expax (section 3.5)

Percentage Accurate: 53.9% → 100.0%

Time: 8.0s

Alternatives: 7

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: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

   1. expm1-define N/A

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

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

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

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 72.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -40000:\\ \;\;\;\;-2 + \frac{-4}{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(a \cdot x\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -40000.0)
   (+ -2.0 (/ -4.0 (* a x)))
   (* (* a x) (+ 1.0 (* (* a x) (+ 0.5 (* a (* x 0.16666666666666666))))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -40000.0) {
		tmp = -2.0 + (-4.0 / (a * x));
	} else {
		tmp = (a * x) * (1.0 + ((a * x) * (0.5 + (a * (x * 0.16666666666666666)))));
	}
	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) <= (-40000.0d0)) then
        tmp = (-2.0d0) + ((-4.0d0) / (a * x))
    else
        tmp = (a * x) * (1.0d0 + ((a * x) * (0.5d0 + (a * (x * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -40000.0) {
		tmp = -2.0 + (-4.0 / (a * x));
	} else {
		tmp = (a * x) * (1.0 + ((a * x) * (0.5 + (a * (x * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -40000.0:
		tmp = -2.0 + (-4.0 / (a * x))
	else:
		tmp = (a * x) * (1.0 + ((a * x) * (0.5 + (a * (x * 0.16666666666666666)))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -40000.0)
		tmp = Float64(-2.0 + Float64(-4.0 / Float64(a * x)));
	else
		tmp = Float64(Float64(a * x) * Float64(1.0 + Float64(Float64(a * x) * Float64(0.5 + Float64(a * Float64(x * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -40000.0)
		tmp = -2.0 + (-4.0 / (a * x));
	else
		tmp = (a * x) * (1.0 + ((a * x) * (0.5 + (a * (x * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -40000.0], N[(-2.0 + N[(-4.0 / N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * x), $MachinePrecision] * N[(1.0 + N[(N[(a * x), $MachinePrecision] * N[(0.5 + N[(a * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -4e4

   1. Initial program 100.0%

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

      1. expm1-define N/A

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

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

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

      3. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. 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. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 0.8%

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

   7. Simplified 0.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 0.8%

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

   9. Applied egg-rr 0.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot x\right)\right)}\right)\right) \]
   11. 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 12.8%

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

   12. Simplified 12.8%

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

   13. Taylor expanded in a around inf

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

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. metadata-eval N/A

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

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

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

       12. metadata-eval N/A

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

       13. *-lowering-*.f64 18.8%

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

   15. Simplified 18.8%

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

   if -4e4 < (*.f64 a x)

   1. Initial program 30.8%

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

      1. expm1-define N/A

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

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

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

      3. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

   7. Simplified 99.0%

      \[\leadsto \color{blue}{\left(a \cdot x\right) \cdot \left(1 + \left(a \cdot x\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2.45:\\ \;\;\;\;-2 + \frac{-4}{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -2.45) (+ -2.0 (/ -4.0 (* a x))) (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2.45) {
		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) <= (-2.45d0)) 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) <= -2.45) {
		tmp = -2.0 + (-4.0 / (a * x));
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -2.45:
		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) <= -2.45)
		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) <= -2.45)
		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], -2.45], 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) < -2.4500000000000002

   1. Initial program 100.0%

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

      1. expm1-define N/A

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

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

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

      3. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. 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. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 0.8%

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

   7. Simplified 0.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 0.8%

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

   9. Applied egg-rr 0.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot x\right)\right)}\right)\right) \]
   11. 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 12.8%

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

   12. Simplified 12.8%

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

   13. Taylor expanded in a around inf

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

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. metadata-eval N/A

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

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

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

       12. metadata-eval N/A

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

       13. *-lowering-*.f64 18.8%

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

   15. Simplified 18.8%

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

   if -2.4500000000000002 < (*.f64 a x)

   1. Initial program 30.8%

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

      1. expm1-define N/A

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

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

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

      3. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 98.0%

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

   7. Simplified 98.0%

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

Alternative 4: 70.9% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

   1. expm1-define N/A

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

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

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

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt1-in N/A

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

   9. *-commutative N/A

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

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

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

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

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

   12. +-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. associate-*l* N/A

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

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

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

   17. *-commutative N/A

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

   18. *-lowering-*.f64 69.7%

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

7. Simplified 69.7%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

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

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

   4. clear-num N/A

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

   5. flip3-+ N/A

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

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

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

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

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

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

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

   9. *-lowering-*.f64 69.7%

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

9. Applied egg-rr 69.7%

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

10. Taylor expanded in a around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot x\right)\right)}\right)\right) \]
11. 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 73.3%

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

12. Simplified 73.3%

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

    7. associate-*r* N/A

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

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

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

    9. *-lowering-*.f64 73.6%

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

14. Applied egg-rr 73.6%

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

15. Final simplification 73.6%

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

Alternative 5: 70.9% accurate, 9.5× speedup

Math

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

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(Float64(a * 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[(N[(a * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

   1. expm1-define N/A

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

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

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

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt1-in N/A

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

   9. *-commutative N/A

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

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

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

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

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

   12. +-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. associate-*l* N/A

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

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

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

   17. *-commutative N/A

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

   18. *-lowering-*.f64 69.7%

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

7. Simplified 69.7%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

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

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

   4. clear-num N/A

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

   5. flip3-+ N/A

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

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

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

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

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

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

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

   9. *-lowering-*.f64 69.7%

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

9. Applied egg-rr 69.7%

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

10. Taylor expanded in a around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot x\right)\right)}\right)\right) \]
11. 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 73.3%

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

12. Simplified 73.3%

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

    1. div-inv N/A

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

    2. *-commutative N/A

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

    3. associate-/l* N/A

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

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

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

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

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

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

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

    7. associate-*r* N/A

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

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

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

    9. *-lowering-*.f64 73.6%

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

14. Applied egg-rr 73.6%

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

Alternative 6: 71.5% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. expm1-define N/A

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

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

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

      3. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. 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. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 0.8%

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

   7. Simplified 0.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 0.8%

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

   9. Applied egg-rr 0.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot x\right)\right)}\right)\right) \]
   11. 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 12.8%

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

   12. Simplified 12.8%

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

   13. Taylor expanded in a around inf

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

   15. Simplified 18.8%

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

   if -2 < (*.f64 a x)

   1. Initial program 30.8%

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

      1. expm1-define N/A

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

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

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

      3. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 98.0%

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

   7. Simplified 98.0%

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

Alternative 7: 8.8% accurate, 105.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, x):
	return -2.0

Julia

function code(a, x)
	return -2.0
end

MATLAB

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

Wolfram

code[a_, x_] := -2.0

Derivation

1. Initial program 51.4%

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

   1. expm1-define N/A

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

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

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

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt1-in N/A

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

   9. *-commutative N/A

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

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

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

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

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

   12. +-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. associate-*l* N/A

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

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

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

   17. *-commutative N/A

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

   18. *-lowering-*.f64 69.7%

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

7. Simplified 69.7%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

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

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

   4. clear-num N/A

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

   5. flip3-+ N/A

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

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

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

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

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

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

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

   9. *-lowering-*.f64 69.7%

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

9. Applied egg-rr 69.7%

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

10. Taylor expanded in a around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot x\right)\right)}\right)\right) \]
11. 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 73.3%

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

12. Simplified 73.3%

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

13. Taylor expanded in a around inf

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

15. Simplified 8.2%

    \[\leadsto \color{blue}{-2} \]
16. 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 2024150 
(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))