expax (section 3.5)

Percentage Accurate: 54.2% → 100.0%

Time: 6.3s

Alternatives: 8

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: 54.2% 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 54.2%

   \[e^{a \cdot x} - 1 \]
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 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: 80.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (exp (* a x)) 0.0)
   (- (* (* (/ (pow a -1.0) a) a) a) 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 = (((pow(a, -1.0) / a) * a) * a) - 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((a ^ -1.0) / a) * a) * a) - 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[Power[a, -1.0], $MachinePrecision] / a), $MachinePrecision] * a), $MachinePrecision] * a), $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 1.2%

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

   7. Applied rewrites 0.6%

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

   8. Taylor expanded in a around -inf

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

   10. Applied rewrites 8.7%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 50.5%

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

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

   1. Initial program 34.1%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 99.6%

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

4. Final simplification 84.6%

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

Alternative 3: 80.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (exp (* a x)) 0.0)
   (- (* (* (/ (pow a -1.0) a) a) a) 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 = (((pow(a, -1.0) / a) * a) * a) - 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((a ^ -1.0) / a) * a) * a) - 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[Power[a, -1.0], $MachinePrecision] / a), $MachinePrecision] * a), $MachinePrecision] * a), $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 1.2%

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

   7. Applied rewrites 0.6%

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

   8. Taylor expanded in a around -inf

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

   10. Applied rewrites 8.7%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 50.5%

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

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

   1. Initial program 34.1%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 99.5%

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

4. Final simplification 84.6%

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

Alternative 4: 71.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double x) {
	return Math.pow((Math.pow((x * a), -1.0) - 0.5), -1.0);
}

Python

def code(a, x):
	return math.pow((math.pow((x * a), -1.0) - 0.5), -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

   \[e^{a \cdot x} - 1 \]
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 23.4%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 23.4%

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

8. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 73.3

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

10. Applied rewrites 73.3%

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

11. Taylor expanded in a around inf

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

13. Applied rewrites 73.5%

    \[\leadsto \frac{1}{\frac{1}{x \cdot a} - \color{blue}{0.5}} \]

14. Final simplification 73.5%

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

Alternative 5: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50000000:\\ \;\;\;\;{-0.5}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, a \cdot x, 1\right) \cdot a\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -50000000.0)
   (pow -0.5 -1.0)
   (* (* (fma 0.5 (* a x) 1.0) a) x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50000000.0) {
		tmp = pow(-0.5, -1.0);
	} else {
		tmp = (fma(0.5, (a * x), 1.0) * a) * x;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -50000000.0)
		tmp = -0.5 ^ -1.0;
	else
		tmp = Float64(Float64(fma(0.5, Float64(a * x), 1.0) * a) * x);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -50000000.0], N[Power[-0.5, -1.0], $MachinePrecision], N[(N[(N[(0.5 * N[(a * x), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -5e7

   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 1.2%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 1.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 18.8

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

   10. Applied rewrites 18.8%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 18.8%

       \[\leadsto \frac{1}{-0.5} \]

   if -5e7 < (*.f64 a x)

   1. Initial program 34.1%

      \[e^{a \cdot x} - 1 \]
   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. Taylor expanded in a around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   7. Applied rewrites 99.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50000000:\\ \;\;\;\;{-0.5}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, a \cdot x, 1\right) \cdot a\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50000000:\\ \;\;\;\;{-0.5}^{-1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -50000000.0) (pow -0.5 -1.0) (* x a)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50000000.0) {
		tmp = pow(-0.5, -1.0);
	} 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) <= (-50000000.0d0)) then
        tmp = (-0.5d0) ** (-1.0d0)
    else
        tmp = x * a
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50000000.0) {
		tmp = Math.pow(-0.5, -1.0);
	} else {
		tmp = x * a;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -50000000.0:
		tmp = math.pow(-0.5, -1.0)
	else:
		tmp = x * a
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -50000000.0)
		tmp = -0.5 ^ -1.0;
	else
		tmp = Float64(x * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -50000000.0)
		tmp = -0.5 ^ -1.0;
	else
		tmp = x * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -50000000.0], N[Power[-0.5, -1.0], $MachinePrecision], N[(x * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -5e7

   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 1.2%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 1.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 18.8

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

   10. Applied rewrites 18.8%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 18.8%

       \[\leadsto \frac{1}{-0.5} \]

   if -5e7 < (*.f64 a x)

   1. Initial program 34.1%

      \[e^{a \cdot x} - 1 \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50000000:\\ \;\;\;\;{-0.5}^{-1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.5% 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 54.2%

   \[e^{a \cdot x} - 1 \]
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 70.0

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

5. Applied rewrites 70.0%

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

6. Final simplification 70.0%

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

Alternative 8: 19.3% 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 54.2%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 22.9%

   \[\leadsto \color{blue}{1} - 1 \]
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 2024318 
(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))