expax (section 3.5)

Percentage Accurate: 54.8% → 100.0%

Time: 5.7s

Alternatives: 6

Speedup: 18.2×

Specification

\[710 > a \cdot x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% 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 57.0%

   \[e^{a \cdot x} - 1 \]
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: 70.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{x}{a} \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a \cdot x, 0.5\right) \cdot a, a \cdot x, a\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -4e+22)
   (- (* (* (/ x a) a) a) 1.0)
   (* (fma (* (fma 0.16666666666666666 (* a x) 0.5) a) (* a x) a) x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -4e+22) {
		tmp = (((x / a) * a) * a) - 1.0;
	} else {
		tmp = fma((fma(0.16666666666666666, (a * x), 0.5) * a), (a * x), a) * x;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -4e+22)
		tmp = Float64(Float64(Float64(Float64(x / a) * a) * a) - 1.0);
	else
		tmp = Float64(fma(Float64(fma(0.16666666666666666, Float64(a * x), 0.5) * a), Float64(a * x), a) * x);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -4e+22], N[(N[(N[(N[(x / a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * N[(a * x), $MachinePrecision] + 0.5), $MachinePrecision] * a), $MachinePrecision] * N[(a * x), $MachinePrecision] + a), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -4e22

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 0.8

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

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 5.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 11.6%

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

   if -4e22 < (*.f64 a x)

   1. Initial program 34.5%

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

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

      1. lower-*.f64 94.7

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 87.7%

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

   12. Applied rewrites 97.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{x}{a} \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a \cdot x, 0.5\right) \cdot a, a \cdot x, a\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500000000000:\\ \;\;\;\;\left(\frac{x}{a} \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, a \cdot x, 1\right) \cdot x\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -500000000000.0)
   (- (* (* (/ x a) a) a) 1.0)
   (* (* (fma 0.5 (* a x) 1.0) x) a)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -500000000000.0) {
		tmp = (((x / a) * a) * a) - 1.0;
	} else {
		tmp = (fma(0.5, (a * x), 1.0) * x) * a;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -500000000000.0)
		tmp = Float64(Float64(Float64(Float64(x / a) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(fma(0.5, Float64(a * x), 1.0) * x) * a);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -500000000000.0], N[(N[(N[(N[(x / a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(0.5 * N[(a * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -5e11

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 0.8

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

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 6.4%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 12.5%

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

   if -5e11 < (*.f64 a x)

   1. Initial program 32.9%

      \[e^{a \cdot x} - 1 \]
   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-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. 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 98.6%

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

   9. Applied rewrites 98.7%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500000000000:\\ \;\;\;\;\left(\frac{x}{a} \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, a \cdot x, 1\right) \cdot x\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, a \cdot x, 1\right) \cdot x\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -2e+27)
   (- (* (* (* (* x x) 0.5) a) a) 1.0)
   (* (* (fma 0.5 (* a x) 1.0) x) a)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2e+27) {
		tmp = ((((x * x) * 0.5) * a) * a) - 1.0;
	} else {
		tmp = (fma(0.5, (a * x), 1.0) * x) * a;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -2e+27)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * 0.5) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(fma(0.5, Float64(a * x), 1.0) * x) * a);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -2e+27], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(0.5 * N[(a * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -2e27

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 0.8

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

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 5.6%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 5.3%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot a\right) \cdot a - 1 \]

   if -2e27 < (*.f64 a x)

   1. Initial program 34.9%

      \[e^{a \cdot x} - 1 \]
   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-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. 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 95.8%

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

   9. Applied rewrites 95.8%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, a \cdot x, 1\right) \cdot x\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 57.0%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 63.7

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

5. Applied rewrites 63.7%

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

Alternative 6: 19.9% 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 57.0%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 19.2%

   \[\leadsto \color{blue}{1} - 1 \]
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 2024339 
(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))