expax (section 3.5)

Percentage Accurate: 65.7% → 100.0%

Time: 6.1s

Alternatives: 8

Speedup: 6.1×

• Unsound rule application detected in e-graph. Results may not be sound.

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.7% 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 69.1%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 71.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;a \cdot x + x \cdot \left(a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -50.0)
   -2.0
   (if (<= (* a x) 4e-5)
     (+ (* a x) (* x (* a (* x (* a 0.5)))))
     (* x (+ a (* x (* a (* a 0.5))))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50.0) {
		tmp = -2.0;
	} else if ((a * x) <= 4e-5) {
		tmp = (a * x) + (x * (a * (x * (a * 0.5))));
	} else {
		tmp = x * (a + (x * (a * (a * 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) <= (-50.0d0)) then
        tmp = -2.0d0
    else if ((a * x) <= 4d-5) then
        tmp = (a * x) + (x * (a * (x * (a * 0.5d0))))
    else
        tmp = x * (a + (x * (a * (a * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50.0) {
		tmp = -2.0;
	} else if ((a * x) <= 4e-5) {
		tmp = (a * x) + (x * (a * (x * (a * 0.5))));
	} else {
		tmp = x * (a + (x * (a * (a * 0.5))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -50.0:
		tmp = -2.0
	elif (a * x) <= 4e-5:
		tmp = (a * x) + (x * (a * (x * (a * 0.5))))
	else:
		tmp = x * (a + (x * (a * (a * 0.5))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -50.0)
		tmp = -2.0;
	elseif (Float64(a * x) <= 4e-5)
		tmp = Float64(Float64(a * x) + Float64(x * Float64(a * Float64(x * Float64(a * 0.5)))));
	else
		tmp = Float64(x * Float64(a + Float64(x * Float64(a * Float64(a * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -50.0)
		tmp = -2.0;
	elseif ((a * x) <= 4e-5)
		tmp = (a * x) + (x * (a * (x * (a * 0.5))));
	else
		tmp = x * (a + (x * (a * (a * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -50.0], -2.0, If[LessEqual[N[(a * x), $MachinePrecision], 4e-5], N[(N[(a * x), $MachinePrecision] + N[(x * N[(a * N[(x * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a + N[(x * N[(a * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a x) < -50

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 0.6%

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

      1. +-commutative 0.6%

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

      2. associate-*r* 0.6%

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

      3. unpow2 0.6%

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

      4. associate-*r* 0.8%

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

      5. distribute-rgt-out 1.0%

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

      6. *-commutative 1.0%

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

      7. *-commutative 1.0%

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

      8. unpow2 1.0%

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

      9. associate-*l* 1.0%

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

   6. Simplified 1.0%

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

      1. distribute-lft-in 0.8%

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

      2. flip-+ 0.5%

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

      3. *-commutative 0.5%

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

      4. *-commutative 0.5%

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

      5. *-commutative 0.5%

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

      6. associate-*l* 0.5%

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

      7. *-commutative 0.5%

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

      8. associate-*l* 0.5%

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

      9. *-commutative 0.5%

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

      10. *-commutative 0.5%

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

      11. associate-*l* 0.4%

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

   8. Applied egg-rr 0.4%

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

   9. Taylor expanded in a around inf 0.3%

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

       1. unpow2 0.3%

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

       2. unpow2 0.3%

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

       3. associate-*r* 0.3%

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

   11. Simplified 0.3%

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

   12. Taylor expanded in a around 0 18.8%

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

   if -50 < (*.f64 a x) < 4.00000000000000033e-5

   1. Initial program 33.5%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 84.0%

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

      1. +-commutative 84.0%

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

      2. associate-*r* 84.0%

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

      3. unpow2 84.0%

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

      4. associate-*r* 88.2%

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

      5. distribute-rgt-out 88.2%

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

      6. *-commutative 88.2%

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

      7. *-commutative 88.2%

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

      8. unpow2 88.2%

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

      9. associate-*l* 88.2%

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

   6. Simplified 88.2%

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

      1. +-commutative 88.2%

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

      2. distribute-lft-in 88.2%

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

      3. *-commutative 88.2%

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

      4. associate-*l* 98.7%

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

      5. *-commutative 98.7%

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

   8. Applied egg-rr 98.7%

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

   if 4.00000000000000033e-5 < (*.f64 a x)

