expax (section 3.5)

Percentage Accurate: 54.9% → 100.0%

Time: 8.1s

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: 54.9% 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 52.1%

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

   1. lift--.f64 N/A

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

   2. 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: 70.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100000:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, 0.16666666666666666, \frac{0.5}{x}\right), \frac{a}{x \cdot x}\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(a \cdot x, x \cdot \mathsf{fma}\left(a, x \cdot \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -100000.0)
   (+
    (*
     x
     (*
      x
      (* x (fma (* a a) (fma a 0.16666666666666666 (/ 0.5 x)) (/ a (* x x))))))
    -1.0)
   (*
    a
    (fma
     (* a x)
     (*
      x
      (fma a (* x (fma a (* x 0.041666666666666664) 0.16666666666666666)) 0.5))
     x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100000.0) {
		tmp = (x * (x * (x * fma((a * a), fma(a, 0.16666666666666666, (0.5 / x)), (a / (x * x)))))) + -1.0;
	} else {
		tmp = a * fma((a * x), (x * fma(a, (x * fma(a, (x * 0.041666666666666664), 0.16666666666666666)), 0.5)), x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -100000.0)
		tmp = Float64(Float64(x * Float64(x * Float64(x * fma(Float64(a * a), fma(a, 0.16666666666666666, Float64(0.5 / x)), Float64(a / Float64(x * x)))))) + -1.0);
	else
		tmp = Float64(a * fma(Float64(a * x), Float64(x * fma(a, Float64(x * fma(a, Float64(x * 0.041666666666666664), 0.16666666666666666)), 0.5)), x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -100000.0], N[(N[(x * N[(x * N[(x * N[(N[(a * a), $MachinePrecision] * N[(a * 0.16666666666666666 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] + N[(a / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(N[(a * x), $MachinePrecision] * N[(x * N[(a * N[(x * N[(a * N[(x * 0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1e5

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 0.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 12.4%

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

   if -1e5 < (*.f64 a x)

   1. Initial program 28.6%

      \[e^{a \cdot x} - 1 \]
   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 91.4%

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

   7. Applied rewrites 99.2%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100000:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, 0.16666666666666666, \frac{0.5}{x}\right), \frac{a}{x \cdot x}\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(a \cdot x, x \cdot \mathsf{fma}\left(a, x \cdot \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.7% accurate, 3.3× speedup

Math

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

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(a, Float64(a * Float64(x * fma(a, Float64(x * 0.16666666666666666), 0.5))), a))
end

Wolfram

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

Derivation

1. Initial program 52.1%

   \[e^{a \cdot x} - 1 \]
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 62.5%

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

7. Applied rewrites 67.7%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 67.6%

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

11. Final simplification 67.6%

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

Alternative 4: 66.2% 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 52.1%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 67.0

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

5. Applied rewrites 67.0%

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

Alternative 5: 19.8% 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 52.1%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 18.0%

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

6. Final simplification 18.0%

   \[\leadsto 1 + -1 \]
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 2024232 
(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))