expax (section 3.5)

Percentage Accurate: 53.5% → 100.0%

Time: 8.4s

Alternatives: 8

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: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.0%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 98.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{if}\;a \cdot x \leq -20000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_0, -1\right)}{t\_0 - -1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(a \cdot x, x \cdot \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (fma a x -1.0))))
   (if (<= (* a x) -20000000000000.0)
     (/ (fma t_0 t_0 -1.0) (- t_0 -1.0))
     (* a (fma (* a x) (* x (fma a (* x 0.16666666666666666) 0.5)) x)))))

C

double code(double a, double x) {
	double t_0 = -1.0 / fma(a, x, -1.0);
	double tmp;
	if ((a * x) <= -20000000000000.0) {
		tmp = fma(t_0, t_0, -1.0) / (t_0 - -1.0);
	} else {
		tmp = a * fma((a * x), (x * fma(a, (x * 0.16666666666666666), 0.5)), x);
	}
	return tmp;
}

Julia

function code(a, x)
	t_0 = Float64(-1.0 / fma(a, x, -1.0))
	tmp = 0.0
	if (Float64(a * x) <= -20000000000000.0)
		tmp = Float64(fma(t_0, t_0, -1.0) / Float64(t_0 - -1.0));
	else
		tmp = Float64(a * fma(Float64(a * x), Float64(x * fma(a, Float64(x * 0.16666666666666666), 0.5)), x));
	end
	return tmp
end

Wolfram

code[a_, x_] := Block[{t$95$0 = N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * x), $MachinePrecision], -20000000000000.0], N[(N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * x), $MachinePrecision] * N[(x * N[(a * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -2e13

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 4.9

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

   5. Applied rewrites 4.9%

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.7%

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

       1. lift--.f64 N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. flip-+ N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

   12. Applied rewrites 99.7%

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

   if -2e13 < (*.f64 a x)

   1. Initial program 32.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 95.7%

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

   7. Applied rewrites 99.3%

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

Alternative 3: 98.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000000000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(a \cdot x, x \cdot \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -20000000000000.0)
   (+ -1.0 (/ -1.0 (fma a x -1.0)))
   (* a (fma (* a x) (* x (fma a (* x 0.16666666666666666) 0.5)) x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -20000000000000.0) {
		tmp = -1.0 + (-1.0 / fma(a, x, -1.0));
	} else {
		tmp = a * fma((a * x), (x * fma(a, (x * 0.16666666666666666), 0.5)), x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -20000000000000.0)
		tmp = Float64(-1.0 + Float64(-1.0 / fma(a, x, -1.0)));
	else
		tmp = Float64(a * fma(Float64(a * x), Float64(x * fma(a, Float64(x * 0.16666666666666666), 0.5)), x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -20000000000000.0], N[(-1.0 + N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * x), $MachinePrecision] * N[(x * N[(a * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -2e13

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 4.9

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

   5. Applied rewrites 4.9%

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.7%

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

   12. Applied rewrites 99.7%

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

   if -2e13 < (*.f64 a x)

   1. Initial program 32.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 95.7%

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

   7. Applied rewrites 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000000000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(a \cdot x, x \cdot \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000000000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a \cdot \left(x \cdot 0.5\right), a, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -20000000000000.0)
   (+ -1.0 (/ -1.0 (fma a x -1.0)))
   (* x (fma (* a (* x 0.5)) a a))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -20000000000000.0) {
		tmp = -1.0 + (-1.0 / fma(a, x, -1.0));
	} else {
		tmp = x * fma((a * (x * 0.5)), a, a);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -20000000000000.0)
		tmp = Float64(-1.0 + Float64(-1.0 / fma(a, x, -1.0)));
	else
		tmp = Float64(x * fma(Float64(a * Float64(x * 0.5)), a, a));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -20000000000000.0], N[(-1.0 + N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(a * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] * a + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -2e13

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 4.9

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

   5. Applied rewrites 4.9%

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.7%

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

   12. Applied rewrites 99.7%

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

   if -2e13 < (*.f64 a x)

   1. Initial program 32.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 30.8

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

   5. Applied rewrites 30.8%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 30.8

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

   7. Applied rewrites 30.8%

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

   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. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 95.7

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

   10. Applied rewrites 95.7%

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

   12. Applied rewrites 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000000000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a \cdot \left(x \cdot 0.5\right), a, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -5 \cdot 10^{-9}:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -5e-9) (+ -1.0 (/ -1.0 (fma a x -1.0))) (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -5e-9) {
		tmp = -1.0 + (-1.0 / fma(a, x, -1.0));
	} else {
		tmp = a * x;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -5e-9)
		tmp = Float64(-1.0 + Float64(-1.0 / fma(a, x, -1.0)));
	else
		tmp = Float64(a * x);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -5e-9], N[(-1.0 + N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -5.0000000000000001e-9

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 6.2

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

   5. Applied rewrites 6.2%

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

   7. Applied rewrites 5.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.7%

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

   12. Applied rewrites 98.7%

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

   if -5.0000000000000001e-9 < (*.f64 a x)

   1. Initial program 31.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -5 \cdot 10^{-9}:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000000000000:\\ \;\;\;\;\frac{1}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -20000000000000.0) (/ 1.0 -0.5) (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -20000000000000.0) {
		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) <= (-20000000000000.0d0)) 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) <= -20000000000000.0) {
		tmp = 1.0 / -0.5;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -20000000000000.0:
		tmp = 1.0 / -0.5
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -20000000000000.0)
		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) <= -20000000000000.0)
		tmp = 1.0 / -0.5;
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -20000000000000.0], N[(1.0 / -0.5), $MachinePrecision], N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -2e13

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 4.9

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

   5. Applied rewrites 4.9%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 4.9

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

   7. Applied rewrites 4.9%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 18.8

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

   10. Applied rewrites 18.8%

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

   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 -2e13 < (*.f64 a x)

   1. Initial program 32.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 7: 67.5% 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 53.0%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 69.0

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

5. Applied rewrites 69.0%

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

Alternative 8: 19.7% 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 53.0%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 20.4%

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

6. Final simplification 20.4%

   \[\leadsto -1 + 1 \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024226 
(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))