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. associate-*r* 76.9%

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

      3. unpow2 76.9%

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

      4. associate-*r* 79.0%

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

      5. distribute-rgt-out 79.0%

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

      6. *-commutative 79.0%

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

      7. *-commutative 79.0%

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

      8. unpow2 79.0%

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

      9. associate-*l* 79.0%

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

   6. Simplified 79.0%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;a \cdot x + x \cdot \left(a \cdot \left(x \cdot \left(a \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 3: 72.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 100000:\\ \;\;\;\;a \cdot \left(x + x \cdot \left(a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -50.0)
   -2.0
   (if (<= (* a x) 100000.0)
     (* a (+ x (* x (* a (* x 0.5)))))
     (* (* 0.5 (* a a)) (* x x)))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50.0) {
		tmp = -2.0;
	} else if ((a * x) <= 100000.0) {
		tmp = a * (x + (x * (a * (x * 0.5))));
	} else {
		tmp = (0.5 * (a * a)) * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-50.0d0)) then
        tmp = -2.0d0
    else if ((a * x) <= 100000.0d0) then
        tmp = a * (x + (x * (a * (x * 0.5d0))))
    else
        tmp = (0.5d0 * (a * a)) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50.0) {
		tmp = -2.0;
	} else if ((a * x) <= 100000.0) {
		tmp = a * (x + (x * (a * (x * 0.5))));
	} else {
		tmp = (0.5 * (a * a)) * (x * x);
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -50.0:
		tmp = -2.0
	elif (a * x) <= 100000.0:
		tmp = a * (x + (x * (a * (x * 0.5))))
	else:
		tmp = (0.5 * (a * a)) * (x * x)
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -50.0)
		tmp = -2.0;
	elseif (Float64(a * x) <= 100000.0)
		tmp = Float64(a * Float64(x + Float64(x * Float64(a * Float64(x * 0.5)))));
	else
		tmp = Float64(Float64(0.5 * Float64(a * a)) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -50.0)
		tmp = -2.0;
	elseif ((a * x) <= 100000.0)
		tmp = a * (x + (x * (a * (x * 0.5))));
	else
		tmp = (0.5 * (a * a)) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -50.0], -2.0, If[LessEqual[N[(a * x), $MachinePrecision], 100000.0], N[(a * N[(x + N[(x * N[(a * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a x) < -50

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 0.6%

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

      1. +-commutative 0.6%

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

      2. associate-*r* 0.6%

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

      3. unpow2 0.6%

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

      4. associate-*r* 0.8%

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

      5. distribute-rgt-out 1.0%

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

      6. *-commutative 1.0%

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

      7. *-commutative 1.0%

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

      8. unpow2 1.0%

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

      9. associate-*l* 1.0%

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

   6. Simplified 1.0%

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

      1. distribute-lft-in 0.8%

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

      2. flip-+ 0.5%

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

      3. *-commutative 0.5%

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

      4. *-commutative 0.5%

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

      5. *-commutative 0.5%

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

      6. associate-*l* 0.5%

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

      7. *-commutative 0.5%

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

      8. associate-*l* 0.5%

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

      9. *-commutative 0.5%

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

      10. *-commutative 0.5%

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

      11. associate-*l* 0.4%

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

   8. Applied egg-rr 0.4%

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

   9. Taylor expanded in a around inf 0.3%

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

       1. unpow2 0.3%

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

       2. unpow2 0.3%

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

       3. associate-*r* 0.3%

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

   11. Simplified 0.3%

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

   12. Taylor expanded in a around 0 18.8%

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

   if -50 < (*.f64 a x) < 1e5

   1. Initial program 34.1%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 83.3%

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

      1. +-commutative 83.3%

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

      2. associate-*r* 83.3%

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

      3. unpow2 83.3%

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

      4. associate-*r* 88.3%

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

      5. distribute-rgt-out 88.3%

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

      6. *-commutative 88.3%

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

      7. *-commutative 88.3%

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

      8. unpow2 88.3%

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

      9. associate-*l* 88.3%

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

   6. Simplified 88.3%

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

      1. +-commutative 88.3%

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

      2. distribute-lft-in 88.3%

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

      3. *-commutative 88.3%

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

      4. associate-*l* 97.9%

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

      5. *-commutative 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. +-commutative 97.9%

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

      2. fma-def 97.9%

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

      3. associate-*r* 97.9%

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

      4. *-commutative 97.9%

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

      5. associate-*l* 97.9%

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

      6. associate-*l* 97.9%

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

      7. *-commutative 97.9%

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

   10. Applied egg-rr 97.9%

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

       1. fma-def 97.9%

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

       2. +-commutative 97.9%

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

       3. distribute-lft-out 97.9%

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

   12. Simplified 97.9%

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

   if 1e5 < (*.f64 a x)

