expax (section 3.5)

Percentage Accurate: 54.0% → 100.0%

Time: 5.8s

Alternatives: 7

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: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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 52.6%

   \[e^{a \cdot x} - 1 \]
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 99.9

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

   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 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a \cdot x} \leq 0:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot x\right) \cdot 0.5\right) \cdot x - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, a, 0.16666666666666666\right), x \cdot a, 0.5\right) \cdot x, a, 1\right) \cdot \left(x \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (exp (* a x)) 0.0)
   (- (* (* (* (* a a) x) 0.5) x) 1.0)
   (*
    (fma
     (*
      (fma (fma (* 0.041666666666666664 x) a 0.16666666666666666) (* x a) 0.5)
      x)
     a
     1.0)
    (* x a))))

C

double code(double a, double x) {
	double tmp;
	if (exp((a * x)) <= 0.0) {
		tmp = ((((a * a) * x) * 0.5) * x) - 1.0;
	} else {
		tmp = fma((fma(fma((0.041666666666666664 * x), a, 0.16666666666666666), (x * a), 0.5) * x), a, 1.0) * (x * a);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (exp(Float64(a * x)) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * x) * 0.5) * x) - 1.0);
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(0.041666666666666664 * x), a, 0.16666666666666666), Float64(x * a), 0.5) * x), a, 1.0) * Float64(x * a));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.041666666666666664 * x), $MachinePrecision] * a + 0.16666666666666666), $MachinePrecision] * N[(x * a), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * a + 1.0), $MachinePrecision] * N[(x * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 a x)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 1.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 7.2%

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

   if 0.0 < (exp.f64 (*.f64 a x))

   1. Initial program 32.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a \cdot \left(x + a \cdot \left(\frac{1}{2} \cdot {x}^{2} + a \cdot \left(\frac{1}{24} \cdot \left(a \cdot {x}^{4}\right) + \frac{1}{6} \cdot {x}^{3}\right)\right)\right)} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, a, 0.16666666666666666\right), x \cdot a, 0.5\right) \cdot x, a, 1\right) \cdot \left(x \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a \cdot x} \leq 0:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot x\right) \cdot 0.5\right) \cdot x - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, a, 0.16666666666666666\right), x \cdot a, 0.5\right) \cdot x, a, 1\right) \cdot \left(x \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a \cdot x} \leq 0:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot x\right) \cdot 0.5\right) \cdot x - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666 \cdot a, x, 0.5\right) \cdot x\right) \cdot a, x \cdot a, x \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (exp (* a x)) 0.0)
   (- (* (* (* (* a a) x) 0.5) x) 1.0)
   (fma (* (* (fma (* 0.16666666666666666 a) x 0.5) x) a) (* x a) (* x a))))

C

double code(double a, double x) {
	double tmp;
	if (exp((a * x)) <= 0.0) {
		tmp = ((((a * a) * x) * 0.5) * x) - 1.0;
	} else {
		tmp = fma(((fma((0.16666666666666666 * a), x, 0.5) * x) * a), (x * a), (x * a));
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (exp(Float64(a * x)) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * x) * 0.5) * x) - 1.0);
	else
		tmp = fma(Float64(Float64(fma(Float64(0.16666666666666666 * a), x, 0.5) * x) * a), Float64(x * a), Float64(x * a));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * a), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * a), $MachinePrecision] * N[(x * a), $MachinePrecision] + N[(x * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 a x)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 1.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 7.2%

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

   if 0.0 < (exp.f64 (*.f64 a x))

   1. Initial program 32.2%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 98.1%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666 \cdot a, x, 0.5\right) \cdot x\right) \cdot a, x \cdot a, x \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a \cdot x} \leq 0:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot x\right) \cdot 0.5\right) \cdot x - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666 \cdot a, x, 0.5\right) \cdot x\right) \cdot a, x \cdot a, x \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a \cdot x} \leq 0:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot x\right) \cdot 0.5\right) \cdot x - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, a, 0.5\right) \cdot a, x, 1\right) \cdot \left(x \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (exp (* a x)) 0.0)
   (- (* (* (* (* a a) x) 0.5) x) 1.0)
   (* (fma (* (fma (* 0.16666666666666666 x) a 0.5) a) x 1.0) (* x a))))

C

double code(double a, double x) {
	double tmp;
	if (exp((a * x)) <= 0.0) {
		tmp = ((((a * a) * x) * 0.5) * x) - 1.0;
	} else {
		tmp = fma((fma((0.16666666666666666 * x), a, 0.5) * a), x, 1.0) * (x * a);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (exp(Float64(a * x)) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * x) * 0.5) * x) - 1.0);
	else
		tmp = Float64(fma(Float64(fma(Float64(0.16666666666666666 * x), a, 0.5) * a), x, 1.0) * Float64(x * a));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(x * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 a x)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 1.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 7.2%

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

   if 0.0 < (exp.f64 (*.f64 a x))

   1. Initial program 32.2%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, a, 0.5\right) \cdot a, x, 1\right) \cdot \left(x \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a \cdot x} \leq 0:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot x\right) \cdot 0.5\right) \cdot x - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, a, 0.5\right) \cdot a, x, 1\right) \cdot \left(x \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a \cdot x} \leq 0:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot x\right) \cdot 0.5\right) \cdot x - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, x \cdot a, 1\right) \cdot x\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (exp (* a x)) 0.0)
   (- (* (* (* (* a a) x) 0.5) x) 1.0)
   (* (* (fma 0.5 (* x a) 1.0) x) a)))

C

double code(double a, double x) {
	double tmp;
	if (exp((a * x)) <= 0.0) {
		tmp = ((((a * a) * x) * 0.5) * x) - 1.0;
	} else {
		tmp = (fma(0.5, (x * a), 1.0) * x) * a;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (exp(Float64(a * x)) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * x) * 0.5) * x) - 1.0);
	else
		tmp = Float64(Float64(fma(0.5, Float64(x * a), 1.0) * x) * a);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(0.5 * N[(x * a), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 a x)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 1.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 7.2%

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

   if 0.0 < (exp.f64 (*.f64 a x))

   1. Initial program 32.2%

      \[e^{a \cdot x} - 1 \]
   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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r* N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot a, x, \frac{1}{2}\right)}, a, x\right) \cdot a \]

      15. lower-*.f64 92.4

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{0.16666666666666666 \cdot a}, x, 0.5\right), a, x\right) \cdot a \]

   7. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.16666666666666666 \cdot a, x, 0.5\right), a, x\right) \cdot a} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.0%

       \[\leadsto \left(\mathsf{fma}\left(0.5, x \cdot a, 1\right) \cdot x\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 66.8% 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 52.6%

   \[e^{a \cdot x} - 1 \]
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 69.4

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

5. Applied rewrites 69.4%

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

6. Final simplification 69.4%

   \[\leadsto x \cdot a \]
7. Add Preprocessing

Alternative 7: 19.5% 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 52.6%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 20.7%

   \[\leadsto \color{blue}{1} - 1 \]
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 2024303 
(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))