exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 5

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, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(e^{2 + x \cdot 2}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (exp (+ 2.0 (* x 2.0))) (* (+ x -1.0) 0.5)))

C

double code(double x) {
	return pow(exp((2.0 + (x * 2.0))), ((x + -1.0) * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp((2.0d0 + (x * 2.0d0))) ** ((x + (-1.0d0)) * 0.5d0)
end function

Java

public static double code(double x) {
	return Math.pow(Math.exp((2.0 + (x * 2.0))), ((x + -1.0) * 0.5));
}

Python

def code(x):
	return math.pow(math.exp((2.0 + (x * 2.0))), ((x + -1.0) * 0.5))

Julia

function code(x)
	return exp(Float64(2.0 + Float64(x * 2.0))) ^ Float64(Float64(x + -1.0) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = exp((2.0 + (x * 2.0))) ^ ((x + -1.0) * 0.5);
end

Wolfram

code[x_] := N[Power[N[Exp[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(x + -1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   3. associate--r- 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. sqr-neg 99.9%

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

3. Simplified 99.9%

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

   1. exp-sum 99.9%

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

   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. Step-by-step derivation

   1. add-exp-log 99.9%

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

   2. log-prod 99.9%

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

   3. pow-exp 99.9%

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

   4. add-log-exp 99.9%

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

   5. add-log-exp 99.9%

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

   6. metadata-eval 99.9%

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

   7. sub-neg 99.9%

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

   8. difference-of-sqr-1 99.9%

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

   9. sub-neg 99.9%

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

   10. metadata-eval 99.9%

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

   11. pow-exp 99.9%

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

   12. sqr-pow 99.9%

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

   13. pow-sqr 99.9%

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

   14. pow-pow 99.9%

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

   15. div-inv 99.9%

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

   16. metadata-eval 99.9%

       \[\leadsto {\left({\left(e^{x + 1}\right)}^{2}\right)}^{\left(\left(x + -1\right) \cdot \color{blue}{0.5}\right)} \]

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left({\left(e^{x + 1}\right)}^{2}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)}} \]
8. Step-by-step derivation

   1. add-exp-log 100.0%

      \[\leadsto {\color{blue}{\left(e^{\log \left({\left(e^{x + 1}\right)}^{2}\right)}\right)}}^{\left(\left(x + -1\right) \cdot 0.5\right)} \]

   2. exp-sum 100.0%

      \[\leadsto {\left(e^{\log \left({\color{blue}{\left(e^{x} \cdot e^{1}\right)}}^{2}\right)}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \]

   3. unpow-prod-down 100.0%

      \[\leadsto {\left(e^{\log \color{blue}{\left({\left(e^{x}\right)}^{2} \cdot {\left(e^{1}\right)}^{2}\right)}}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \]

   4. log-prod 100.0%

      \[\leadsto {\left(e^{\color{blue}{\log \left({\left(e^{x}\right)}^{2}\right) + \log \left({\left(e^{1}\right)}^{2}\right)}}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \]

   5. pow-exp 100.0%

      \[\leadsto {\left(e^{\log \color{blue}{\left(e^{x \cdot 2}\right)} + \log \left({\left(e^{1}\right)}^{2}\right)}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \]

   6. add-log-exp 100.0%

      \[\leadsto {\left(e^{\color{blue}{x \cdot 2} + \log \left({\left(e^{1}\right)}^{2}\right)}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \]

   7. pow-exp 100.0%

      \[\leadsto {\left(e^{x \cdot 2 + \log \color{blue}{\left(e^{1 \cdot 2}\right)}}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \]

   8. metadata-eval 100.0%

      \[\leadsto {\left(e^{x \cdot 2 + \log \left(e^{\color{blue}{2}}\right)}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \]

   9. add-log-exp 100.0%

      \[\leadsto {\left(e^{x \cdot 2 + \color{blue}{2}}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \]

9. Applied egg-rr 100.0%

   \[\leadsto {\color{blue}{\left(e^{x \cdot 2 + 2}\right)}}^{\left(\left(x + -1\right) \cdot 0.5\right)} \]

10. Final simplification 100.0%

    \[\leadsto {\left(e^{2 + x \cdot 2}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \]

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   3. associate--r- 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. sqr-neg 99.9%

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

3. Simplified 99.9%

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

   1. difference-of-sqr--1 99.9%

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

   2. exp-prod 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   3. associate--r- 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. sqr-neg 99.9%

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

3. Simplified 99.9%

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

   1. exp-sum 99.9%

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

   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. Step-by-step derivation

   1. add-exp-log 99.9%

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

   2. log-prod 99.9%

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

   3. pow-exp 99.9%

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

   4. add-log-exp 99.9%

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

   5. add-log-exp 99.9%

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

   6. metadata-eval 99.9%

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

   7. sub-neg 99.9%

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

   8. difference-of-sqr-1 99.9%

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

   9. sub-neg 99.9%

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

   10. metadata-eval 99.9%

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

   11. pow-exp 99.9%

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

   12. add-sqr-sqrt 99.2%

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

   13. pow2 99.2%

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

   14. pow-pow 99.2%

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

7. Applied egg-rr 99.2%

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

8. Taylor expanded in x around -inf 99.9%

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

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

Alternative 4: 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 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   3. associate--r- 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. sqr-neg 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 5: 51.4% 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 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   3. associate--r- 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. sqr-neg 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 49.7%

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

5. Final simplification 49.7%

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

Reproduce

herbie shell --seed 2023274 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))