expax (section 3.5)

Percentage Accurate: 54.3% → 100.0%

Time: 8.5s

Alternatives: 10

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: 54.3% 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 51.7%

   \[e^{a \cdot x} - 1 \]
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: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{if}\;a \cdot x \leq -4000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_0, -1\right)}{t\_0 - -1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(x, a \cdot \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (fma a x -1.0))))
   (if (<= (* a x) -4000000.0)
     (/ (fma t_0 t_0 -1.0) (- t_0 -1.0))
     (*
      a
      (fma
       (*
        (* a x)
        (fma
         x
         (* a (fma a (* x 0.041666666666666664) 0.16666666666666666))
         0.5))
       x
       x)))))

C

double code(double a, double x) {
	double t_0 = -1.0 / fma(a, x, -1.0);
	double tmp;
	if ((a * x) <= -4000000.0) {
		tmp = fma(t_0, t_0, -1.0) / (t_0 - -1.0);
	} else {
		tmp = a * fma(((a * x) * fma(x, (a * fma(a, (x * 0.041666666666666664), 0.16666666666666666)), 0.5)), x, 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) <= -4000000.0)
		tmp = Float64(fma(t_0, t_0, -1.0) / Float64(t_0 - -1.0));
	else
		tmp = Float64(a * fma(Float64(Float64(a * x) * fma(x, Float64(a * fma(a, Float64(x * 0.041666666666666664), 0.16666666666666666)), 0.5)), x, 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], -4000000.0], N[(N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(a * x), $MachinePrecision] * N[(x * N[(a * N[(a * N[(x * 0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -4e6

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 5.0

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

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.3%

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

       1. lift--.f64 N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. flip-+ N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

   12. Applied rewrites 99.3%

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

   if -4e6 < (*.f64 a x)

   1. Initial program 30.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a \cdot \left(x + a \cdot \left(\frac{1}{2} \cdot {x}^{2} + a \cdot \left(\frac{1}{24} \cdot \left(a \cdot {x}^{4}\right) + \frac{1}{6} \cdot {x}^{3}\right)\right)\right)} \]

   5. Applied rewrites 95.2%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(a, x \cdot \left(x \cdot \mathsf{fma}\left(a \cdot x, \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.2%

      \[\leadsto a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(x, a \cdot \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(x, a \cdot \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -4000000.0)
   (+ -1.0 (/ -1.0 (fma a x -1.0)))
   (*
    a
    (fma
     (*
      (* a x)
      (fma x (* a (fma a (* x 0.041666666666666664) 0.16666666666666666)) 0.5))
     x
     x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -4000000.0) {
		tmp = -1.0 + (-1.0 / fma(a, x, -1.0));
	} else {
		tmp = a * fma(((a * x) * fma(x, (a * fma(a, (x * 0.041666666666666664), 0.16666666666666666)), 0.5)), x, x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -4000000.0)
		tmp = Float64(-1.0 + Float64(-1.0 / fma(a, x, -1.0)));
	else
		tmp = Float64(a * fma(Float64(Float64(a * x) * fma(x, Float64(a * fma(a, Float64(x * 0.041666666666666664), 0.16666666666666666)), 0.5)), x, x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -4000000.0], N[(-1.0 + N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(a * x), $MachinePrecision] * N[(x * N[(a * N[(a * N[(x * 0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -4e6

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 5.0

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

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.3%

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

   if -4e6 < (*.f64 a x)

   1. Initial program 30.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a \cdot \left(x + a \cdot \left(\frac{1}{2} \cdot {x}^{2} + a \cdot \left(\frac{1}{24} \cdot \left(a \cdot {x}^{4}\right) + \frac{1}{6} \cdot {x}^{3}\right)\right)\right)} \]

