expax (section 3.5)

Percentage Accurate: 53.4% → 100.0%

Time: 4.8s

Alternatives: 6

Speedup: 18.2×

Specification

\[710 > a \cdot x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{e^{a \cdot x} - 1} \]

   2. lift-exp.f64 N/A

      \[\leadsto \color{blue}{e^{a \cdot x}} - 1 \]

   3. lower-expm1.f64 100.0

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a \cdot x} \leq 10^{-9}:\\ \;\;\;\;\frac{x}{-0.5 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot x\right) \cdot a, 0.5, a\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (exp (* a x)) 1e-9) (/ x (* -0.5 x)) (* (fma (* (* a x) a) 0.5 a) x)))

C

double code(double a, double x) {
	double tmp;
	if (exp((a * x)) <= 1e-9) {
		tmp = x / (-0.5 * x);
	} else {
		tmp = fma(((a * x) * a), 0.5, a) * x;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (exp(Float64(a * x)) <= 1e-9)
		tmp = Float64(x / Float64(-0.5 * x));
	else
		tmp = Float64(fma(Float64(Float64(a * x) * a), 0.5, a) * x);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision], 1e-9], N[(x / N[(-0.5 * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * x), $MachinePrecision] * a), $MachinePrecision] * 0.5 + a), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 a x)) < 1.00000000000000006e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 1.1%

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

   7. Applied rewrites 1.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\frac{-1}{2} \cdot x + \color{blue}{\frac{1}{a}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.8%

       \[\leadsto \frac{x}{\mathsf{fma}\left(-0.5, \color{blue}{x}, \frac{1}{a}\right)} \]

   11. Taylor expanded in a around inf

       \[\leadsto \frac{x}{\frac{-1}{2} \cdot x} \]
   12. Step-by-step derivation

   13. Applied rewrites 18.7%

       \[\leadsto \frac{x}{-0.5 \cdot x} \]

   if 1.00000000000000006e-9 < (exp.f64 (*.f64 a x))

   1. Initial program 37.1%

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

   3. Taylor expanded in a around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 99.2%

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

Alternative 3: 72.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{fma}\left(-0.5, x, {a}^{-1}\right)} \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (/ x (fma -0.5 x (pow a -1.0))))

C

double code(double a, double x) {
	return x / fma(-0.5, x, pow(a, -1.0));
}

Julia

function code(a, x)
	return Float64(x / fma(-0.5, x, (a ^ -1.0)))
end

Wolfram

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in a around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt1-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-rgt1-in N/A

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

   12. associate-*r* N/A

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

   13. associate-*l* N/A

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

   14. unpow2 N/A

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

   15. associate-*r* N/A

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

   16. *-commutative N/A

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

   17. distribute-lft-in N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

5. Applied rewrites 58.6%

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

7. Applied rewrites 58.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{x}{\frac{-1}{2} \cdot x + \color{blue}{\frac{1}{a}}} \]
9. Step-by-step derivation

10. Applied rewrites 69.1%

    \[\leadsto \frac{x}{\mathsf{fma}\left(-0.5, \color{blue}{x}, \frac{1}{a}\right)} \]

11. Final simplification 69.1%

    \[\leadsto \frac{x}{\mathsf{fma}\left(-0.5, x, {a}^{-1}\right)} \]
12. Add Preprocessing

Alternative 4: 72.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20:\\ \;\;\;\;\frac{x}{-0.5 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -20.0) (/ x (* -0.5 x)) (* x a)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -20.0) {
		tmp = x / (-0.5 * x);
	} else {
		tmp = x * a;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-20.0d0)) then
        tmp = x / ((-0.5d0) * x)
    else
        tmp = x * a
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -20.0) {
		tmp = x / (-0.5 * x);
	} else {
		tmp = x * a;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -20.0:
		tmp = x / (-0.5 * x)
	else:
		tmp = x * a
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -20.0)
		tmp = Float64(x / Float64(-0.5 * x));
	else
		tmp = Float64(x * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -20.0)
		tmp = x / (-0.5 * x);
	else
		tmp = x * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -20.0], N[(x / N[(-0.5 * x), $MachinePrecision]), $MachinePrecision], N[(x * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -20

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 1.1%

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

   7. Applied rewrites 1.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\frac{-1}{2} \cdot x + \color{blue}{\frac{1}{a}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.8%

       \[\leadsto \frac{x}{\mathsf{fma}\left(-0.5, \color{blue}{x}, \frac{1}{a}\right)} \]

   11. Taylor expanded in a around inf

       \[\leadsto \frac{x}{\frac{-1}{2} \cdot x} \]
   12. Step-by-step derivation

   13. Applied rewrites 18.7%

       \[\leadsto \frac{x}{-0.5 \cdot x} \]

   if -20 < (*.f64 a x)

   1. Initial program 37.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 5: 67.6% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 63.5

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

5. Applied rewrites 63.5%

   \[\leadsto \color{blue}{x \cdot a} \]
6. Add Preprocessing

Alternative 6: 19.7% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double x) {
	return 1.0 - 1.0;
}

Python

def code(a, x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

code[a_, x_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 22.6%

   \[\leadsto \color{blue}{1} - 1 \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024307 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :pre (> 710.0 (* a x))

  :alt
  (! :herbie-platform default (expm1 (* a x)))

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