expax (section 3.5)

Percentage Accurate: 53.6% → 100.0%

Time: 9.9s

Alternatives: 4

Speedup: 35.0×

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: 53.6% 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 52.7%

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

   1. expm1-define 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 69.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50000:\\ \;\;\;\;a \cdot \frac{\frac{\frac{1}{a}}{a} - x \cdot x}{\frac{1}{a} - x} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(1 + \left(a \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -50000.0)
   (+ (* a (/ (- (/ (/ 1.0 a) a) (* x x)) (- (/ 1.0 a) x))) -1.0)
   (* a (* x (+ 1.0 (* (* a x) 0.5))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50000.0) {
		tmp = (a * ((((1.0 / a) / a) - (x * x)) / ((1.0 / a) - x))) + -1.0;
	} else {
		tmp = a * (x * (1.0 + ((a * x) * 0.5)));
	}
	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) <= (-50000.0d0)) then
        tmp = (a * ((((1.0d0 / a) / a) - (x * x)) / ((1.0d0 / a) - x))) + (-1.0d0)
    else
        tmp = a * (x * (1.0d0 + ((a * x) * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -50000.0) {
		tmp = (a * ((((1.0 / a) / a) - (x * x)) / ((1.0 / a) - x))) + -1.0;
	} else {
		tmp = a * (x * (1.0 + ((a * x) * 0.5)));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -50000.0:
		tmp = (a * ((((1.0 / a) / a) - (x * x)) / ((1.0 / a) - x))) + -1.0
	else:
		tmp = a * (x * (1.0 + ((a * x) * 0.5)))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -50000.0)
		tmp = Float64(Float64(a * Float64(Float64(Float64(Float64(1.0 / a) / a) - Float64(x * x)) / Float64(Float64(1.0 / a) - x))) + -1.0);
	else
		tmp = Float64(a * Float64(x * Float64(1.0 + Float64(Float64(a * x) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -50000.0)
		tmp = (a * ((((1.0 / a) / a) - (x * x)) / ((1.0 / a) - x))) + -1.0;
	else
		tmp = a * (x * (1.0 + ((a * x) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -50000.0], N[(N[(a * N[(N[(N[(N[(1.0 / a), $MachinePrecision] / a), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / a), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(x * N[(1.0 + N[(N[(a * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -5e4

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0 5.3%

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

   4. Taylor expanded in a around inf 5.3%

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

      1. +-commutative 5.3%

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

      2. flip-+ 9.0%

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

      3. frac-times 9.0%

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

      4. metadata-eval 9.0%

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

   6. Applied egg-rr 9.0%

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

      1. associate-/r* 9.0%

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

   8. Simplified 9.0%

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

   if -5e4 < (*.f64 a x)

   1. Initial program 30.4%

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

      1. expm1-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 85.1%

      \[\leadsto \color{blue}{a \cdot \left(x + a \cdot \left(0.16666666666666666 \cdot \left(a \cdot {x}^{3}\right) + 0.5 \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 85.1%

         \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(0.16666666666666666 \cdot \left(a \cdot {x}^{3}\right) + 0.5 \cdot {x}^{2}\right) + x\right)} \]

      2. fma-define 85.1%

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(a, 0.16666666666666666 \cdot \left(a \cdot {x}^{3}\right) + 0.5 \cdot {x}^{2}, x\right)} \]

      3. associate-*r* 85.1%

         \[\leadsto a \cdot \mathsf{fma}\left(a, \color{blue}{\left(0.16666666666666666 \cdot a\right) \cdot {x}^{3}} + 0.5 \cdot {x}^{2}, x\right) \]

      4. cube-mult 85.1%

         \[\leadsto a \cdot \mathsf{fma}\left(a, \left(0.16666666666666666 \cdot a\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + 0.5 \cdot {x}^{2}, x\right) \]

      5. unpow2 85.1%

         \[\leadsto a \cdot \mathsf{fma}\left(a, \left(0.16666666666666666 \cdot a\right) \cdot \left(x \cdot \color{blue}{{x}^{2}}\right) + 0.5 \cdot {x}^{2}, x\right) \]

      6. associate-*r* 92.6%

         \[\leadsto a \cdot \mathsf{fma}\left(a, \color{blue}{\left(\left(0.16666666666666666 \cdot a\right) \cdot x\right) \cdot {x}^{2}} + 0.5 \cdot {x}^{2}, x\right) \]

      7. distribute-rgt-out 92.6%

         \[\leadsto a \cdot \mathsf{fma}\left(a, \color{blue}{{x}^{2} \cdot \left(\left(0.16666666666666666 \cdot a\right) \cdot x + 0.5\right)}, x\right) \]

      8. unpow2 92.6%

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

      9. *-commutative 92.6%

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

      10. *-commutative 92.6%

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

   7. Simplified 92.6%

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

   8. Taylor expanded in x around 0 97.6%

      \[\leadsto a \cdot \color{blue}{\left(x \cdot \left(1 + 0.5 \cdot \left(a \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -50000:\\ \;\;\;\;a \cdot \frac{\frac{\frac{1}{a}}{a} - x \cdot x}{\frac{1}{a} - x} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(1 + \left(a \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.0% accurate, 35.0× 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 52.7%

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

   1. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0 67.3%

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

Alternative 4: 3.1% accurate, 105.0× speedup

Math

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

FPCore

(FPCore (a x) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(a, x):
	return 1.0

Julia

function code(a, x)
	return 1.0
end

MATLAB

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

Wolfram

code[a_, x_] := 1.0

Derivation

1. Initial program 52.7%

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

3. Taylor expanded in a around 0 20.6%

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

   1. associate--l+ 20.6%

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

   2. +-commutative 20.6%

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

   3. sub-neg 20.6%

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

   4. metadata-eval 20.6%

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

5. Applied egg-rr 20.6%

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

6. Taylor expanded in a around inf 4.4%

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

7. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Developer target: 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 2024097 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :pre (> 710.0 (* a x))

  :alt
  (expm1 (* a x))

  (- (exp (* a x)) 1.0))