exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 3.3s

Alternatives: 3

Speedup: 1.0×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{-\left(1 - x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))

C

double code(double x) {
	return exp(-(1.0 - (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(-(1.0d0 - (x * x)))
end function

Java

public static double code(double x) {
	return Math.exp(-(1.0 - (x * x)));
}

Python

def code(x):
	return math.exp(-(1.0 - (x * x)))

Julia

function code(x)
	return exp(Float64(-Float64(1.0 - Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = exp(-(1.0 - (x * x)));
end

Wolfram

code[x_] := N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-\left(1 - x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))

C

double code(double x) {
	return exp(-(1.0 - (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(-(1.0d0 - (x * x)))
end function

Java

public static double code(double x) {
	return Math.exp(-(1.0 - (x * x)));
}

Python

def code(x):
	return math.exp(-(1.0 - (x * x)))

Julia

function code(x)
	return exp(Float64(-Float64(1.0 - Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = exp(-(1.0 - (x * x)));
end

Wolfram

code[x_] := N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x + -1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (+ (* x x) -1.0)))

C

double code(double x) {
	return exp(((x * x) + -1.0));
}

Fortran

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

Java

public static double code(double x) {
	return Math.exp(((x * x) + -1.0));
}

Python

def code(x):
	return math.exp(((x * x) + -1.0))

Julia

function code(x)
	return exp(Float64(Float64(x * x) + -1.0))
end

MATLAB

function tmp = code(x)
	tmp = exp(((x * x) + -1.0));
end

Wolfram

code[x_] := N[Exp[N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. neg-sub0 100.0%

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   2. associate--r- 100.0%

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) + x \cdot x}} \]

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto e^{x \cdot x + -1} \]

Alternative 2: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1:\\ \;\;\;\;e^{-1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 1.0) (exp -1.0) (* x (/ x E))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 1.0) {
		tmp = exp(-1.0);
	} else {
		tmp = x * (x / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 1.0) {
		tmp = Math.exp(-1.0);
	} else {
		tmp = x * (x / Math.E);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 1.0:
		tmp = math.exp(-1.0)
	else:
		tmp = x * (x / math.e)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 1.0)
		tmp = exp(-1.0);
	else
		tmp = Float64(x * Float64(x / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 1.0)
		tmp = exp(-1.0);
	else
		tmp = x * (x / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.0], N[Exp[-1.0], $MachinePrecision], N[(x * N[(x / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

      2. associate--r- 100.0%

         \[\leadsto e^{\color{blue}{\left(0 - 1\right) + x \cdot x}} \]

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.5%

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

   if 1 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

      2. associate--r- 100.0%

         \[\leadsto e^{\color{blue}{\left(0 - 1\right) + x \cdot x}} \]

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Step-by-step derivation

      1. exp-sum 100.0%

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

      2. exp-prod 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 54.4%

      \[\leadsto \color{blue}{e^{-1} \cdot {x}^{2} + e^{-1}} \]
   7. Step-by-step derivation

      1. *-commutative 54.4%

         \[\leadsto \color{blue}{{x}^{2} \cdot e^{-1}} + e^{-1} \]

      2. distribute-lft1-in 54.4%

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

      3. metadata-eval 54.4%

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

      4. rec-exp 54.4%

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

      5. e-exp-1 54.4%

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

      6. associate-*r/ 54.4%

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

      7. distribute-lft1-in 54.4%

         \[\leadsto \frac{\color{blue}{{x}^{2} \cdot 1 + 1}}{e} \]

      8. *-rgt-identity 54.4%

         \[\leadsto \frac{\color{blue}{{x}^{2}} + 1}{e} \]

      9. unpow2 54.4%

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

   8. Simplified 54.4%

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

   9. Taylor expanded in x around inf 54.4%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{e}} \]
   10. Step-by-step derivation

       1. unpow2 54.4%

          \[\leadsto \frac{\color{blue}{x \cdot x}}{e} \]

       2. *-rgt-identity 54.4%

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

       3. associate-*r/ 54.4%

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

       4. e-exp-1 54.4%

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

       5. rec-exp 54.4%

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

       6. metadata-eval 54.4%

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

       7. associate-*l* 54.4%

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

       8. metadata-eval 54.4%

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

       9. rec-exp 54.4%

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

       10. e-exp-1 54.4%

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

       11. associate-*r/ 54.4%

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

       12. *-rgt-identity 54.4%

           \[\leadsto x \cdot \frac{\color{blue}{x}}{e} \]

   11. Simplified 54.4%

       \[\leadsto \color{blue}{x \cdot \frac{x}{e}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1:\\ \;\;\;\;e^{-1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \end{array} \]

Alternative 3: 51.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp -1.0))

C

double code(double x) {
	return exp(-1.0);
}

Fortran

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

Java

public static double code(double x) {
	return Math.exp(-1.0);
}

Python

def code(x):
	return math.exp(-1.0)

Julia

function code(x)
	return exp(-1.0)
end

MATLAB

function tmp = code(x)
	tmp = exp(-1.0);
end

Wolfram

code[x_] := N[Exp[-1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. neg-sub0 100.0%

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   2. associate--r- 100.0%

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) + x \cdot x}} \]

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 49.0%

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

5. Final simplification 49.0%

   \[\leadsto e^{-1} \]

Reproduce

herbie shell --seed 2023174 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))