expax (section 3.5)

Percentage Accurate: 54.8% → 100.0%

Time: 9.5s

Alternatives: 8

Speedup: 6.4×

Specification

\[710 > a \cdot x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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 55.3%

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

   1. accelerator-lowering-expm1.f64 N/A

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

   2. *-lowering-*.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 71.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. flip-- N/A

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

   6. /-lowering-/.f64 N/A

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

   7. accelerator-lowering-expm1.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

   8. *-lowering-*.f64 99.4

      \[\leadsto \frac{1}{\frac{1}{\mathsf{expm1}\left(\color{blue}{a \cdot x}\right)}} \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2} \cdot x + \frac{1}{a}}{x}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. /-lowering-/.f64 72.3

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

7. Simplified 72.3%

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

   1. associate-/r/ N/A

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

   2. *-lowering-*.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. /-lowering-/.f64 73.0

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

9. Applied egg-rr 73.0%

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

10. Final simplification 73.0%

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

Alternative 3: 71.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2:\\ \;\;\;\;-2 + \frac{-4}{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(a \cdot \left(x \cdot 0.5\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -2.0)
   (+ -2.0 (/ -4.0 (* a x)))
   (* a (fma (* a (* x 0.5)) x x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2.0) {
		tmp = -2.0 + (-4.0 / (a * x));
	} else {
		tmp = a * fma((a * (x * 0.5)), x, x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -2.0)
		tmp = Float64(-2.0 + Float64(-4.0 / Float64(a * x)));
	else
		tmp = Float64(a * fma(Float64(a * Float64(x * 0.5)), x, x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -2.0], N[(-2.0 + N[(-4.0 / N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip-- N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

      8. *-lowering-*.f64 100.0

         \[\leadsto \frac{1}{\frac{1}{\mathsf{expm1}\left(\color{blue}{a \cdot x}\right)}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2} \cdot x + \frac{1}{a}}{x}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 18.8

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

   7. Simplified 18.8%

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

   8. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(2 + 4 \cdot \frac{1}{a \cdot x}\right)\right)} \]

      2. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(2\right)\right) + \left(\mathsf{neg}\left(4 \cdot \frac{1}{a \cdot x}\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \color{blue}{-2} + \left(\mathsf{neg}\left(4 \cdot \frac{1}{a \cdot x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{-2 + \left(\mathsf{neg}\left(4 \cdot \frac{1}{a \cdot x}\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto -2 + \left(\mathsf{neg}\left(\color{blue}{\frac{4 \cdot 1}{a \cdot x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto -2 + \left(\mathsf{neg}\left(\frac{\color{blue}{4}}{a \cdot x}\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto -2 + \color{blue}{\frac{\mathsf{neg}\left(4\right)}{a \cdot x}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto -2 + \color{blue}{\frac{\mathsf{neg}\left(4\right)}{a \cdot x}} \]

      9. metadata-eval N/A

         \[\leadsto -2 + \frac{\color{blue}{-4}}{a \cdot x} \]

      10. *-lowering-*.f64 18.8

          \[\leadsto -2 + \frac{-4}{\color{blue}{a \cdot x}} \]

   10. Simplified 18.8%

       \[\leadsto \color{blue}{-2 + \frac{-4}{a \cdot x}} \]

   if -2 < (*.f64 a x)

   1. Initial program 34.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 92.0

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

   5. Simplified 92.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 98.7

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

   7. Applied egg-rr 98.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2:\\ \;\;\;\;-2 + \frac{-4}{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(a \cdot \left(x \cdot 0.5\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2:\\ \;\;\;\;-2 + \frac{-4}{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -2.0) (+ -2.0 (/ -4.0 (* a x))) (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2.0) {
		tmp = -2.0 + (-4.0 / (a * x));
	} else {
		tmp = a * x;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-2.0d0)) then
        tmp = (-2.0d0) + ((-4.0d0) / (a * x))
    else
        tmp = a * x
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2.0) {
		tmp = -2.0 + (-4.0 / (a * x));
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -2.0:
		tmp = -2.0 + (-4.0 / (a * x))
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -2.0)
		tmp = Float64(-2.0 + Float64(-4.0 / Float64(a * x)));
	else
		tmp = Float64(a * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -2.0)
		tmp = -2.0 + (-4.0 / (a * x));
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -2.0], N[(-2.0 + N[(-4.0 / N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip-- N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

      8. *-lowering-*.f64 100.0

         \[\leadsto \frac{1}{\frac{1}{\mathsf{expm1}\left(\color{blue}{a \cdot x}\right)}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2} \cdot x + \frac{1}{a}}{x}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 18.8

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

   7. Simplified 18.8%

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

   8. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(2 + 4 \cdot \frac{1}{a \cdot x}\right)\right)} \]