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 78.2%

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

      1. +-commutative 78.2%

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

      2. associate-*r* 78.2%

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

      3. unpow2 78.2%

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

      4. associate-*r* 78.6%

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

      5. distribute-rgt-out 78.6%

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

      6. *-commutative 78.6%

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

      7. *-commutative 78.6%

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

      8. unpow2 78.6%

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

      9. associate-*l* 78.6%

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

   6. Simplified 78.6%

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

      1. +-commutative 78.6%

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

      2. distribute-lft-in 78.6%

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

      3. *-commutative 78.6%

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

      4. associate-*l* 70.9%

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

      5. *-commutative 70.9%

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

   8. Applied egg-rr 70.9%

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

   9. Taylor expanded in x around inf 78.2%

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

       1. unpow2 78.2%

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

       2. unpow2 78.2%

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

       3. associate-*r* 78.2%

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

   11. Simplified 78.2%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 100000:\\ \;\;\;\;a \cdot \left(x + x \cdot \left(a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 4: 71.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;a \cdot \left(x + x \cdot \left(a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -50.0)
   -2.0
   (if (<= (* a x) 4e-5)
     (* a (+ x (* x (* a (* x 0.5)))))
     (* x (+ a (* x (* a (* a 0.5))))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50.0) {
		tmp = -2.0;
	} else if ((a * x) <= 4e-5) {
		tmp = a * (x + (x * (a * (x * 0.5))));
	} else {
		tmp = x * (a + (x * (a * (a * 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) <= (-50.0d0)) then
        tmp = -2.0d0
    else if ((a * x) <= 4d-5) then
        tmp = a * (x + (x * (a * (x * 0.5d0))))
    else
        tmp = x * (a + (x * (a * (a * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50.0) {
		tmp = -2.0;
	} else if ((a * x) <= 4e-5) {
		tmp = a * (x + (x * (a * (x * 0.5))));
	} else {
		tmp = x * (a + (x * (a * (a * 0.5))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -50.0:
		tmp = -2.0
	elif (a * x) <= 4e-5:
		tmp = a * (x + (x * (a * (x * 0.5))))
	else:
		tmp = x * (a + (x * (a * (a * 0.5))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -50.0)
		tmp = -2.0;
	elseif (Float64(a * x) <= 4e-5)
		tmp = Float64(a * Float64(x + Float64(x * Float64(a * Float64(x * 0.5)))));
	else
		tmp = Float64(x * Float64(a + Float64(x * Float64(a * Float64(a * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -50.0)
		tmp = -2.0;
	elseif ((a * x) <= 4e-5)
		tmp = a * (x + (x * (a * (x * 0.5))));
	else
		tmp = x * (a + (x * (a * (a * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -50.0], -2.0, If[LessEqual[N[(a * x), $MachinePrecision], 4e-5], N[(a * N[(x + N[(x * N[(a * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a + N[(x * N[(a * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a x) < -50

