expax (section 3.5)

Percentage Accurate: 54.9% → 100.0%

Time: 9.4s

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: 54.9% 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 54.6%

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

   1. lift-*.f64 N/A

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

   2. lower-expm1.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 71.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -200:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{1 + a \cdot \mathsf{fma}\left(a, x \cdot x, x\right)}, -1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot x\right) \cdot \left(a \cdot \mathsf{fma}\left(a \cdot x, 0.16666666666666666, 0.5\right)\right), x, a \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -200.0)
   (+ (fma 2.0 (/ 1.0 (+ 1.0 (* a (fma a (* x x) x)))) -1.0) -1.0)
   (fma (* (* a x) (* a (fma (* a x) 0.16666666666666666 0.5))) x (* a x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -200.0) {
		tmp = fma(2.0, (1.0 / (1.0 + (a * fma(a, (x * x), x)))), -1.0) + -1.0;
	} else {
		tmp = fma(((a * x) * (a * fma((a * x), 0.16666666666666666, 0.5))), x, (a * x));
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -200.0)
		tmp = Float64(fma(2.0, Float64(1.0 / Float64(1.0 + Float64(a * fma(a, Float64(x * x), x)))), -1.0) + -1.0);
	else
		tmp = fma(Float64(Float64(a * x) * Float64(a * fma(Float64(a * x), 0.16666666666666666, 0.5))), x, Float64(a * x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -200.0], N[(N[(2.0 * N[(1.0 / N[(1.0 + N[(a * N[(a * N[(x * x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(a * x), $MachinePrecision] * N[(a * N[(N[(a * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(a * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -200

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 4.8

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

   5. Simplified 4.8%

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

   7. Applied egg-rr 3.0%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 18.8%

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

   if -200 < (*.f64 a x)

   1. Initial program 33.6%

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

      1. lift-*.f64 N/A

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

      2. lower-expm1.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around 0

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

   7. Simplified 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -200:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{1 + a \cdot \mathsf{fma}\left(a, x \cdot x, x\right)}, -1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot x\right) \cdot \left(a \cdot \mathsf{fma}\left(a \cdot x, 0.16666666666666666, 0.5\right)\right), x, a \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(a \cdot x\right) \cdot \left(a \cdot \mathsf{fma}\left(a \cdot x, 0.16666666666666666, 0.5\right)\right), x, a \cdot x\right) \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (fma (* (* a x) (* a (fma (* a x) 0.16666666666666666 0.5))) x (* a x)))

C

double code(double a, double x) {
	return fma(((a * x) * (a * fma((a * x), 0.16666666666666666, 0.5))), x, (a * x));
}

Julia

function code(a, x)
	return fma(Float64(Float64(a * x) * Float64(a * fma(Float64(a * x), 0.16666666666666666, 0.5))), x, Float64(a * x))
end

Wolfram

code[a_, x_] := N[(N[(N[(a * x), $MachinePrecision] * N[(a * N[(N[(a * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(a * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. lift-*.f64 N/A

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

   2. lower-expm1.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in a around 0

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

7. Simplified 69.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. distribute-rgt-in N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

9. Applied egg-rr 69.5%

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

10. Final simplification 69.5%

    \[\leadsto \mathsf{fma}\left(\left(a \cdot x\right) \cdot \left(a \cdot \mathsf{fma}\left(a \cdot x, 0.16666666666666666, 0.5\right)\right), x, a \cdot x\right) \]
11. Add Preprocessing

Alternative 4: 66.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(a \cdot \left(a \cdot x\right), \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), a\right) \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (* x (fma (* a (* a x)) (fma a (* x 0.16666666666666666) 0.5) a)))

C

double code(double a, double x) {
	return x * fma((a * (a * x)), fma(a, (x * 0.16666666666666666), 0.5), a);
}

Julia

function code(a, x)
	return Float64(x * fma(Float64(a * Float64(a * x)), fma(a, Float64(x * 0.16666666666666666), 0.5), a))
end

Wolfram

code[a_, x_] := N[(x * N[(N[(a * N[(a * x), $MachinePrecision]), $MachinePrecision] * N[(a * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. lift-*.f64 N/A

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

   2. lower-expm1.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in a around 0

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

7. Simplified 69.5%

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

Alternative 5: 65.6% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (a x) :precision binary64 (fma x a (* a (* x (* a (* x 0.5))))))

C

double code(double a, double x) {
	return fma(x, a, (a * (x * (a * (x * 0.5)))));
}

Julia

function code(a, x)
	return fma(x, a, Float64(a * Float64(x * Float64(a * Float64(x * 0.5)))))
end

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 65.2

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

5. Simplified 65.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 63.4

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

7. Applied egg-rr 63.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 68.5

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

9. Applied egg-rr 68.5%

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

10. Final simplification 68.5%

    \[\leadsto \mathsf{fma}\left(x, a, a \cdot \left(x \cdot \left(a \cdot \left(x \cdot 0.5\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 6: 66.4% 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 54.6%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 68.5

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

5. Simplified 68.5%

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

Alternative 7: 20.0% accurate, 109.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a x) :precision binary64 0.0)

C

double code(double a, double x) {
	return 0.0;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double a, double x) {
	return 0.0;
}

Python

def code(a, x):
	return 0.0

Julia

function code(a, x)
	return 0.0
end

MATLAB

function tmp = code(a, x)
	tmp = 0.0;
end

Wolfram

code[a_, x_] := 0.0

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in a around 0

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

5. Simplified 21.6%

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

   1. metadata-eval 21.6

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 21.6%

   \[\leadsto \color{blue}{0} \]
8. 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 2024208 
(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))