exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 2.2s

Alternatives: 4

Speedup: 1.0×

• 50.4% 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

\[\begin{array}{l} \\ e^{-0.6666666666666666} \cdot \left(e^{-0.3333333333333333} \cdot {\left(e^{x}\right)}^{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (exp -0.6666666666666666) (* (exp -0.3333333333333333) (pow (exp x) x))))

C

double code(double x) {
	return exp(-0.6666666666666666) * (exp(-0.3333333333333333) * pow(exp(x), x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp((-0.6666666666666666d0)) * (exp((-0.3333333333333333d0)) * (exp(x) ** x))
end function

Java

public static double code(double x) {
	return Math.exp(-0.6666666666666666) * (Math.exp(-0.3333333333333333) * Math.pow(Math.exp(x), x));
}

Python

def code(x):
	return math.exp(-0.6666666666666666) * (math.exp(-0.3333333333333333) * math.pow(math.exp(x), x))

Julia

function code(x)
	return Float64(exp(-0.6666666666666666) * Float64(exp(-0.3333333333333333) * (exp(x) ^ x)))
end

MATLAB

function tmp = code(x)
	tmp = exp(-0.6666666666666666) * (exp(-0.3333333333333333) * (exp(x) ^ x));
end

Wolfram

code[x_] := N[(N[Exp[-0.6666666666666666], $MachinePrecision] * N[(N[Exp[-0.3333333333333333], $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], 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. 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. pow-exp 100.0%

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

   2. metadata-eval 100.0%

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

   3. sub-neg 100.0%

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

   4. difference-of-sqr-1 100.0%

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

   5. sub-neg 100.0%

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

   6. add-log-exp 100.0%

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

   7. pow-exp 100.0%

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

   8. metadata-eval 100.0%

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

   9. add-log-exp 100.0%

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

   10. log-prod 100.0%

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

   11. add-exp-log 100.0%

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

   12. *-commutative 100.0%

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

   13. add-cube-cbrt 98.4%

       \[\leadsto \color{blue}{\left(\left(\sqrt[3]{e^{-1}} \cdot \sqrt[3]{e^{-1}}\right) \cdot \sqrt[3]{e^{-1}}\right)} \cdot {\left(e^{x}\right)}^{x} \]

   14. associate-*l* 98.4%

       \[\leadsto \color{blue}{\left(\sqrt[3]{e^{-1}} \cdot \sqrt[3]{e^{-1}}\right) \cdot \left(\sqrt[3]{e^{-1}} \cdot {\left(e^{x}\right)}^{x}\right)} \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{-0.6666666666666666} \cdot \left(e^{-0.3333333333333333} \cdot {\left(e^{x}\right)}^{x}\right)} \]

8. Final simplification 100.0%

   \[\leadsto e^{-0.6666666666666666} \cdot \left(e^{-0.3333333333333333} \cdot {\left(e^{x}\right)}^{x}\right) \]

Alternative 2: 100.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[Exp[-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. 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. Final simplification 100.0%

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

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: 50.4% 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 51.4%

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

5. Final simplification 51.4%

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

Reproduce

herbie shell --seed 2023230 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))