exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 3.4s

Alternatives: 4

Speedup: 1.0×

• 50.3% 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. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto e^{x \cdot x + -1} \]
4. Add Preprocessing

Alternative 2: 75.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (fma x x 1.0) E))

C

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

Julia

function code(x)
	return Float64(fma(x, x, 1.0) / exp(1))
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 78.0%

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

   1. distribute-rgt1-in 78.0%

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

   2. metadata-eval 78.0%

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

   3. rec-exp 78.0%

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

   4. associate-*r/ 78.0%

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

   5. *-rgt-identity 78.0%

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

   6. unpow2 78.0%

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

   7. fma-define 78.0%

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

   8. exp-1-e 78.0%

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

5. Simplified 78.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{e}} \]
6. Add Preprocessing

Alternative 3: 75.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.005:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.005) (/ 1.0 E) (* x (/ x E))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.005) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = x * (x / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.005) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = x * (x / Math.E);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.005:
		tmp = 1.0 / math.e
	else:
		tmp = x * (x / math.e)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.005)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(x * Float64(x / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.005)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = x * (x / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.005], N[(1.0 / E), $MachinePrecision], N[(x * N[(x / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.0050000000000000001

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.1%

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

      1. distribute-rgt1-in 99.1%

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

      2. metadata-eval 99.1%

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

      3. rec-exp 99.1%

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

      4. associate-*r/ 99.1%

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

      5. *-rgt-identity 99.1%

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

      6. unpow2 99.1%

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

      7. fma-define 99.1%

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

      8. exp-1-e 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0 98.3%

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

   if 0.0050000000000000001 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 52.5%

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

      1. distribute-rgt1-in 52.5%

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

      2. metadata-eval 52.5%

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

      3. rec-exp 52.5%

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

      4. associate-*r/ 52.5%

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

      5. *-rgt-identity 52.5%

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

      6. unpow2 52.5%

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

      7. fma-define 52.5%

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

      8. exp-1-e 52.5%

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

   5. Simplified 52.5%

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

   6. Taylor expanded in x around inf 52.5%

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

      1. unpow2 52.5%

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

      2. associate-/l* 52.5%

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

   8. Applied egg-rr 52.5%

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

Alternative 4: 50.3% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 E))

C

double code(double x) {
	return 1.0 / ((double) M_E);
}

Java

public static double code(double x) {
	return 1.0 / Math.E;
}

Python

def code(x):
	return 1.0 / math.e

Julia

function code(x)
	return Float64(1.0 / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(1.0 / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 78.0%

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

   1. distribute-rgt1-in 78.0%

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

   2. metadata-eval 78.0%

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

   3. rec-exp 78.0%

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

   4. associate-*r/ 78.0%

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

   5. *-rgt-identity 78.0%

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

   6. unpow2 78.0%

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

   7. fma-define 78.0%

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

   8. exp-1-e 78.0%

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

5. Simplified 78.0%

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

6. Taylor expanded in x around 0 55.2%

   \[\leadsto \color{blue}{\frac{1}{e}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))