expax (section 3.5)

Percentage Accurate: 53.9% → 100.0%

Time: 9.0s

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: 53.9% 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 52.4%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lower-expm1.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 99.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, x \cdot \mathsf{fma}\left(a, x \cdot 0.5, -1\right), 1\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(a \cdot x, 0.16666666666666666, 0.5\right), a \cdot x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -2.0)
   (+ (/ 1.0 (fma a (* x (fma a (* x 0.5) -1.0)) 1.0)) -1.0)
   (* a (fma (* x (fma (* a x) 0.16666666666666666 0.5)) (* a x) x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -2.0) {
		tmp = (1.0 / fma(a, (x * fma(a, (x * 0.5), -1.0)), 1.0)) + -1.0;
	} else {
		tmp = a * fma((x * fma((a * x), 0.16666666666666666, 0.5)), (a * x), x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -2.0)
		tmp = Float64(Float64(1.0 / fma(a, Float64(x * fma(a, Float64(x * 0.5), -1.0)), 1.0)) + -1.0);
	else
		tmp = Float64(a * fma(Float64(x * fma(Float64(a * x), 0.16666666666666666, 0.5)), Float64(a * x), x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -2.0], N[(N[(1.0 / N[(a * N[(x * N[(a * N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(N[(x * N[(N[(a * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(a * x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 3.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot \left(x \cdot x\right), \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)} - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.1%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 99.0%

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

   if -2 < (*.f64 a x)

   1. Initial program 30.8%

      \[e^{a \cdot x} - 1 \]
   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 92.7%

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

   7. Applied rewrites 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, x \cdot \mathsf{fma}\left(a, x \cdot 0.5, -1\right), 1\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(a \cdot x, 0.16666666666666666, 0.5\right), a \cdot x, x\right)\\ \end{array} \]
5. 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 2024223 
(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))