expax (section 3.5)

Percentage Accurate: 54.5% → 100.0%

Time: 7.5s

Alternatives: 13

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: 54.5% 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 52.7%

   \[e^{a \cdot x} - 1 \]
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: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -200:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-a\right) \cdot x, a, a\right), x, -1\right) \cdot x, a, 1\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, a, 0.16666666666666666\right), a \cdot x, 0.5\right) \cdot x, a, 1\right) \cdot \left(a \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -200.0)
   (- (/ 1.0 (fma (* (fma (fma (* (- a) x) a a) x -1.0) x) a 1.0)) 1.0)
   (*
    (fma
     (*
      (fma (fma (* 0.041666666666666664 x) a 0.16666666666666666) (* a x) 0.5)
      x)
     a
     1.0)
    (* a x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -200.0) {
		tmp = (1.0 / fma((fma(fma((-a * x), a, a), x, -1.0) * x), a, 1.0)) - 1.0;
	} else {
		tmp = fma((fma(fma((0.041666666666666664 * x), a, 0.16666666666666666), (a * x), 0.5) * x), a, 1.0) * (a * x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -200.0)
		tmp = Float64(Float64(1.0 / fma(Float64(fma(fma(Float64(Float64(-a) * x), a, a), x, -1.0) * x), a, 1.0)) - 1.0);
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(0.041666666666666664 * x), a, 0.16666666666666666), Float64(a * x), 0.5) * x), a, 1.0) * Float64(a * x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -200.0], N[(N[(1.0 / N[(N[(N[(N[(N[((-a) * x), $MachinePrecision] * a + a), $MachinePrecision] * x + -1.0), $MachinePrecision] * x), $MachinePrecision] * a + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.041666666666666664 * x), $MachinePrecision] * a + 0.16666666666666666), $MachinePrecision] * N[(a * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * a + 1.0), $MachinePrecision] * 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. *-commutative N/A

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

      3. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.5%

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

   if -200 < (*.f64 a x)

   1. Initial program 30.9%

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

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -200:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-a\right) \cdot x, a, a\right), x, -1\right) \cdot x, a, 1\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, a, 0.16666666666666666\right), a \cdot x, 0.5\right) \cdot x, a, 1\right) \cdot \left(a \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -200:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(\left(\left(\left(-a\right) \cdot x\right) \cdot x\right) \cdot a\right) \cdot x, a, 1\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, a, 0.16666666666666666\right), a \cdot x, 0.5\right) \cdot x, a, 1\right) \cdot \left(a \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -200.0)
   (- (/ 1.0 (fma (* (* (* (* (- a) x) x) a) x) a 1.0)) 1.0)
   (*
    (fma
     (*
      (fma (fma (* 0.041666666666666664 x) a 0.16666666666666666) (* a x) 0.5)
      x)
     a
     1.0)
    (* a x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -200.0) {
		tmp = (1.0 / fma(((((-a * x) * x) * a) * x), a, 1.0)) - 1.0;
	} else {
		tmp = fma((fma(fma((0.041666666666666664 * x), a, 0.16666666666666666), (a * x), 0.5) * x), a, 1.0) * (a * x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -200.0)
		tmp = Float64(Float64(1.0 / fma(Float64(Float64(Float64(Float64(Float64(-a) * x) * x) * a) * x), a, 1.0)) - 1.0);
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(0.041666666666666664 * x), a, 0.16666666666666666), Float64(a * x), 0.5) * x), a, 1.0) * Float64(a * x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -200.0], N[(N[(1.0 / N[(N[(N[(N[(N[((-a) * x), $MachinePrecision] * x), $MachinePrecision] * a), $MachinePrecision] * x), $MachinePrecision] * a + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.041666666666666664 * x), $MachinePrecision] * a + 0.16666666666666666), $MachinePrecision] * N[(a * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * a + 1.0), $MachinePrecision] * 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. *-commutative N/A

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

      3. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around inf

       \[\leadsto \frac{1}{\mathsf{fma}\left(-1 \cdot \left({a}^{2} \cdot {x}^{3}\right), a, 1\right)} - 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 99.5%

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

   if -200 < (*.f64 a x)

   1. Initial program 30.9%

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

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -200:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(\left(\left(\left(-a\right) \cdot x\right) \cdot x\right) \cdot a\right) \cdot x, a, 1\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, a, 0.16666666666666666\right), a \cdot x, 0.5\right) \cdot x, a, 1\right) \cdot \left(a \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      3. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around inf

       \[\leadsto \frac{1}{\mathsf{fma}\left(-1 \cdot \left({a}^{2} \cdot {x}^{3}\right), a, 1\right)} - 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 99.5%

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

   if -200 < (*.f64 a x)

   1. Initial program 30.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.7%

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

Alternative 5: 99.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      3. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 99.5%

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

   if -200 < (*.f64 a x)

   1. Initial program 30.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.7%

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

Alternative 6: 98.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      3. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.4%

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

   if -200 < (*.f64 a x)

   1. Initial program 30.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.4%

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

Alternative 7: 98.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      3. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.4%

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

   if -200 < (*.f64 a x)

   1. Initial program 30.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.4%

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

Alternative 8: 98.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      3. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.5%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 99.1%

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

   if -200 < (*.f64 a x)

   1. Initial program 30.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.6%

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

Alternative 9: 98.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -200.0) {
		tmp = (1.0 / fma((fma(x, a, -1.0) * x), a, 1.0)) - 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) <= -200.0)
		tmp = Float64(Float64(1.0 / fma(Float64(fma(x, a, -1.0) * x), a, 1.0)) - 1.0);
	else
		tmp = Float64(fma(Float64(0.5 * a), x, 1.0) * Float64(a * x));
	end
	return tmp
end

Wolfram

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

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

      3. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.5%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 99.1%

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

   if -200 < (*.f64 a x)

   1. Initial program 30.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.4%

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

4. Final simplification 99.3%

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

Alternative 10: 98.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -200.0) {
		tmp = (1.0 / fma(a, x, 1.0)) - 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) <= -200.0)
		tmp = Float64(Float64(1.0 / fma(a, x, 1.0)) - 1.0);
	else
		tmp = Float64(fma(Float64(0.5 * a), x, 1.0) * Float64(a * x));
	end
	return tmp
end

Wolfram

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

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

      3. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.5%

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

   12. Applied rewrites 97.6%

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

   if -200 < (*.f64 a x)

   1. Initial program 30.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.4%

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

4. Final simplification 98.8%

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

Alternative 11: 97.6% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -200.0], N[(N[(1.0 / N[(a * x + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(a * x), $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. *-commutative N/A

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

      3. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.5%

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

   12. Applied rewrites 97.6%

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

   if -200 < (*.f64 a x)

   1. Initial program 30.9%

      \[e^{a \cdot x} - 1 \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 98.1%

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

Alternative 12: 66.6% 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 52.7%

   \[e^{a \cdot x} - 1 \]
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 68.8

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

5. Applied rewrites 68.8%

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

6. Final simplification 68.8%

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

Alternative 13: 19.6% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double x) {
	return 1.0 - 1.0;
}

Python

def code(a, x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

code[a_, x_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 52.7%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 20.3%

   \[\leadsto \color{blue}{1} - 1 \]
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 2024235 
(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))