exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 2.4s

Alternatives: 3

Speedup: 1.0×

• 49.5% 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. sqr-neg 100.0%

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

   3. associate--r- 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.94:\\ \;\;\;\;e^{-1}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.94) (exp -1.0) (exp (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.94) {
		tmp = exp(-1.0);
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 0.94d0) then
        tmp = exp((-1.0d0))
    else
        tmp = exp((x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.94) {
		tmp = Math.exp(-1.0);
	} else {
		tmp = Math.exp((x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.94:
		tmp = math.exp(-1.0)
	else:
		tmp = math.exp((x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.94)
		tmp = exp(-1.0);
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.94)
		tmp = exp(-1.0);
	else
		tmp = exp((x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.94], N[Exp[-1.0], $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.93999999999999995

   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. sqr-neg 100.0%

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

      3. associate--r- 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 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 97.9%

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

   if 0.93999999999999995 < (*.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. sqr-neg 100.0%

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

      3. associate--r- 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 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. difference-of-sqr--1 100.0%

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

      2. exp-prod 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow-prod-up 50.0%

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

      3. add-exp-log 50.0%

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

      4. pow-exp 50.0%

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

      5. prod-exp 50.5%

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

      6. exp-sum 50.5%

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

      7. unpow-prod-down 50.5%

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

      8. log-prod 50.5%

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

      9. unpow-1 50.5%

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

      10. neg-log 50.5%

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

      11. add-log-exp 100.0%

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

      12. pow-exp 100.0%

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

      13. metadata-eval 100.0%

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

      14. add-log-exp 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. unpow2 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.94:\\ \;\;\;\;e^{-1}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \]

Alternative 3: 51.3% 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. sqr-neg 100.0%

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

   3. associate--r- 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 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 51.2%

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

5. Final simplification 51.2%

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

Reproduce

herbie shell --seed 2023290 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))