expax (section 3.5)

Percentage Accurate: 53.3% → 100.0%

Time: 5.9s

Alternatives: 3

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: 53.3% 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 52.3%

   \[e^{a \cdot x} - 1 \]
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. Final simplification 100.0%

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

Alternative 2: 68.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(x \cdot \left(1 + \left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (* a (* x (+ 1.0 (* (* a x) (+ 0.5 (* (* a x) 0.16666666666666666)))))))

C

double code(double a, double x) {
	return a * (x * (1.0 + ((a * x) * (0.5 + ((a * x) * 0.16666666666666666)))));
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = a * (x * (1.0d0 + ((a * x) * (0.5d0 + ((a * x) * 0.16666666666666666d0)))))
end function

Java

public static double code(double a, double x) {
	return a * (x * (1.0 + ((a * x) * (0.5 + ((a * x) * 0.16666666666666666)))));
}

Python

def code(a, x):
	return a * (x * (1.0 + ((a * x) * (0.5 + ((a * x) * 0.16666666666666666)))))

Julia

function code(a, x)
	return Float64(a * Float64(x * Float64(1.0 + Float64(Float64(a * x) * Float64(0.5 + Float64(Float64(a * x) * 0.16666666666666666))))))
end

MATLAB

function tmp = code(a, x)
	tmp = a * (x * (1.0 + ((a * x) * (0.5 + ((a * x) * 0.16666666666666666)))));
end

Wolfram

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

Derivation

1. Initial program 52.3%

   \[e^{a \cdot x} - 1 \]
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 58.6%

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

   1. +-commutative 58.6%

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

   2. fma-define 58.6%

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

   3. *-commutative 58.6%

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

   4. associate-*r* 58.6%

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

   5. *-commutative 58.6%

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

   6. unpow3 58.6%

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

   7. unpow2 58.6%

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

   8. associate-*l* 58.6%

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

   9. associate-*l* 61.6%

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

   10. *-commutative 61.6%

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

   11. distribute-lft-out 62.1%

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

7. Simplified 62.1%

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

8. Taylor expanded in x around 0 61.8%

   \[\leadsto a \cdot \color{blue}{\left(x \cdot \left(1 + x \cdot \left(0.16666666666666666 \cdot \left({a}^{2} \cdot x\right) + 0.5 \cdot a\right)\right)\right)} \]

9. Taylor expanded in a around 0 62.0%

   \[\leadsto a \cdot \left(x \cdot \left(1 + \color{blue}{a \cdot \left(0.16666666666666666 \cdot \left(a \cdot {x}^{2}\right) + 0.5 \cdot x\right)}\right)\right) \]
10. Step-by-step derivation

    1. +-commutative 62.0%

       \[\leadsto a \cdot \left(x \cdot \left(1 + a \cdot \color{blue}{\left(0.5 \cdot x + 0.16666666666666666 \cdot \left(a \cdot {x}^{2}\right)\right)}\right)\right) \]

    2. *-commutative 62.0%

       \[\leadsto a \cdot \left(x \cdot \left(1 + a \cdot \left(\color{blue}{x \cdot 0.5} + 0.16666666666666666 \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right) \]

    3. distribute-rgt-in 61.8%

       \[\leadsto a \cdot \left(x \cdot \left(1 + \color{blue}{\left(\left(x \cdot 0.5\right) \cdot a + \left(0.16666666666666666 \cdot \left(a \cdot {x}^{2}\right)\right) \cdot a\right)}\right)\right) \]

    4. *-commutative 61.8%

       \[\leadsto a \cdot \left(x \cdot \left(1 + \left(\color{blue}{a \cdot \left(x \cdot 0.5\right)} + \left(0.16666666666666666 \cdot \left(a \cdot {x}^{2}\right)\right) \cdot a\right)\right)\right) \]