   5. Applied rewrites 95.2%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(a, x \cdot \left(x \cdot \mathsf{fma}\left(a \cdot x, \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.2%

      \[\leadsto a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(x, a \cdot \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(x, a \cdot \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), a \cdot \left(x \cdot \left(a \cdot x\right)\right), a \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -4000000.0)
   (+ -1.0 (/ -1.0 (fma a x -1.0)))
   (fma (fma a (* x 0.16666666666666666) 0.5) (* a (* x (* a x))) (* a x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -4000000.0) {
		tmp = -1.0 + (-1.0 / fma(a, x, -1.0));
	} else {
		tmp = fma(fma(a, (x * 0.16666666666666666), 0.5), (a * (x * (a * x))), (a * x));
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -4000000.0)
		tmp = Float64(-1.0 + Float64(-1.0 / fma(a, x, -1.0)));
	else
		tmp = fma(fma(a, Float64(x * 0.16666666666666666), 0.5), Float64(a * Float64(x * Float64(a * x))), Float64(a * x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -4000000.0], N[(-1.0 + N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] * N[(a * N[(x * N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -4e6

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 5.0

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

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.3%

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

   if -4e6 < (*.f64 a x)

   1. Initial program 30.1%

      \[e^{a \cdot x} - 1 \]
   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 95.0%

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

   7. Applied rewrites 98.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), a \cdot \left(x \cdot \left(a \cdot x\right)\right), a \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(a \cdot x, x \cdot \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -4000000.0)
   (+ -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) <= -4000000.0) {
		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) <= -4000000.0)
		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], -4000000.0], 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) < -4e6

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 5.0

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

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.3%

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

   if -4e6 < (*.f64 a x)

   1. Initial program 30.1%

      \[e^{a \cdot x} - 1 \]
   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 95.0%

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

   7. Applied rewrites 98.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(a \cdot x, x \cdot \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, a \cdot \left(x \cdot \left(a \cdot x\right)\right), a \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -4000000.0)
   (+ -1.0 (/ -1.0 (fma a x -1.0)))
   (fma 0.5 (* a (* x (* a x))) (* a x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -4000000.0) {
		tmp = -1.0 + (-1.0 / fma(a, x, -1.0));
	} else {
		tmp = fma(0.5, (a * (x * (a * x))), (a * x));
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -4000000.0)
		tmp = Float64(-1.0 + Float64(-1.0 / fma(a, x, -1.0)));
	else
		tmp = fma(0.5, Float64(a * Float64(x * Float64(a * x))), Float64(a * x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -4000000.0], N[(-1.0 + N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(a * N[(x * N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -4e6

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 5.0

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

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.3%

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

   if -4e6 < (*.f64 a x)

   1. Initial program 30.1%

      \[e^{a \cdot x} - 1 \]
   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 95.0%

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, a \cdot \left(x \cdot \left(a \cdot x\right)\right), a \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(a \cdot x, x \cdot 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -4000000.0)
   (+ -1.0 (/ -1.0 (fma a x -1.0)))
   (* a (fma (* a x) (* x 0.5) x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -4000000.0) {
		tmp = -1.0 + (-1.0 / fma(a, x, -1.0));
	} else {
		tmp = a * fma((a * x), (x * 0.5), x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -4000000.0)
		tmp = Float64(-1.0 + Float64(-1.0 / fma(a, x, -1.0)));
	else
		tmp = Float64(a * fma(Float64(a * x), Float64(x * 0.5), x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -4000000.0], N[(-1.0 + N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * x), $MachinePrecision] * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -4e6

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 5.0

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

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.3%

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

   if -4e6 < (*.f64 a x)

   1. Initial program 30.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. prod-exp N/A

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

      7. metadata-eval N/A

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

      8. lower-expm1.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 96.5

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in a around 0

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

   10. Applied rewrites 98.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -4000000:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(a \cdot x, x \cdot 0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2 \cdot 10^{-5}:\\ \;\;\;\;-1 + \frac{-1}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -2e-5) (+ -1.0 (/ -1.0 (fma a x -1.0))) (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2e-5) {
		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) <= -2e-5)
		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], -2e-5], 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) < -2.00000000000000016e-5

   1. Initial program 99.6%

      \[e^{a \cdot x} - 1 \]
   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 6.1

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

   5. Applied rewrites 6.1%

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

   7. Applied rewrites 4.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.9%

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

   if -2.00000000000000016e-5 < (*.f64 a x)

   1. Initial program 29.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

4. Final simplification 97.3%

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

Alternative 9: 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 51.7%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 68.3

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

5. Applied rewrites 68.3%

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

Alternative 10: 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 51.7%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 18.2%

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

6. Final simplification 18.2%

   \[\leadsto -1 + 1 \]
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 2024233 
(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))