expax (section 3.5)

Percentage Accurate: 53.7% → 100.0%

Time: 9.1s

Alternatives: 5

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: 53.7% 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 52.2%

   \[e^{a \cdot x} - 1 \]
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: 69.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1 \cdot 10^{+36}:\\ \;\;\;\;a \cdot \frac{x \cdot \left(-x\right)}{\mathsf{fma}\left(a, x \cdot \left(x \cdot 0.5\right), -x\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a \cdot \left(a \cdot x\right), \mathsf{fma}\left(a \cdot x, 0.16666666666666666, 0.5\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1e+36)
   (* a (/ (* x (- x)) (fma a (* x (* x 0.5)) (- x))))
   (* x (fma (* a (* a x)) (fma (* a x) 0.16666666666666666 0.5) a))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1e+36) {
		tmp = a * ((x * -x) / fma(a, (x * (x * 0.5)), -x));
	} else {
		tmp = x * fma((a * (a * x)), fma((a * x), 0.16666666666666666, 0.5), a);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -1e+36)
		tmp = Float64(a * Float64(Float64(x * Float64(-x)) / fma(a, Float64(x * Float64(x * 0.5)), Float64(-x))));
	else
		tmp = Float64(x * fma(Float64(a * Float64(a * x)), fma(Float64(a * x), 0.16666666666666666, 0.5), a));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -1e+36], N[(a * N[(N[(x * (-x)), $MachinePrecision] / N[(a * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(a * N[(a * x), $MachinePrecision]), $MachinePrecision] * N[(N[(a * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1.00000000000000004e36

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 1.1

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

   5. Applied rewrites 1.1%

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

   7. Applied rewrites 0.5%

      \[\leadsto \left(a \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \left(a \cdot a\right) \cdot 0.25, -x \cdot x\right)\right) \cdot \color{blue}{\frac{1}{a \cdot \left(0.5 \cdot \left(x \cdot x\right)\right) - x}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 8.8%

       \[\leadsto \left(a \cdot \left(x \cdot \left(-x\right)\right)\right) \cdot \frac{1}{a \cdot \left(0.5 \cdot \left(x \cdot x\right)\right) - x} \]
   11. Step-by-step derivation

   12. Applied rewrites 9.1%

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

   if -1.00000000000000004e36 < (*.f64 a x)

   1. Initial program 33.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 94.9

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.5%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1 \cdot 10^{+36}:\\ \;\;\;\;a \cdot \frac{x \cdot \left(-x\right)}{\mathsf{fma}\left(a, x \cdot \left(x \cdot 0.5\right), -x\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a \cdot \left(a \cdot x\right), \mathsf{fma}\left(a \cdot x, 0.16666666666666666, 0.5\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(a \cdot \left(a \cdot x\right), \mathsf{fma}\left(a \cdot x, 0.16666666666666666, 0.5\right), a\right) \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (* x (fma (* a (* a x)) (fma (* a x) 0.16666666666666666 0.5) a)))

C

double code(double a, double x) {
	return x * fma((a * (a * x)), fma((a * x), 0.16666666666666666, 0.5), a);
}

Julia

function code(a, x)
	return Float64(x * fma(Float64(a * Float64(a * x)), fma(Float64(a * x), 0.16666666666666666, 0.5), a))
end

Wolfram

code[a_, x_] := N[(x * N[(N[(a * N[(a * x), $MachinePrecision]), $MachinePrecision] * N[(N[(a * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 69.6

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

5. Applied rewrites 69.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 69.6%

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

Alternative 4: 67.2% 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 52.2%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 69.6

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

5. Applied rewrites 69.6%

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

Alternative 5: 19.5% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ 1 + -1 \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (+ 1.0 -1.0))

C

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

Fortran

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

Java

public static double code(double a, double x) {
	return 1.0 + -1.0;
}

Python

def code(a, x):
	return 1.0 + -1.0

Julia

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

MATLAB

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

Wolfram

code[a_, x_] := N[(1.0 + -1.0), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 20.6%

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

6. Final simplification 20.6%

   \[\leadsto 1 + -1 \]
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 2024214 
(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))