expax (section 3.5)

Percentage Accurate: 55.2% → 100.0%

Time: 9.4s

Alternatives: 7

Speedup: 18.2×

Specification

\[710 > a \cdot x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% 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 48.9%

   \[e^{a \cdot x} - 1 \]
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: 70.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{-0.5}\\ \mathbf{elif}\;a \cdot x \leq -4 \cdot 10^{+18}:\\ \;\;\;\;a \cdot \left(a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x \cdot \left(a \cdot 0.5\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -2e+101)
   (/ 1.0 -0.5)
   (if (<= (* a x) -4e+18)
     (+ (* a (* a (* x (* x 0.5)))) -1.0)
     (* a (fma (* x (* a 0.5)) x x)))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2e+101) {
		tmp = 1.0 / -0.5;
	} else if ((a * x) <= -4e+18) {
		tmp = (a * (a * (x * (x * 0.5)))) + -1.0;
	} else {
		tmp = a * fma((x * (a * 0.5)), x, x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -2e+101)
		tmp = Float64(1.0 / -0.5);
	elseif (Float64(a * x) <= -4e+18)
		tmp = Float64(Float64(a * Float64(a * Float64(x * Float64(x * 0.5)))) + -1.0);
	else
		tmp = Float64(a * fma(Float64(x * Float64(a * 0.5)), x, x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -2e+101], N[(1.0 / -0.5), $MachinePrecision], If[LessEqual[N[(a * x), $MachinePrecision], -4e+18], N[(N[(a * N[(a * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(N[(x * N[(a * 0.5), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a x) < -2e101

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 100.0

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lower-expm1.f64 100.0

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

   4. Applied rewrites 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. lower-/.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. lower-fma.f64 N/A

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

      4. lower-/.f64 18.8

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

   7. Applied rewrites 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

      \[\leadsto \frac{1}{\frac{-1}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.8%

       \[\leadsto \frac{1}{-0.5} \]

   if -2e101 < (*.f64 a x) < -4e18

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 2.0

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

   5. Applied rewrites 2.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 19.7%

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

   if -4e18 < (*.f64 a x)

   1. Initial program 29.4%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 29.4

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lower-expm1.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 92.9

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

   7. Applied rewrites 92.9%

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

   9. Applied rewrites 98.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{-0.5}\\ \mathbf{elif}\;a \cdot x \leq -4 \cdot 10^{+18}:\\ \;\;\;\;a \cdot \left(a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x \cdot \left(a \cdot 0.5\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+237}:\\ \;\;\;\;a \cdot \left(a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-0.5 + \frac{1}{a \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= a -2.5e+237)
   (+ (* a (* a (* x (* x 0.5)))) -1.0)
   (/ 1.0 (+ -0.5 (/ 1.0 (* a x))))))

C

double code(double a, double x) {
	double tmp;
	if (a <= -2.5e+237) {
		tmp = (a * (a * (x * (x * 0.5)))) + -1.0;
	} else {
		tmp = 1.0 / (-0.5 + (1.0 / (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 <= (-2.5d+237)) then
        tmp = (a * (a * (x * (x * 0.5d0)))) + (-1.0d0)
    else
        tmp = 1.0d0 / ((-0.5d0) + (1.0d0 / (a * x)))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if (a <= -2.5e+237) {
		tmp = (a * (a * (x * (x * 0.5)))) + -1.0;
	} else {
		tmp = 1.0 / (-0.5 + (1.0 / (a * x)));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if a <= -2.5e+237:
		tmp = (a * (a * (x * (x * 0.5)))) + -1.0
	else:
		tmp = 1.0 / (-0.5 + (1.0 / (a * x)))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (a <= -2.5e+237)
		tmp = Float64(Float64(a * Float64(a * Float64(x * Float64(x * 0.5)))) + -1.0);
	else
		tmp = Float64(1.0 / Float64(-0.5 + Float64(1.0 / Float64(a * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if (a <= -2.5e+237)
		tmp = (a * (a * (x * (x * 0.5)))) + -1.0;
	else
		tmp = 1.0 / (-0.5 + (1.0 / (a * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[a, -2.5e+237], N[(N[(a * N[(a * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 / N[(-0.5 + N[(1.0 / N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5000000000000001e237

   1. Initial program 90.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 42.8%

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

   if -2.5000000000000001e237 < a

   1. Initial program 47.3%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 47.3

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lower-expm1.f64 99.3

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

   4. Applied rewrites 99.3%

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

   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. lower-/.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. lower-fma.f64 N/A

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

      4. lower-/.f64 77.4

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

   7. Applied rewrites 77.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 77.6%

       \[\leadsto \frac{1}{\frac{1}{a \cdot x} + \color{blue}{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+237}:\\ \;\;\;\;a \cdot \left(a \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-0.5 + \frac{1}{a \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x \cdot \left(a \cdot 0.5\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -4e+18) (/ 1.0 -0.5) (* a (fma (* x (* a 0.5)) x x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -4e+18) {
		tmp = 1.0 / -0.5;
	} else {
		tmp = a * fma((x * (a * 0.5)), x, x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -4e+18)
		tmp = Float64(1.0 / -0.5);
	else
		tmp = Float64(a * fma(Float64(x * Float64(a * 0.5)), x, x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -4e+18], N[(1.0 / -0.5), $MachinePrecision], N[(a * N[(N[(x * N[(a * 0.5), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -4e18

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 100.0

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lower-expm1.f64 100.0

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

   4. Applied rewrites 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. lower-/.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. lower-fma.f64 N/A

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

      4. lower-/.f64 18.8

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

   7. Applied rewrites 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

      \[\leadsto \frac{1}{\frac{-1}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.8%

       \[\leadsto \frac{1}{-0.5} \]

   if -4e18 < (*.f64 a x)

   1. Initial program 29.4%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 29.4

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lower-expm1.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 92.9

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

   7. Applied rewrites 92.9%

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

   9. Applied rewrites 98.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x \cdot \left(a \cdot 0.5\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -4e+18) (/ 1.0 -0.5) (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -4e+18) {
		tmp = 1.0 / -0.5;
	} 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) <= (-4d+18)) then
        tmp = 1.0d0 / (-0.5d0)
    else
        tmp = a * x
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -4e+18) {
		tmp = 1.0 / -0.5;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -4e+18:
		tmp = 1.0 / -0.5
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -4e+18)
		tmp = Float64(1.0 / -0.5);
	else
		tmp = Float64(a * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -4e+18)
		tmp = 1.0 / -0.5;
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -4e+18], N[(1.0 / -0.5), $MachinePrecision], N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -4e18

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 100.0

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lower-expm1.f64 100.0

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

   4. Applied rewrites 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. lower-/.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. lower-fma.f64 N/A

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

      4. lower-/.f64 18.8

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

   7. Applied rewrites 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

      \[\leadsto \frac{1}{\frac{-1}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.8%

       \[\leadsto \frac{1}{-0.5} \]

   if -4e18 < (*.f64 a x)

   1. Initial program 29.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

Alternative 6: 65.7% 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 48.9%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 71.9

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

5. Applied rewrites 71.9%

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

Alternative 7: 19.5% 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 48.9%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 19.9%

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

6. Final simplification 19.9%

   \[\leadsto 1 + -1 \]
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 2024238 
(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))