exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 2.6s

Alternatives: 4

Speedup: 1.0×

• 50.0% 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, 0.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (pow E (fma x x -1.0)))

C

double code(double x) {
	return pow(((double) M_E), fma(x, x, -1.0));
}

Julia

function code(x)
	return exp(1) ^ fma(x, x, -1.0)
end

Wolfram

code[x_] := N[Power[E, N[(x * x + -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. Step-by-step derivation

   1. *-un-lft-identity 100.0%

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

   2. exp-prod 100.0%

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

   3. fma-def 100.0%

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

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\mathsf{fma}\left(x, x, -1\right)\right)}} \]
6. Step-by-step derivation

   1. exp-1-e 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

   \[\leadsto {e}^{\left(\mathsf{fma}\left(x, x, -1\right)\right)} \]

Alternative 2: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.05:\\ \;\;\;\;e^{-1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.05) (exp -1.0) (* x (/ x E))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.05) {
		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) <= 0.05) {
		tmp = Math.exp(-1.0);
	} else {
		tmp = x * (x / Math.E);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.05:
		tmp = math.exp(-1.0)
	else:
		tmp = x * (x / math.e)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.05)
		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) <= 0.05)
		tmp = exp(-1.0);
	else
		tmp = x * (x / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.05], N[Exp[-1.0], $MachinePrecision], N[(x * N[(x / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.050000000000000003

   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 99.1%

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

   if 0.050000000000000003 < (*.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. Taylor expanded in x around 0 54.5%

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

      1. *-rgt-identity 54.5%

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

      2. distribute-lft-in 54.5%

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

      3. unpow2 54.5%

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

      4. fma-def 54.5%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in x around inf 54.5%

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

      1. unpow2 54.5%

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

      2. *-commutative 54.5%

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

      3. associate-*l* 54.5%

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

      4. metadata-eval 54.5%

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

      5. rec-exp 54.5%

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

      6. e-exp-1 54.5%

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

      7. associate-*r/ 54.5%

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

      8. *-rgt-identity 54.5%

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

   9. Simplified 54.5%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.05:\\ \;\;\;\;e^{-1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

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(-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]

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%

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

Alternative 4: 50.7% 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 51.5%

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

5. Final simplification 51.5%

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

Reproduce

herbie shell --seed 2023171 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))