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 0.6%

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

      1. +-commutative 0.6%

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

      2. associate-*r* 0.6%

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

      3. unpow2 0.6%

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

      4. associate-*r* 0.8%

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

      5. distribute-rgt-out 1.0%

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

      6. *-commutative 1.0%

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

      7. *-commutative 1.0%

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

      8. unpow2 1.0%

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

      9. associate-*l* 1.0%

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

   6. Simplified 1.0%

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

      1. distribute-lft-in 0.8%

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

      2. flip-+ 0.5%

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

      3. *-commutative 0.5%

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

      4. *-commutative 0.5%

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

      5. *-commutative 0.5%

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

      6. associate-*l* 0.5%

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

      7. *-commutative 0.5%

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

      8. associate-*l* 0.5%

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

      9. *-commutative 0.5%

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

      10. *-commutative 0.5%

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

      11. associate-*l* 0.4%

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

   8. Applied egg-rr 0.4%

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

   9. Taylor expanded in a around inf 0.3%

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

       1. unpow2 0.3%

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

       2. unpow2 0.3%

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

       3. associate-*r* 0.3%

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

   11. Simplified 0.3%

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

   12. Taylor expanded in a around 0 18.8%

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

   if -50 < (*.f64 a x) < 4.00000000000000033e-5

   1. Initial program 33.5%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 84.0%

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

      1. +-commutative 84.0%

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

      2. associate-*r* 84.0%

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

      3. unpow2 84.0%

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

      4. associate-*r* 88.2%

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

      5. distribute-rgt-out 88.2%

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

      6. *-commutative 88.2%

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

      7. *-commutative 88.2%

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

      8. unpow2 88.2%

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

      9. associate-*l* 88.2%

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

   6. Simplified 88.2%

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

      1. +-commutative 88.2%

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

      2. distribute-lft-in 88.2%

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

      3. *-commutative 88.2%

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

      4. associate-*l* 98.7%

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

      5. *-commutative 98.7%

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

   8. Applied egg-rr 98.7%

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

      1. +-commutative 98.7%

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

      2. fma-def 98.7%

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

      3. associate-*r* 98.7%

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

      4. *-commutative 98.7%

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

      5. associate-*l* 98.7%

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

      6. associate-*l* 98.7%

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

      7. *-commutative 98.7%

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

   10. Applied egg-rr 98.7%

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

       1. fma-def 98.7%

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

       2. +-commutative 98.7%

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

       3. distribute-lft-out 98.7%

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

   12. Simplified 98.7%

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

   if 4.00000000000000033e-5 < (*.f64 a x)

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. associate-*r* 76.9%

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

      3. unpow2 76.9%

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

      4. associate-*r* 79.0%

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

      5. distribute-rgt-out 79.0%

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

      6. *-commutative 79.0%

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

      7. *-commutative 79.0%

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

      8. unpow2 79.0%

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

      9. associate-*l* 79.0%

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

   6. Simplified 79.0%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;a \cdot \left(x + x \cdot \left(a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 5: 71.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 100000:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(a \cdot \left(a \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -50.0)
   -2.0
   (if (<= (* a x) 100000.0) (* a x) (* 0.5 (* a (* a (* x x)))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50.0) {
		tmp = -2.0;
	} else if ((a * x) <= 100000.0) {
		tmp = a * x;
	} else {
		tmp = 0.5 * (a * (a * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-50.0d0)) then
        tmp = -2.0d0
    else if ((a * x) <= 100000.0d0) then
        tmp = a * x
    else
        tmp = 0.5d0 * (a * (a * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50.0) {
		tmp = -2.0;
	} else if ((a * x) <= 100000.0) {
		tmp = a * x;
	} else {
		tmp = 0.5 * (a * (a * (x * x)));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -50.0:
		tmp = -2.0
	elif (a * x) <= 100000.0:
		tmp = a * x
	else:
		tmp = 0.5 * (a * (a * (x * x)))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -50.0)
		tmp = -2.0;
	elseif (Float64(a * x) <= 100000.0)
		tmp = Float64(a * x);
	else
		tmp = Float64(0.5 * Float64(a * Float64(a * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -50.0)
		tmp = -2.0;
	elseif ((a * x) <= 100000.0)
		tmp = a * x;
	else
		tmp = 0.5 * (a * (a * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -50.0], -2.0, If[LessEqual[N[(a * x), $MachinePrecision], 100000.0], N[(a * x), $MachinePrecision], N[(0.5 * N[(a * N[(a * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a x) < -50

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 0.6%

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

      1. +-commutative 0.6%

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

      2. associate-*r* 0.6%

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

      3. unpow2 0.6%

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

      4. associate-*r* 0.8%

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

      5. distribute-rgt-out 1.0%

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

      6. *-commutative 1.0%

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

      7. *-commutative 1.0%

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

      8. unpow2 1.0%

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

      9. associate-*l* 1.0%

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

   6. Simplified 1.0%

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

      1. distribute-lft-in 0.8%

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

      2. flip-+ 0.5%

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

      3. *-commutative 0.5%

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

      4. *-commutative 0.5%

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

      5. *-commutative 0.5%

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

      6. associate-*l* 0.5%

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

      7. *-commutative 0.5%

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

      8. associate-*l* 0.5%

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

      9. *-commutative 0.5%

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

      10. *-commutative 0.5%

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

      11. associate-*l* 0.4%

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

   8. Applied egg-rr 0.4%

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

   9. Taylor expanded in a around inf 0.3%

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

       1. unpow2 0.3%

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

       2. unpow2 0.3%

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

       3. associate-*r* 0.3%

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

   11. Simplified 0.3%

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

   12. Taylor expanded in a around 0 18.8%

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

   if -50 < (*.f64 a x) < 1e5

   1. Initial program 34.1%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 96.5%

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

   if 1e5 < (*.f64 a x)

