expax (section 3.5)

Percentage Accurate: 53.5% → 100.0%

Time: 10.0s

Alternatives: 8

Speedup: 10.5×

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: 53.5% 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 50.6%

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

   1. accelerator-lowering-expm1.f64 N/A

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

   2. *-lowering-*.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 72.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;-2 + \frac{-4}{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + a \cdot \left(a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1000.0)
   (+ -2.0 (/ -4.0 (* a x)))
   (* x (+ a (* a (* a (* x 0.5)))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1000.0) {
		tmp = -2.0 + (-4.0 / (a * x));
	} else {
		tmp = x * (a + (a * (a * (x * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-1000.0d0)) then
        tmp = (-2.0d0) + ((-4.0d0) / (a * x))
    else
        tmp = x * (a + (a * (a * (x * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1000.0) {
		tmp = -2.0 + (-4.0 / (a * x));
	} else {
		tmp = x * (a + (a * (a * (x * 0.5))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -1000.0:
		tmp = -2.0 + (-4.0 / (a * x))
	else:
		tmp = x * (a + (a * (a * (x * 0.5))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -1000.0)
		tmp = Float64(-2.0 + Float64(-4.0 / Float64(a * x)));
	else
		tmp = Float64(x * Float64(a + Float64(a * Float64(a * Float64(x * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -1000.0)
		tmp = -2.0 + (-4.0 / (a * x));
	else
		tmp = x * (a + (a * (a * (x * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -1000.0], N[(-2.0 + N[(-4.0 / N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a + N[(a * N[(a * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1e3

   1. Initial program 100.0%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right) + \frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{x}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) + x\right) \]

      3. associate-+l+ N/A

         \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + x\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto a \cdot \left(a \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot {x}^{3}\right) + \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{3} + \left(\color{blue}{a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   5. Simplified 3.7%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}}\right)\right)\right) \]

      5. flip3-+ N/A

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

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

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

   7. Applied egg-rr 3.7%

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

   8. Taylor expanded in a around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 15.8%

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

   10. Simplified 15.8%

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

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{-1 \cdot \left(2 + 4 \cdot \frac{1}{a \cdot x}\right)} \]
   12. 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. metadata-eval N/A

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

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

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. distribute-neg-frac N/A

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

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

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

       9. metadata-eval N/A

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

       10. *-lowering-*.f64 18.8%

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

   13. Simplified 18.8%

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

   if -1e3 < (*.f64 a x)

   1. Initial program 26.5%

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

      1. flip3-- N/A

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

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

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

      3. pow-exp N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\left(a \cdot x\right) \cdot 3\right)\right), \left(\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x}} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(a \cdot \left(x \cdot 3\right)\right)\right), \left(\color{blue}{e^{a \cdot x}} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \left(x \cdot 3\right)\right)\right), \left(\color{blue}{e^{a \cdot x}} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 3\right)\right)\right), \left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 3\right)\right)\right), \left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + \color{blue}{e^{a \cdot x}} \cdot 1\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 3\right)\right)\right), \left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)\right)\right) \]

      11. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 3\right)\right)\right), \left(\left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right) + \color{blue}{e^{a \cdot x}}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 3\right)\right)\right), \left(e^{a \cdot x} + \color{blue}{\left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 3\right)\right)\right), \mathsf{+.f64}\left(\left(e^{a \cdot x}\right), \color{blue}{\left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 3\right)\right)\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(a \cdot x\right)\right), \left(\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x}} + 1\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 3\right)\right)\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), \left(\color{blue}{e^{a \cdot x}} \cdot e^{a \cdot x} + 1\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 3\right)\right)\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), \left(1 + \color{blue}{e^{a \cdot x} \cdot e^{a \cdot x}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 3\right)\right)\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a \cdot x} \cdot e^{a \cdot x}\right)}\right)\right)\right) \]

