expax (section 3.5)

Percentage Accurate: 54.3% → 100.0%

Time: 8.5s

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: 54.3% 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 54.4%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 N/A

      \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

   3. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, x\right)\right) \]

3. Simplified 100.0%

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

Alternative 2: 70.0% accurate, 9.5× speedup

Math

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

FPCore

(FPCore (a x) :precision binary64 (/ (* a x) (- 1.0 (* a (* x 0.5)))))

C

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

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)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 N/A

      \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

   3. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, x\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in a around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt1-in N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(a \cdot x\right), \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right)}\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot x} + 1\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(1 + \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot x}\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x\right)}\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{+.f64}\left(1, \left(\left(a \cdot \frac{1}{2}\right) \cdot x\right)\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{+.f64}\left(1, \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   18. *-lowering-*.f64 67.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

7. Simplified 67.9%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(1 + a \cdot \left(x \cdot \frac{1}{2}\right)\right) \cdot a\right), \color{blue}{x}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(1 + a \cdot \left(x \cdot \frac{1}{2}\right)\right), a\right), x\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right), a\right), x\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(x \cdot \frac{1}{2}\right)\right)\right), a\right), x\right) \]

   7. *-lowering-*.f64 67.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), a\right), x\right) \]

9. Applied egg-rr 67.9%

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

    1. associate-*l* N/A

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

    2. flip-+ N/A

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

    3. associate-*l/ N/A

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

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right) \cdot \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right) \cdot \left(a \cdot x\right)\right), \color{blue}{\left(1 - a \cdot \left(x \cdot \frac{1}{2}\right)\right)}\right) \]

11. Applied egg-rr 63.8%

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

12. Taylor expanded in a around 0

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(a \cdot x\right)}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right) \]
13. Step-by-step derivation

    1. *-lowering-*.f64 72.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right) \]

14. Simplified 72.1%

    \[\leadsto \frac{\color{blue}{a \cdot x}}{1 - a \cdot \left(x \cdot 0.5\right)} \]
15. Add Preprocessing

Alternative 3: 66.3% 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 54.4%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 N/A

      \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

   3. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, x\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in a around 0

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

   1. *-lowering-*.f64 69.1%

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{x}\right) \]

7. Simplified 69.1%

   \[\leadsto \color{blue}{a \cdot x} \]
8. 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 2024140 
(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))