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*l* 78.2%

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

      3. unpow2 78.2%

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

      4. associate-*l* 77.0%

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

      5. distribute-lft-out 77.0%

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

      6. unpow2 77.0%

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

      7. associate-*l* 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in a around inf 78.2%

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

      1. unpow2 78.2%

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

      2. unpow2 78.2%

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

      3. associate-*r* 76.9%

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

   9. Simplified 76.9%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 100000:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(a \cdot \left(a \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]

Alternative 6: 71.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 100000:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -50.0)
   -2.0
   (if (<= (* a x) 100000.0) (* a x) (* (* 0.5 (* a a)) (* x x)))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50.0) {
		tmp = -2.0;
	} else if ((a * x) <= 100000.0) {
		tmp = a * x;
	} else {
		tmp = (0.5 * (a * a)) * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-50.0d0)) then
        tmp = -2.0d0
    else if ((a * x) <= 100000.0d0) then
        tmp = a * x
    else
        tmp = (0.5d0 * (a * a)) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50.0) {
		tmp = -2.0;
	} else if ((a * x) <= 100000.0) {
		tmp = a * x;
	} else {
		tmp = (0.5 * (a * a)) * (x * x);
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -50.0:
		tmp = -2.0
	elif (a * x) <= 100000.0:
		tmp = a * x
	else:
		tmp = (0.5 * (a * a)) * (x * x)
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -50.0)
		tmp = -2.0;
	elseif (Float64(a * x) <= 100000.0)
		tmp = Float64(a * x);
	else
		tmp = Float64(Float64(0.5 * Float64(a * a)) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -50.0)
		tmp = -2.0;
	elseif ((a * x) <= 100000.0)
		tmp = a * x;
	else
		tmp = (0.5 * (a * a)) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -50.0], -2.0, If[LessEqual[N[(a * x), $MachinePrecision], 100000.0], N[(a * x), $MachinePrecision], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a x) < -50

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 0.6%

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

      1. +-commutative 0.6%

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

      2. associate-*r* 0.6%

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

      3. unpow2 0.6%

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

      4. associate-*r* 0.8%

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

      5. distribute-rgt-out 1.0%

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

      6. *-commutative 1.0%

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

      7. *-commutative 1.0%

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

      8. unpow2 1.0%

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

      9. associate-*l* 1.0%

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

   6. Simplified 1.0%

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

      1. distribute-lft-in 0.8%

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

      2. flip-+ 0.5%

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

      3. *-commutative 0.5%

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

      4. *-commutative 0.5%

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

      5. *-commutative 0.5%

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

      6. associate-*l* 0.5%

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

      7. *-commutative 0.5%

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

      8. associate-*l* 0.5%

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

      9. *-commutative 0.5%

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

      10. *-commutative 0.5%

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

      11. associate-*l* 0.4%

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

   8. Applied egg-rr 0.4%

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

   9. Taylor expanded in a around inf 0.3%

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

       1. unpow2 0.3%

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

       2. unpow2 0.3%

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

       3. associate-*r* 0.3%

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

   11. Simplified 0.3%

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

   12. Taylor expanded in a around 0 18.8%

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

   if -50 < (*.f64 a x) < 1e5

   1. Initial program 34.1%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 96.5%

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

   if 1e5 < (*.f64 a x)

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 78.2%

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

      1. +-commutative 78.2%

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

      2. associate-*r* 78.2%

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

      3. unpow2 78.2%

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

      4. associate-*r* 78.6%

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

      5. distribute-rgt-out 78.6%

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

      6. *-commutative 78.6%

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

      7. *-commutative 78.6%

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

      8. unpow2 78.6%

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

      9. associate-*l* 78.6%

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

   6. Simplified 78.6%

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

      1. +-commutative 78.6%

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

      2. distribute-lft-in 78.6%

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

      3. *-commutative 78.6%

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

      4. associate-*l* 70.9%

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

      5. *-commutative 70.9%

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

   8. Applied egg-rr 70.9%

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

   9. Taylor expanded in x around inf 78.2%

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

       1. unpow2 78.2%

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

       2. unpow2 78.2%

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

       3. associate-*r* 78.2%

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

   11. Simplified 78.2%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50:\\ \;\;\;\;-2\\ \mathbf{elif}\;a \cdot x \leq 100000:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 7: 60.9% accurate, 14.9× 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-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 0.6%

