exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 2.3s

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{{x}^{2}}}{e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp (pow x 2.0)) E))

C

double code(double x) {
	return exp(pow(x, 2.0)) / ((double) M_E);
}

Java

public static double code(double x) {
	return Math.exp(Math.pow(x, 2.0)) / Math.E;
}

Python

def code(x):
	return math.exp(math.pow(x, 2.0)) / math.e

Julia

function code(x)
	return Float64(exp((x ^ 2.0)) / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = exp((x ^ 2.0)) / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / E), $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. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 100.0%

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

   2. exp-prod 100.0%

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

   3. expm1-log1p-u 100.0%

      \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + -1\right)\right)\right)}} \]

   4. expm1-undefine 100.0%

      \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot x + -1\right)} - 1\right)}} \]

   5. pow-sub 100.0%

      \[\leadsto \color{blue}{\frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(x \cdot x + -1\right)}\right)}}{{\left(e^{1}\right)}^{1}}} \]

   6. fma-define 100.0%

      \[\leadsto \frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(x, x, -1\right)}\right)}\right)}}{{\left(e^{1}\right)}^{1}} \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(x, x, \color{blue}{-1}\right)\right)}\right)}}{{\left(e^{1}\right)}^{1}} \]

   8. fma-neg 100.0%

      \[\leadsto \frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(\color{blue}{x \cdot x - 1}\right)}\right)}}{{\left(e^{1}\right)}^{1}} \]

   9. pow1 100.0%

      \[\leadsto \frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(x \cdot x\right)}^{1}} - 1\right)}\right)}}{{\left(e^{1}\right)}^{1}} \]

   10. pow-to-exp 100.0%

       \[\leadsto \frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(\color{blue}{e^{\log \left(x \cdot x\right) \cdot 1}} - 1\right)}\right)}}{{\left(e^{1}\right)}^{1}} \]

   11. expm1-define 100.0%

       \[\leadsto \frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x \cdot x\right) \cdot 1\right)}\right)}\right)}}{{\left(e^{1}\right)}^{1}} \]

   12. log1p-expm1-u 100.0%

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

   13. pow-to-exp 100.0%

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

   14. pow1 100.0%

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

   15. exp-prod 100.0%

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

   16. *-un-lft-identity 100.0%

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

   17. pow2 100.0%

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

   18. pow1 100.0%

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

   19. exp-1-e 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{e}} \]
7. Add Preprocessing

Alternative 2: 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. Add Preprocessing
5. Add Preprocessing

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. Add Preprocessing

5. Taylor expanded in x around 0 49.7%

   \[\leadsto \color{blue}{e^{-1}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))