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. distribute-lft-out-- N/A

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

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

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

      16. --lowering--.f64 N/A

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

      17. *-lowering-*.f64 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 99.0%

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

   10. Simplified 99.0%

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

Alternative 3: 72.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;-2 + \frac{-4}{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(1 + \left(a \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1000.0)
   (+ -2.0 (/ -4.0 (* a x)))
   (* a (* x (+ 1.0 (* (* a x) 0.5))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1000.0) {
		tmp = -2.0 + (-4.0 / (a * x));
	} else {
		tmp = a * (x * (1.0 + ((a * x) * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-1000.0d0)) then
        tmp = (-2.0d0) + ((-4.0d0) / (a * x))
    else
        tmp = a * (x * (1.0d0 + ((a * x) * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1000.0) {
		tmp = -2.0 + (-4.0 / (a * x));
	} else {
		tmp = a * (x * (1.0 + ((a * x) * 0.5)));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -1000.0:
		tmp = -2.0 + (-4.0 / (a * x))
	else:
		tmp = a * (x * (1.0 + ((a * x) * 0.5)))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -1000.0)
		tmp = Float64(-2.0 + Float64(-4.0 / Float64(a * x)));
	else
		tmp = Float64(a * Float64(x * Float64(1.0 + Float64(Float64(a * x) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -1000.0)
		tmp = -2.0 + (-4.0 / (a * x));
	else
		tmp = a * (x * (1.0 + ((a * x) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -1000.0], N[(-2.0 + N[(-4.0 / N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * N[(1.0 + N[(N[(a * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1e3

   1. Initial program 100.0%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right) + \frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{x}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) + x\right) \]

      3. associate-+l+ N/A

         \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + x\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto a \cdot \left(a \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot {x}^{3}\right) + \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{3} + \left(\color{blue}{a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   5. Simplified 3.7%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}}\right)\right)\right) \]

      5. flip3-+ N/A

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

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

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

   7. Applied egg-rr 3.7%

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

   8. Taylor expanded in a around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 15.8%

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

   10. Simplified 15.8%

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

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{-1 \cdot \left(2 + 4 \cdot \frac{1}{a \cdot x}\right)} \]
   12. 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. metadata-eval N/A

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

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

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. distribute-neg-frac N/A

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

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

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

       9. metadata-eval N/A

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

       10. *-lowering-*.f64 18.8%

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

   13. Simplified 18.8%

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

   if -1e3 < (*.f64 a x)

   1. Initial program 26.5%

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

      1. accelerator-lowering-expm1.f64 N/A

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

      2. *-lowering-*.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. *-lowering-*.f64 99.0%

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

   7. Simplified 99.0%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;-2 + \frac{-4}{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(1 + \left(a \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2.4:\\ \;\;\;\;-2 + \frac{-4}{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -2.4) (+ -2.0 (/ -4.0 (* a x))) (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2.4) {
		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) <= (-2.4d0)) 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) <= -2.4) {
		tmp = -2.0 + (-4.0 / (a * x));
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -2.4:
		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) <= -2.4)
		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) <= -2.4)
		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], -2.4], 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) < -2.39999999999999991

   1. Initial program 100.0%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right) + \frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{x}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) + x\right) \]

      3. associate-+l+ N/A

         \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + x\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto a \cdot \left(a \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot {x}^{3}\right) + \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{3} + \left(\color{blue}{a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   5. Simplified 3.7%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}}\right)\right)\right) \]

      5. flip3-+ N/A

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

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

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

   7. Applied egg-rr 3.7%

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

   8. Taylor expanded in a around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 15.8%

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

   10. Simplified 15.8%

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

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{-1 \cdot \left(2 + 4 \cdot \frac{1}{a \cdot x}\right)} \]
   12. 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. metadata-eval N/A

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

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

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. distribute-neg-frac N/A

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

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

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

       9. metadata-eval N/A

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

       10. *-lowering-*.f64 18.8%

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

   13. Simplified 18.8%

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

   if -2.39999999999999991 < (*.f64 a x)

   1. Initial program 26.5%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 98.5%

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

   5. Simplified 98.5%

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

Alternative 5: 71.0% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.6%

   \[e^{a \cdot x} - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right) + \frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{x}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) + x\right) \]

