expax (section 3.5)

Percentage Accurate: 53.8% → 100.0%

Time: 6.3s

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: 53.8% 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 50.9%

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

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

   5. *-commutative N/A

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

   6. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 71.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;\frac{1}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5 \cdot x, a, 1\right) \cdot a\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1000.0) (/ 1.0 -0.5) (* (* (fma (* 0.5 x) a 1.0) a) x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1000.0) {
		tmp = 1.0 / -0.5;
	} else {
		tmp = (fma((0.5 * x), a, 1.0) * a) * x;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -1000.0)
		tmp = Float64(1.0 / -0.5);
	else
		tmp = Float64(Float64(fma(Float64(0.5 * x), a, 1.0) * a) * x);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -1000.0], N[(1.0 / -0.5), $MachinePrecision], N[(N[(N[(N[(0.5 * x), $MachinePrecision] * a + 1.0), $MachinePrecision] * a), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1e3

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 5.0

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

   5. Applied rewrites 5.0%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 5.0

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

   7. Applied rewrites 5.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2} \cdot x + \frac{1}{a}}{x}}} \]
   9. 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. lower-fma.f64 N/A

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

      3. lower-/.f64 18.8

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

   10. Applied rewrites 18.8%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 18.8%

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

   if -1e3 < (*.f64 a x)

   1. Initial program 30.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 29.5

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

   5. Applied rewrites 29.5%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 29.5

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in a around 0

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   10. Applied rewrites 99.1%

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

Alternative 3: 71.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;\frac{1}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot a, x, 1\right) \cdot \left(a \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1000.0) (/ 1.0 -0.5) (* (fma (* 0.5 a) x 1.0) (* a x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1000.0) {
		tmp = 1.0 / -0.5;
	} else {
		tmp = fma((0.5 * a), x, 1.0) * (a * x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -1000.0)
		tmp = Float64(1.0 / -0.5);
	else
		tmp = Float64(fma(Float64(0.5 * a), x, 1.0) * Float64(a * x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -1000.0], N[(1.0 / -0.5), $MachinePrecision], N[(N[(N[(0.5 * a), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(a * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1e3

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 5.0

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

   5. Applied rewrites 5.0%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 5.0

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

   7. Applied rewrites 5.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2} \cdot x + \frac{1}{a}}{x}}} \]
   9. 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. lower-fma.f64 N/A

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

      3. lower-/.f64 18.8

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

   10. Applied rewrites 18.8%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 18.8%

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

   if -1e3 < (*.f64 a x)

   1. Initial program 30.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;\frac{1}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot a, x, 1\right) \cdot \left(a \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{a \cdot x} - 0.5} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = 1.0d0 / ((1.0d0 / (a * x)) - 0.5d0)
end function

Java

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

Python

def code(a, x):
	return 1.0 / ((1.0 / (a * x)) - 0.5)

Julia

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

MATLAB

function tmp = code(a, x)
	tmp = 1.0 / ((1.0 / (a * x)) - 0.5);
end

Wolfram

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

Derivation

1. Initial program 50.9%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 22.3

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

5. Applied rewrites 22.3%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 22.3

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

7. Applied rewrites 22.3%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2} \cdot x + \frac{1}{a}}{x}}} \]
9. 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. lower-fma.f64 N/A

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

   3. lower-/.f64 74.7

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

10. Applied rewrites 74.7%

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

11. Taylor expanded in a around inf

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

13. Applied rewrites 74.8%

    \[\leadsto \frac{1}{\frac{1}{x \cdot a} - \color{blue}{0.5}} \]

14. Final simplification 74.8%

    \[\leadsto \frac{1}{\frac{1}{a \cdot x} - 0.5} \]
15. Add Preprocessing

Alternative 5: 71.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;\frac{1}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1000.0) (/ 1.0 -0.5) (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1000.0) {
		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) <= (-1000.0d0)) 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) <= -1000.0) {
		tmp = 1.0 / -0.5;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -1000.0:
		tmp = 1.0 / -0.5
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -1000.0)
		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) <= -1000.0)
		tmp = 1.0 / -0.5;
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -1000.0], N[(1.0 / -0.5), $MachinePrecision], N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1e3

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 5.0

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

   5. Applied rewrites 5.0%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 5.0

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

   7. Applied rewrites 5.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{2} \cdot x + \frac{1}{a}}{x}}} \]
   9. 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. lower-fma.f64 N/A

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

      3. lower-/.f64 18.8

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

   10. Applied rewrites 18.8%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 18.8%

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

   if -1e3 < (*.f64 a x)

   1. Initial program 30.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;\frac{1}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.7% 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 50.9%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 70.6

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

5. Applied rewrites 70.6%

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

6. Final simplification 70.6%

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

Alternative 7: 19.3% 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 50.9%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 20.2%

   \[\leadsto \color{blue}{1} - 1 \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024242 
(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))