expax (section 3.5)

Percentage Accurate: 65.8% → 100.0%

Time: 4.2s

Alternatives: 4

Speedup: 8.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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: 65.8% 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 65.8%

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

   1. expm1-def 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. Final simplification 100.0%

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

Alternative 2: 66.2% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq 2 \cdot 10^{+41}:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) 2e+41) (* a x) (* 0.5 (* x (* x (* a a))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= 2e+41) {
		tmp = a * x;
	} else {
		tmp = 0.5 * (x * (x * (a * a)));
	}
	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) <= 2d+41) then
        tmp = a * x
    else
        tmp = 0.5d0 * (x * (x * (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= 2e+41) {
		tmp = a * x;
	} else {
		tmp = 0.5 * (x * (x * (a * a)));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= 2e+41:
		tmp = a * x
	else:
		tmp = 0.5 * (x * (x * (a * a)))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= 2e+41)
		tmp = Float64(a * x);
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= 2e+41)
		tmp = a * x;
	else
		tmp = 0.5 * (x * (x * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], 2e+41], N[(a * x), $MachinePrecision], N[(0.5 * N[(x * N[(x * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < 2.00000000000000001e41

   1. Initial program 57.9%

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

      1. expm1-def 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. Taylor expanded in a around 0 64.0%

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

   if 2.00000000000000001e41 < (*.f64 a x)

   1. Initial program 100.0%

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

      1. expm1-def 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. Taylor expanded in a around 0 64.0%

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

      1. associate-*r* 64.0%

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

      2. unpow2 64.0%

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

      3. associate-*r* 66.6%

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

      4. distribute-rgt-out 66.6%

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

      5. *-commutative 66.6%

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

      6. unpow2 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in x around 0 66.6%

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

      1. *-commutative 66.6%

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

      2. unpow2 66.6%

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

      3. associate-*r* 66.6%

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

      4. associate-*l* 64.7%

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

   9. Simplified 64.7%

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

   10. Taylor expanded in x around inf 64.0%

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

       1. unpow2 64.0%

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

       2. unpow2 64.0%

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

       3. *-commutative 64.0%

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

       4. associate-*l* 66.6%

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

   12. Simplified 66.6%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq 2 \cdot 10^{+41}:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 3: 64.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(a \cdot \left(1 + x \cdot \left(a \cdot 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (* x (* a (+ 1.0 (* x (* a 0.5))))))

C

double code(double a, double x) {
	return x * (a * (1.0 + (x * (a * 0.5))));
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = x * (a * (1.0d0 + (x * (a * 0.5d0))))
end function

Java

public static double code(double a, double x) {
	return x * (a * (1.0 + (x * (a * 0.5))));
}

Python

def code(a, x):
	return x * (a * (1.0 + (x * (a * 0.5))))

Julia

function code(a, x)
	return Float64(x * Float64(a * Float64(1.0 + Float64(x * Float64(a * 0.5)))))
end

MATLAB

function tmp = code(a, x)
	tmp = x * (a * (1.0 + (x * (a * 0.5))));
end

Wolfram

code[a_, x_] := N[(x * N[(a * N[(1.0 + N[(x * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

   1. expm1-def 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. Taylor expanded in a around 0 57.1%

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

   1. associate-*r* 57.1%

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

   2. unpow2 57.1%

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

   3. associate-*r* 62.5%

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

   4. distribute-rgt-out 62.5%

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

   5. *-commutative 62.5%

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

   6. unpow2 62.5%

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

6. Simplified 62.5%

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

7. Taylor expanded in x around 0 62.5%

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

   1. *-commutative 62.5%

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

   2. unpow2 62.5%

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

   3. associate-*r* 62.5%

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

   4. associate-*l* 63.8%

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

9. Simplified 63.8%

   \[\leadsto x \cdot \left(\color{blue}{a \cdot \left(a \cdot \left(x \cdot 0.5\right)\right)} + a\right) \]
10. Step-by-step derivation

    1. +-commutative 63.8%

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

    2. *-un-lft-identity 63.8%

       \[\leadsto x \cdot \left(\color{blue}{1 \cdot a} + a \cdot \left(a \cdot \left(x \cdot 0.5\right)\right)\right) \]

    3. *-commutative 63.8%

       \[\leadsto x \cdot \left(1 \cdot a + \color{blue}{\left(a \cdot \left(x \cdot 0.5\right)\right) \cdot a}\right) \]

    4. distribute-rgt-out 63.8%

       \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(1 + a \cdot \left(x \cdot 0.5\right)\right)\right)} \]

    5. *-commutative 63.8%

       \[\leadsto x \cdot \left(a \cdot \left(1 + \color{blue}{\left(x \cdot 0.5\right) \cdot a}\right)\right) \]

    6. associate-*l* 63.8%

       \[\leadsto x \cdot \left(a \cdot \left(1 + \color{blue}{x \cdot \left(0.5 \cdot a\right)}\right)\right) \]

    7. *-commutative 63.8%

       \[\leadsto x \cdot \left(a \cdot \left(1 + x \cdot \color{blue}{\left(a \cdot 0.5\right)}\right)\right) \]

11. Applied egg-rr 63.8%

    \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(1 + x \cdot \left(a \cdot 0.5\right)\right)\right)} \]

12. Final simplification 63.8%

    \[\leadsto x \cdot \left(a \cdot \left(1 + x \cdot \left(a \cdot 0.5\right)\right)\right) \]

Alternative 4: 56.6% 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 65.8%

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

   1. expm1-def 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. Taylor expanded in a around 0 57.3%

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

5. Final simplification 57.3%

   \[\leadsto a \cdot x \]

Developer target: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (< (fabs (* a x)) 0.1)
   (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0))))
   (- (exp (* a x)) 1.0)))

C

double code(double a, double x) {
	double tmp;
	if (fabs((a * x)) < 0.1) {
		tmp = (a * x) * (1.0 + (((a * x) / 2.0) + (pow((a * x), 2.0) / 6.0)));
	} else {
		tmp = exp((a * x)) - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs((a * x)) < 0.1d0) then
        tmp = (a * x) * (1.0d0 + (((a * x) / 2.0d0) + (((a * x) ** 2.0d0) / 6.0d0)))
    else
        tmp = exp((a * x)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if (Math.abs((a * x)) < 0.1) {
		tmp = (a * x) * (1.0 + (((a * x) / 2.0) + (Math.pow((a * x), 2.0) / 6.0)));
	} else {
		tmp = Math.exp((a * x)) - 1.0;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if math.fabs((a * x)) < 0.1:
		tmp = (a * x) * (1.0 + (((a * x) / 2.0) + (math.pow((a * x), 2.0) / 6.0)))
	else:
		tmp = math.exp((a * x)) - 1.0
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (abs(Float64(a * x)) < 0.1)
		tmp = Float64(Float64(a * x) * Float64(1.0 + Float64(Float64(Float64(a * x) / 2.0) + Float64((Float64(a * x) ^ 2.0) / 6.0))));
	else
		tmp = Float64(exp(Float64(a * x)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if (abs((a * x)) < 0.1)
		tmp = (a * x) * (1.0 + (((a * x) / 2.0) + (((a * x) ^ 2.0) / 6.0)));
	else
		tmp = exp((a * x)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[Less[N[Abs[N[(a * x), $MachinePrecision]], $MachinePrecision], 0.1], N[(N[(a * x), $MachinePrecision] * N[(1.0 + N[(N[(N[(a * x), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Power[N[(a * x), $MachinePrecision], 2.0], $MachinePrecision] / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]

Reproduce

herbie shell --seed 2023293 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

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