   3. associate-+l+ N/A

      \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + x\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto a \cdot \left(a \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot {x}^{3}\right) + \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{3} + \left(\color{blue}{a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

   6. unpow3 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. distribute-lft1-in N/A

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

   14. distribute-rgt-out N/A

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

5. Simplified 67.8%

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

   1. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}\right)}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}}\right)\right)\right) \]

   5. flip3-+ N/A

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

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

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

7. Applied egg-rr 67.8%

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

8. Taylor expanded in a around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 71.7%

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

10. Simplified 71.7%

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

    1. clear-num N/A

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

    2. un-div-inv N/A

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

    3. div-inv N/A

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

    4. metadata-eval N/A

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

    5. times-frac N/A

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

    6. /-rgt-identity N/A

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

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

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

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

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

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

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

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

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

    11. *-lowering-*.f64 71.9%

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

12. Applied egg-rr 71.9%

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

13. Final simplification 71.9%

    \[\leadsto x \cdot \frac{a}{1 + a \cdot \left(x \cdot -0.5\right)} \]
14. Add Preprocessing

Alternative 6: 71.6% 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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right) + \frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{x}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) + x\right) \]

      3. associate-+l+ N/A

         \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + x\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto a \cdot \left(a \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot {x}^{3}\right) + \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{3} + \left(\color{blue}{a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   5. Simplified 3.7%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}}\right)\right)\right) \]

      5. flip3-+ N/A

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

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

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

   7. Applied egg-rr 3.7%

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

   8. Taylor expanded in a around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 15.8%

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

   10. Simplified 15.8%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 18.8%

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

   if -2 < (*.f64 a x)

   1. Initial program 26.5%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 98.5%

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

   5. Simplified 98.5%

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

Alternative 7: 21.2% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (if (<= a -3.2e-7) -2.0 0.0))

C

double code(double a, double x) {
	double tmp;
	if (a <= -3.2e-7) {
		tmp = -2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if (a <= (-3.2d-7)) then
        tmp = -2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if (a <= -3.2e-7) {
		tmp = -2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if a <= -3.2e-7:
		tmp = -2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (a <= -3.2e-7)
		tmp = -2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if (a <= -3.2e-7)
		tmp = -2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[a, -3.2e-7], -2.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if a < -3.2000000000000001e-7

   1. Initial program 76.1%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right) + \frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{x}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) + x\right) \]

      3. associate-+l+ N/A

         \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + x\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto a \cdot \left(a \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot {x}^{3}\right) + \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{3} + \left(\color{blue}{a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   5. Simplified 28.0%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}}\right)\right)\right) \]

      5. flip3-+ N/A

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

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

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

   7. Applied egg-rr 28.0%

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

   8. Taylor expanded in a around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 36.0%

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

   10. Simplified 36.0%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 14.7%

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

   if -3.2000000000000001e-7 < a

   1. Initial program 44.7%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, 1\right) \]
   4. Step-by-step derivation

   5. Simplified 20.6%

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

      1. metadata-eval 20.6%

         \[\leadsto 0 \]

   7. Applied egg-rr 20.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 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 50.6%

   \[e^{a \cdot x} - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right) + \frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{x}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) + x\right) \]

   3. associate-+l+ N/A

      \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + \color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + x\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto a \cdot \left(a \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot {x}^{3}\right) + \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{3} + \left(\color{blue}{a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)} + x\right)\right) \]

   6. unpow3 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. distribute-lft1-in N/A

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

   14. distribute-rgt-out N/A

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

5. Simplified 67.8%

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

   1. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}\right)}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(a \cdot \frac{1}{6}\right)\right)\right)\right)}}}\right)\right)\right) \]

   5. flip3-+ N/A

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

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

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

7. Applied egg-rr 67.8%

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

8. Taylor expanded in a around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 71.7%

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

10. Simplified 71.7%

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

11. Taylor expanded in a around inf

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

13. Simplified 8.6%

    \[\leadsto \color{blue}{-2} \]
14. 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 2024191 
(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))