exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 3

Speedup: 1.0×

• 49.9% 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

\[\begin{array}{l} \\ {\left(e^{2 + 2 \cdot x}\right)}^{\left(2 \cdot \left(0.25 \cdot \left(x + -1\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (exp (+ 2.0 (* 2.0 x))) (* 2.0 (* 0.25 (+ x -1.0)))))

C

double code(double x) {
	return pow(exp((2.0 + (2.0 * x))), (2.0 * (0.25 * (x + -1.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp((2.0d0 + (2.0d0 * x))) ** (2.0d0 * (0.25d0 * (x + (-1.0d0))))
end function

Java

public static double code(double x) {
	return Math.pow(Math.exp((2.0 + (2.0 * x))), (2.0 * (0.25 * (x + -1.0))));
}

Python

def code(x):
	return math.pow(math.exp((2.0 + (2.0 * x))), (2.0 * (0.25 * (x + -1.0))))

Julia

function code(x)
	return exp(Float64(2.0 + Float64(2.0 * x))) ^ Float64(2.0 * Float64(0.25 * Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = exp((2.0 + (2.0 * x))) ^ (2.0 * (0.25 * (x + -1.0)));
end

Wolfram

code[x_] := N[Power[N[Exp[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(2.0 * N[(0.25 * N[(x + -1.0), $MachinePrecision]), $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. 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. sqr-pow 100.0%

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

   2. pow-prod-down 100.0%

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

   3. sqr-pow 100.0%

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

   4. pow2 100.0%

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

   5. associate-/l/ 100.0%

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

   6. metadata-eval 100.0%

      \[\leadsto {\left({\left(e^{x + 1}\right)}^{2}\right)}^{\left(\frac{x + -1}{\color{blue}{4}}\right)} \cdot {\left(e^{x + 1} \cdot e^{x + 1}\right)}^{\left(\frac{\frac{x + -1}{2}}{2}\right)} \]

   7. pow2 100.0%

      \[\leadsto {\left({\left(e^{x + 1}\right)}^{2}\right)}^{\left(\frac{x + -1}{4}\right)} \cdot {\color{blue}{\left({\left(e^{x + 1}\right)}^{2}\right)}}^{\left(\frac{\frac{x + -1}{2}}{2}\right)} \]

   8. associate-/l/ 100.0%

      \[\leadsto {\left({\left(e^{x + 1}\right)}^{2}\right)}^{\left(\frac{x + -1}{4}\right)} \cdot {\left({\left(e^{x + 1}\right)}^{2}\right)}^{\color{blue}{\left(\frac{x + -1}{2 \cdot 2}\right)}} \]

   9. metadata-eval 100.0%

      \[\leadsto {\left({\left(e^{x + 1}\right)}^{2}\right)}^{\left(\frac{x + -1}{4}\right)} \cdot {\left({\left(e^{x + 1}\right)}^{2}\right)}^{\left(\frac{x + -1}{\color{blue}{4}}\right)} \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left({\left(e^{x + 1}\right)}^{2}\right)}^{\left(\frac{x + -1}{4}\right)} \cdot {\left({\left(e^{x + 1}\right)}^{2}\right)}^{\left(\frac{x + -1}{4}\right)}} \]
8. Step-by-step derivation

   1. pow-sqr 100.0%

      \[\leadsto \color{blue}{{\left({\left(e^{x + 1}\right)}^{2}\right)}^{\left(2 \cdot \frac{x + -1}{4}\right)}} \]

   2. unpow2 100.0%

      \[\leadsto {\color{blue}{\left(e^{x + 1} \cdot e^{x + 1}\right)}}^{\left(2 \cdot \frac{x + -1}{4}\right)} \]

   3. prod-exp 100.0%

      \[\leadsto {\color{blue}{\left(e^{\left(x + 1\right) + \left(x + 1\right)}\right)}}^{\left(2 \cdot \frac{x + -1}{4}\right)} \]

   4. count-2 100.0%

      \[\leadsto {\left(e^{\color{blue}{2 \cdot \left(x + 1\right)}}\right)}^{\left(2 \cdot \frac{x + -1}{4}\right)} \]

   5. +-commutative 100.0%

      \[\leadsto {\left(e^{2 \cdot \color{blue}{\left(1 + x\right)}}\right)}^{\left(2 \cdot \frac{x + -1}{4}\right)} \]

   6. distribute-lft-in 100.0%

      \[\leadsto {\left(e^{\color{blue}{2 \cdot 1 + 2 \cdot x}}\right)}^{\left(2 \cdot \frac{x + -1}{4}\right)} \]

   7. metadata-eval 100.0%

      \[\leadsto {\left(e^{\color{blue}{2} + 2 \cdot x}\right)}^{\left(2 \cdot \frac{x + -1}{4}\right)} \]

   8. metadata-eval 100.0%

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

   9. associate-/l* 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-/l* 100.0%

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

   12. associate-/r/ 100.0%

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

   13. metadata-eval 100.0%

       \[\leadsto {\left(e^{2 + 2 \cdot x}\right)}^{\left(2 \cdot \left(\color{blue}{0.25} \cdot \left(x + -1\right)\right)\right)} \]

   14. +-commutative 100.0%

       \[\leadsto {\left(e^{2 + 2 \cdot x}\right)}^{\left(2 \cdot \left(0.25 \cdot \color{blue}{\left(-1 + x\right)}\right)\right)} \]

9. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(e^{2 + 2 \cdot x}\right)}^{\left(2 \cdot \left(0.25 \cdot \left(-1 + x\right)\right)\right)}} \]

10. Final simplification 100.0%

    \[\leadsto {\left(e^{2 + 2 \cdot x}\right)}^{\left(2 \cdot \left(0.25 \cdot \left(x + -1\right)\right)\right)} \]

Alternative 2: 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. 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%

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

Alternative 3: 50.8% 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 52.4%

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

5. Final simplification 52.4%

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

Reproduce

herbie shell --seed 2023301 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))