expax (section 3.5)

Percentage Accurate: 52.7% → 100.0%

Time: 6.4s

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: 52.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.4%

   \[e^{a \cdot x} - 1 \]
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. Final simplification 100.0%

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

Alternative 2: 67.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(x, a, \left(\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, a, 0.5\right) \cdot a\right) \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot x\right) \cdot 0.5\right) \cdot x - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= x 4.2e+206)
   (fma x a (* (* (* (fma (* 0.16666666666666666 x) a 0.5) a) x) (* a x)))
   (- (* (* (* (* a a) x) 0.5) x) 1.0)))

C

double code(double a, double x) {
	double tmp;
	if (x <= 4.2e+206) {
		tmp = fma(x, a, (((fma((0.16666666666666666 * x), a, 0.5) * a) * x) * (a * x)));
	} else {
		tmp = ((((a * a) * x) * 0.5) * x) - 1.0;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (x <= 4.2e+206)
		tmp = fma(x, a, Float64(Float64(Float64(fma(Float64(0.16666666666666666 * x), a, 0.5) * a) * x) * Float64(a * x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * x) * 0.5) * x) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[x, 4.2e+206], N[(x * a + N[(N[(N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * x), $MachinePrecision] * N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.19999999999999974e206

   1. Initial program 55.5%

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

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

   7. Applied rewrites 69.4%

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

   if 4.19999999999999974e206 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{1} - 1 \]

   6. Taylor expanded in a around 0

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. associate-+r+ N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 0.8%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 18.7%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(x, a, \left(\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, a, 0.5\right) \cdot a\right) \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot x\right) \cdot 0.5\right) \cdot x - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.4%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 24.3

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

5. Applied rewrites 24.3%

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

7. Applied rewrites 24.3%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-- N/A

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

9. Applied rewrites 24.3%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2} \cdot x + \frac{1}{a}}{x}}} \]
11. 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 71.2

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

12. Applied rewrites 71.2%

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

Alternative 4: 67.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, a, 0.5\right) \cdot a, x, 1\right) \cdot \left(a \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot x\right) \cdot 0.5\right) \cdot x - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= x 4.2e+206)
   (* (fma (* (fma (* 0.16666666666666666 x) a 0.5) a) x 1.0) (* a x))
   (- (* (* (* (* a a) x) 0.5) x) 1.0)))

C

double code(double a, double x) {
	double tmp;
	if (x <= 4.2e+206) {
		tmp = fma((fma((0.16666666666666666 * x), a, 0.5) * a), x, 1.0) * (a * x);
	} else {
		tmp = ((((a * a) * x) * 0.5) * x) - 1.0;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (x <= 4.2e+206)
		tmp = Float64(fma(Float64(fma(Float64(0.16666666666666666 * x), a, 0.5) * a), x, 1.0) * Float64(a * x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * x) * 0.5) * x) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[x, 4.2e+206], N[(N[(N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(a * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.19999999999999974e206

   1. Initial program 55.5%

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, a, 0.5\right) \cdot a, x, 1\right) \cdot \left(x \cdot a\right)} \]

   if 4.19999999999999974e206 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{1} - 1 \]

   6. Taylor expanded in a around 0

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. associate-+r+ N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 0.8%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 18.7%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, a, 0.5\right) \cdot a, x, 1\right) \cdot \left(a \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot x\right) \cdot 0.5\right) \cdot x - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+206}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, a, 0.5\right) \cdot a, x, 1\right) \cdot a\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot x\right) \cdot 0.5\right) \cdot x - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= x 4.2e+206)
   (* (* (fma (* (fma (* 0.16666666666666666 x) a 0.5) a) x 1.0) a) x)
   (- (* (* (* (* a a) x) 0.5) x) 1.0)))

C

double code(double a, double x) {
	double tmp;
	if (x <= 4.2e+206) {
		tmp = (fma((fma((0.16666666666666666 * x), a, 0.5) * a), x, 1.0) * a) * x;
	} else {
		tmp = ((((a * a) * x) * 0.5) * x) - 1.0;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (x <= 4.2e+206)
		tmp = Float64(Float64(fma(Float64(fma(Float64(0.16666666666666666 * x), a, 0.5) * a), x, 1.0) * a) * x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * x) * 0.5) * x) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[x, 4.2e+206], N[(N[(N[(N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * x + 1.0), $MachinePrecision] * a), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.19999999999999974e206

   1. Initial program 55.5%

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

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

   7. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, a, 0.5\right) \cdot a, x, 1\right) \cdot a\right) \cdot x} \]

   if 4.19999999999999974e206 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{1} - 1 \]

   6. Taylor expanded in a around 0

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. associate-+r+ N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 0.8%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 18.7%

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

Alternative 6: 67.5% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.4%

   \[e^{a \cdot x} - 1 \]
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 66.6

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

5. Applied rewrites 66.6%

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

6. Final simplification 66.6%

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

Alternative 7: 18.9% 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 57.4%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 22.8%

   \[\leadsto \color{blue}{1} - 1 \]
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 2024270 
(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))