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

      1. +-commutative 0.6%

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

      2. associate-*r* 0.6%

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

      3. unpow2 0.6%

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

      4. associate-*r* 0.8%

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

      5. distribute-rgt-out 1.0%

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

      6. *-commutative 1.0%

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

      7. *-commutative 1.0%

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

      8. unpow2 1.0%

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

      9. associate-*l* 1.0%

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

   6. Simplified 1.0%

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

      1. distribute-lft-in 0.8%

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

      2. flip-+ 0.5%

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

      3. *-commutative 0.5%

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

      4. *-commutative 0.5%

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

      5. *-commutative 0.5%

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

      6. associate-*l* 0.5%

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

      7. *-commutative 0.5%

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

      8. associate-*l* 0.5%

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

      9. *-commutative 0.5%

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

      10. *-commutative 0.5%

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

      11. associate-*l* 0.4%

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

   8. Applied egg-rr 0.4%

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

   9. Taylor expanded in a around inf 0.3%

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

       1. unpow2 0.3%

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

       2. unpow2 0.3%

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

       3. associate-*r* 0.3%

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

   11. Simplified 0.3%

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

   12. Taylor expanded in a around 0 18.8%

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

   if -2 < (*.f64 a x)

   1. Initial program 56.5%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 72.2%

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

4. Final simplification 56.7%

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

Alternative 8: 6.6% 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 69.1%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in a around 0 58.2%

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

   1. +-commutative 58.2%

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

   2. associate-*r* 58.2%

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

   3. unpow2 58.2%

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

   4. associate-*r* 60.7%

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

   5. distribute-rgt-out 60.7%

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

   6. *-commutative 60.7%

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

   7. *-commutative 60.7%

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

   8. unpow2 60.7%

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

   9. associate-*l* 60.7%

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

6. Simplified 60.7%

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

   1. distribute-lft-in 60.7%

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

   2. flip-+ 20.6%

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

   3. *-commutative 20.6%

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

   4. *-commutative 20.6%

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

   5. *-commutative 20.6%

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

   6. associate-*l* 20.6%

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

   7. *-commutative 20.6%

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

   8. associate-*l* 20.8%

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

   9. *-commutative 20.8%

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

   10. *-commutative 20.8%

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

   11. associate-*l* 26.4%

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

8. Applied egg-rr 26.4%

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

9. Taylor expanded in a around inf 2.1%

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

    1. unpow2 2.1%

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

    2. unpow2 2.1%

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

    3. associate-*r* 2.1%

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

11. Simplified 2.1%

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

12. Taylor expanded in a around 0 7.3%

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

13. Final simplification 7.3%

    \[\leadsto -2 \]

Developer target: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (< (fabs (* a x)) 0.1)
   (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0))))
   (- (exp (* a x)) 1.0)))

C

double code(double a, double x) {
	double tmp;
	if (fabs((a * x)) < 0.1) {
		tmp = (a * x) * (1.0 + (((a * x) / 2.0) + (pow((a * x), 2.0) / 6.0)));
	} else {
		tmp = exp((a * x)) - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs((a * x)) < 0.1d0) then
        tmp = (a * x) * (1.0d0 + (((a * x) / 2.0d0) + (((a * x) ** 2.0d0) / 6.0d0)))
    else
        tmp = exp((a * x)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if (Math.abs((a * x)) < 0.1) {
		tmp = (a * x) * (1.0 + (((a * x) / 2.0) + (Math.pow((a * x), 2.0) / 6.0)));
	} else {
		tmp = Math.exp((a * x)) - 1.0;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if math.fabs((a * x)) < 0.1:
		tmp = (a * x) * (1.0 + (((a * x) / 2.0) + (math.pow((a * x), 2.0) / 6.0)))
	else:
		tmp = math.exp((a * x)) - 1.0
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (abs(Float64(a * x)) < 0.1)
		tmp = Float64(Float64(a * x) * Float64(1.0 + Float64(Float64(Float64(a * x) / 2.0) + Float64((Float64(a * x) ^ 2.0) / 6.0))));
	else
		tmp = Float64(exp(Float64(a * x)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if (abs((a * x)) < 0.1)
		tmp = (a * x) * (1.0 + (((a * x) / 2.0) + (((a * x) ^ 2.0) / 6.0)));
	else
		tmp = exp((a * x)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[Less[N[Abs[N[(a * x), $MachinePrecision]], $MachinePrecision], 0.1], N[(N[(a * x), $MachinePrecision] * N[(1.0 + N[(N[(N[(a * x), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Power[N[(a * x), $MachinePrecision], 2.0], $MachinePrecision] / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]

Reproduce

herbie shell --seed 2023173 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

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