      2. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(2\right)\right) + \left(\mathsf{neg}\left(4 \cdot \frac{1}{a \cdot x}\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \color{blue}{-2} + \left(\mathsf{neg}\left(4 \cdot \frac{1}{a \cdot x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{-2 + \left(\mathsf{neg}\left(4 \cdot \frac{1}{a \cdot x}\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto -2 + \left(\mathsf{neg}\left(\color{blue}{\frac{4 \cdot 1}{a \cdot x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto -2 + \left(\mathsf{neg}\left(\frac{\color{blue}{4}}{a \cdot x}\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto -2 + \color{blue}{\frac{\mathsf{neg}\left(4\right)}{a \cdot x}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto -2 + \color{blue}{\frac{\mathsf{neg}\left(4\right)}{a \cdot x}} \]

      9. metadata-eval N/A

         \[\leadsto -2 + \frac{\color{blue}{-4}}{a \cdot x} \]

      10. *-lowering-*.f64 18.8

          \[\leadsto -2 + \frac{-4}{\color{blue}{a \cdot x}} \]

   10. Simplified 18.8%

       \[\leadsto \color{blue}{-2 + \frac{-4}{a \cdot x}} \]

   if -2 < (*.f64 a x)

   1. Initial program 34.2%

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

   3. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 97.8

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

   5. Simplified 97.8%

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

Alternative 5: 71.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. flip-- N/A

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

   6. /-lowering-/.f64 N/A

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

   7. accelerator-lowering-expm1.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

   8. *-lowering-*.f64 99.4

      \[\leadsto \frac{1}{\frac{1}{\mathsf{expm1}\left(\color{blue}{a \cdot x}\right)}} \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2} \cdot x + \frac{1}{a}}{x}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. /-lowering-/.f64 72.3

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

7. Simplified 72.3%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. /-lowering-/.f64 72.8

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

9. Applied egg-rr 72.8%

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

Alternative 6: 70.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (if (<= (* a x) -2.0) -2.0 (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2.0) {
		tmp = -2.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-2.0d0)) then
        tmp = -2.0d0
    else
        tmp = a * x
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2.0) {
		tmp = -2.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -2.0:
		tmp = -2.0
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -2.0)
		tmp = -2.0;
	else
		tmp = Float64(a * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -2.0)
		tmp = -2.0;
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -2.0], -2.0, N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip-- N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

      8. *-lowering-*.f64 100.0

         \[\leadsto \frac{1}{\frac{1}{\mathsf{expm1}\left(\color{blue}{a \cdot x}\right)}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2} \cdot x + \frac{1}{a}}{x}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 18.8

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

   7. Simplified 18.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-2} \]
   9. Step-by-step derivation

   10. Simplified 18.7%

       \[\leadsto \color{blue}{-2} \]

   if -2 < (*.f64 a x)

   1. Initial program 34.2%

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

   3. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 97.8

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

   5. Simplified 97.8%

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

Alternative 7: 25.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1.5 \cdot 10^{-154}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (if (<= (* a x) -1.5e-154) -2.0 0.0))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1.5e-154) {
		tmp = -2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-1.5d-154)) then
        tmp = -2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1.5e-154) {
		tmp = -2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -1.5e-154:
		tmp = -2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -1.5e-154)
		tmp = -2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -1.5e-154)
		tmp = -2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -1.5e-154], -2.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1.5000000000000001e-154

   1. Initial program 76.4%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip-- N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

      8. *-lowering-*.f64 99.9

         \[\leadsto \frac{1}{\frac{1}{\mathsf{expm1}\left(\color{blue}{a \cdot x}\right)}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2} \cdot x + \frac{1}{a}}{x}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 39.5

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

   7. Simplified 39.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-2} \]
   9. Step-by-step derivation

   10. Simplified 15.6%

       \[\leadsto \color{blue}{-2} \]

   if -1.5000000000000001e-154 < (*.f64 a x)

   1. Initial program 39.1%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 36.6%

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

      1. metadata-eval 36.6

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

   7. Applied egg-rr 36.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 8.9% accurate, 109.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (a x) :precision binary64 -2.0)

C

double code(double a, double x) {
	return -2.0;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = -2.0d0
end function

Java

public static double code(double a, double x) {
	return -2.0;
}

Python

def code(a, x):
	return -2.0

Julia

function code(a, x)
	return -2.0
end

MATLAB

function tmp = code(a, x)
	tmp = -2.0;
end

Wolfram

code[a_, x_] := -2.0

Derivation

1. Initial program 55.3%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. flip-- N/A

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

   6. /-lowering-/.f64 N/A

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

   7. accelerator-lowering-expm1.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

   8. *-lowering-*.f64 99.4

      \[\leadsto \frac{1}{\frac{1}{\mathsf{expm1}\left(\color{blue}{a \cdot x}\right)}} \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{expm1}\left(a \cdot x\right)}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2} \cdot x + \frac{1}{a}}{x}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. /-lowering-/.f64 72.3

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

7. Simplified 72.3%

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

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-2} \]
9. Step-by-step derivation

10. Simplified 8.3%

    \[\leadsto \color{blue}{-2} \]
11. 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 2024205 
(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))