    5. associate-*r* 61.8%

       \[\leadsto a \cdot \left(x \cdot \left(1 + \left(\color{blue}{\left(a \cdot x\right) \cdot 0.5} + \left(0.16666666666666666 \cdot \left(a \cdot {x}^{2}\right)\right) \cdot a\right)\right)\right) \]

    6. *-commutative 61.8%

       \[\leadsto a \cdot \left(x \cdot \left(1 + \left(\color{blue}{0.5 \cdot \left(a \cdot x\right)} + \left(0.16666666666666666 \cdot \left(a \cdot {x}^{2}\right)\right) \cdot a\right)\right)\right) \]

    7. associate-*l* 61.8%

       \[\leadsto a \cdot \left(x \cdot \left(1 + \left(0.5 \cdot \left(a \cdot x\right) + \color{blue}{0.16666666666666666 \cdot \left(\left(a \cdot {x}^{2}\right) \cdot a\right)}\right)\right)\right) \]

    8. *-commutative 61.8%

       \[\leadsto a \cdot \left(x \cdot \left(1 + \left(0.5 \cdot \left(a \cdot x\right) + 0.16666666666666666 \cdot \left(\color{blue}{\left({x}^{2} \cdot a\right)} \cdot a\right)\right)\right)\right) \]

    9. associate-*r* 57.2%

       \[\leadsto a \cdot \left(x \cdot \left(1 + \left(0.5 \cdot \left(a \cdot x\right) + 0.16666666666666666 \cdot \color{blue}{\left({x}^{2} \cdot \left(a \cdot a\right)\right)}\right)\right)\right) \]

    10. unpow2 57.2%

        \[\leadsto a \cdot \left(x \cdot \left(1 + \left(0.5 \cdot \left(a \cdot x\right) + 0.16666666666666666 \cdot \left({x}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right)\right) \]

    11. *-commutative 57.2%

        \[\leadsto a \cdot \left(x \cdot \left(1 + \left(0.5 \cdot \left(a \cdot x\right) + 0.16666666666666666 \cdot \color{blue}{\left({a}^{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

    12. unpow2 57.2%

        \[\leadsto a \cdot \left(x \cdot \left(1 + \left(0.5 \cdot \left(a \cdot x\right) + 0.16666666666666666 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {x}^{2}\right)\right)\right)\right) \]

    13. unpow2 57.2%

        \[\leadsto a \cdot \left(x \cdot \left(1 + \left(0.5 \cdot \left(a \cdot x\right) + 0.16666666666666666 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

    14. swap-sqr 66.0%

        \[\leadsto a \cdot \left(x \cdot \left(1 + \left(0.5 \cdot \left(a \cdot x\right) + 0.16666666666666666 \cdot \color{blue}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)}\right)\right)\right) \]

    15. associate-*r* 66.0%

        \[\leadsto a \cdot \left(x \cdot \left(1 + \left(0.5 \cdot \left(a \cdot x\right) + \color{blue}{\left(0.16666666666666666 \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)}\right)\right)\right) \]

    16. distribute-rgt-out 66.2%

        \[\leadsto a \cdot \left(x \cdot \left(1 + \color{blue}{\left(a \cdot x\right) \cdot \left(0.5 + 0.16666666666666666 \cdot \left(a \cdot x\right)\right)}\right)\right) \]

11. Simplified 66.2%

    \[\leadsto a \cdot \left(x \cdot \left(1 + \color{blue}{\left(a \cdot x\right) \cdot \left(0.5 + 0.16666666666666666 \cdot \left(a \cdot x\right)\right)}\right)\right) \]

12. Final simplification 66.2%

    \[\leadsto a \cdot \left(x \cdot \left(1 + \left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.16666666666666666\right)\right)\right) \]
13. Add Preprocessing

Alternative 3: 67.4% 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 52.3%

   \[e^{a \cdot x} - 1 \]
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 66.0%

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

6. Final simplification 66.0%

   \[\leadsto a \cdot x \]
7. 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 2024072 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :pre (> 710.0 (* a x))

  :alt
  (expm1 (* a x))

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