\color{blue}{a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)}\right)} - 1\right)\right) \]

   8. Applied egg-rr 40.6%

      \[\leadsto x \cdot \left(a + \color{blue}{\left(e^{\mathsf{log1p}\left(a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)} - 1\right)}\right) \]
   9. Step-by-step derivation

      1. expm1-def 41.0%

         \[\leadsto x \cdot \left(a + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\right)}\right) \]

      2. expm1-log1p 63.8%

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

      3. associate-*r* 63.8%

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

      4. *-commutative 63.8%

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

      5. associate-*r* 63.8%

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

   10. Simplified 63.8%

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

       1. *-un-lft-identity 63.8%

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

       2. *-commutative 63.8%

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

       3. associate-*r* 63.8%

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

       4. *-commutative 63.8%

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

       5. associate-*r* 63.8%

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

       6. distribute-rgt-out 63.8%

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

       7. associate-*r* 63.8%

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

       8. *-commutative 63.8%

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

       9. associate-*r* 63.8%

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

   12. Applied egg-rr 63.8%

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

   if 3.4999999999999997e-278 < a

   1. Initial program 72.3%

      \[e^{a \cdot x} - 1 \]
   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 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*l* 61.6%

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

      3. unpow2 61.6%

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

      4. associate-*l* 63.6%

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

      5. distribute-lft-out 63.7%

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

      6. unpow2 63.7%

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

      7. associate-*l* 63.7%

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

   6. Simplified 63.7%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-278}:\\ \;\;\;\;x \cdot \left(a \cdot \left(1 + a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 5: 57.7% 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 73.6%

   \[e^{a \cdot x} - 1 \]
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 53.0%

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

5. Final simplification 53.0%

   \[\leadsto a \cdot x \]

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 2023250 
(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))