expax (section 3.5)

Percentage Accurate: 53.4% → 100.0%

Time: 8.9s

Alternatives: 6

Speedup: 18.2×

Specification

\[710 > a \cdot x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 - a \cdot x}\\ \mathbf{if}\;a \cdot x \leq -200:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_0, -1\right)}{t\_0 - -1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a \cdot x, a \cdot \mathsf{fma}\left(a \cdot x, 0.16666666666666666, 0.5\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- 1.0 (* a x)))))
   (if (<= (* a x) -200.0)
     (/ (fma t_0 t_0 -1.0) (- t_0 -1.0))
     (* x (fma (* a x) (* a (fma (* a x) 0.16666666666666666 0.5)) a)))))

C

double code(double a, double x) {
	double t_0 = 1.0 / (1.0 - (a * x));
	double tmp;
	if ((a * x) <= -200.0) {
		tmp = fma(t_0, t_0, -1.0) / (t_0 - -1.0);
	} else {
		tmp = x * fma((a * x), (a * fma((a * x), 0.16666666666666666, 0.5)), a);
	}
	return tmp;
}

Julia

function code(a, x)
	t_0 = Float64(1.0 / Float64(1.0 - Float64(a * x)))
	tmp = 0.0
	if (Float64(a * x) <= -200.0)
		tmp = Float64(fma(t_0, t_0, -1.0) / Float64(t_0 - -1.0));
	else
		tmp = Float64(x * fma(Float64(a * x), Float64(a * fma(Float64(a * x), 0.16666666666666666, 0.5)), a));
	end
	return tmp
end

Wolfram

code[a_, x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * x), $MachinePrecision], -200.0], N[(N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(a * x), $MachinePrecision] * N[(a * N[(N[(a * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + a), $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 5.2

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

   5. Applied rewrites 5.2%

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

   7. Applied rewrites 4.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.3%

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

       1. lift--.f64 N/A

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

       2. sub-neg N/A

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

       3. flip-+ N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

   12. Applied rewrites 97.3%

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

   if -200 < (*.f64 a x)

   1. Initial program 33.9%

      \[e^{a \cdot x} - 1 \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 99.1%

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

   9. Applied rewrites 99.0%

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

4. Final simplification 98.4%

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

Alternative 3: 98.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -200.0], N[(N[(1.0 / N[(1.0 - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x * N[(N[(a * x), $MachinePrecision] * N[(a * N[(N[(a * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + a), $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 5.2

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

   5. Applied rewrites 5.2%

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

   7. Applied rewrites 4.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.3%

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

   if -200 < (*.f64 a x)

   1. Initial program 33.9%

      \[e^{a \cdot x} - 1 \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 99.1%

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

   9. Applied rewrites 99.0%

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

4. Final simplification 98.4%

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

Alternative 4: 98.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -200.0], N[(N[(1.0 / N[(1.0 - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x * N[(N[(a * N[(a * x), $MachinePrecision]), $MachinePrecision] * 0.5 + a), $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 5.2

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

   5. Applied rewrites 5.2%

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

   7. Applied rewrites 4.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.3%

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

   if -200 < (*.f64 a x)

   1. Initial program 33.9%

      \[e^{a \cdot x} - 1 \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.9%

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

   12. Applied rewrites 98.9%

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

4. Final simplification 98.3%

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

Alternative 5: 67.0% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 64.0

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

5. Applied rewrites 64.0%

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

Alternative 6: 19.1% accurate, 27.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 20.4%

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

6. Final simplification 20.4%

   \[\leadsto -1 + 1 \]
7. 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 2024221 
(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))