exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 5

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, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{-1 + x\_m}\\ {t\_0}^{\left(x\_m + 0.5\right)} \cdot \sqrt{t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (exp (+ -1.0 x_m)))) (* (pow t_0 (+ x_m 0.5)) (sqrt t_0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = exp((-1.0 + x_m));
	return pow(t_0, (x_m + 0.5)) * sqrt(t_0);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    t_0 = exp(((-1.0d0) + x_m))
    code = (t_0 ** (x_m + 0.5d0)) * sqrt(t_0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.exp((-1.0 + x_m));
	return Math.pow(t_0, (x_m + 0.5)) * Math.sqrt(t_0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.exp((-1.0 + x_m))
	return math.pow(t_0, (x_m + 0.5)) * math.sqrt(t_0)

Julia

x_m = abs(x)
function code(x_m)
	t_0 = exp(Float64(-1.0 + x_m))
	return Float64((t_0 ^ Float64(x_m + 0.5)) * sqrt(t_0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	t_0 = exp((-1.0 + x_m));
	tmp = (t_0 ^ (x_m + 0.5)) * sqrt(t_0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Exp[N[(-1.0 + x$95$m), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[t$95$0, N[(x$95$m + 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$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. 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 99.9%

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

   3. expm1-log1p-u 99.9%

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

   4. expm1-undefine 99.9%

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

   5. pow-sub 99.9%

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

   6. fma-define 99.9%

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

   7. metadata-eval 99.9%

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

   8. fma-neg 99.9%

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

   9. pow1 99.9%

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

   10. pow-to-exp 99.9%

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

   11. expm1-define 99.9%

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

   12. log1p-expm1-u 99.9%

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

   13. pow-to-exp 99.9%

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

   14. pow1 99.9%

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

   15. exp-prod 99.9%

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

   16. *-un-lft-identity 99.9%

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

   17. pow2 99.9%

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

   18. pow1 99.9%

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

   19. exp-1-e 99.9%

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

6. Applied egg-rr 99.9%

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

   1. e-exp-1 99.9%

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

   2. div-exp 100.0%

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

   3. unpow2 100.0%

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

   4. difference-of-sqr-1 99.9%

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

   5. sub-neg 99.9%

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

   6. metadata-eval 99.9%

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

   7. add-log-exp 99.9%

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

   8. log-pow 99.9%

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

   9. add-exp-log 100.0%

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

   10. pow-plus 76.9%

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

   11. add-sqr-sqrt 76.9%

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

   12. associate-*r* 76.9%

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

   13. +-commutative 76.9%

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

   14. +-commutative 76.9%

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

   15. +-commutative 76.9%

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

8. Applied egg-rr 76.9%

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

   1. add-log-exp 76.5%

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

   2. *-un-lft-identity 76.5%

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

   3. log-prod 76.5%

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

   4. metadata-eval 76.5%

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

   5. add-log-exp 76.9%

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

   6. pow1/2 76.9%

      \[\leadsto \left(0 + {\left(e^{-1 + x}\right)}^{x} \cdot \color{blue}{{\left(e^{-1 + x}\right)}^{0.5}}\right) \cdot \sqrt{e^{-1 + x}} \]

   7. pow-prod-up 76.9%

      \[\leadsto \left(0 + \color{blue}{{\left(e^{-1 + x}\right)}^{\left(x + 0.5\right)}}\right) \cdot \sqrt{e^{-1 + x}} \]

10. Applied egg-rr 76.9%

    \[\leadsto \color{blue}{\left(0 + {\left(e^{-1 + x}\right)}^{\left(x + 0.5\right)}\right)} \cdot \sqrt{e^{-1 + x}} \]
11. Step-by-step derivation

    1. +-lft-identity 76.9%

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

12. Simplified 76.9%

    \[\leadsto \color{blue}{{\left(e^{-1 + x}\right)}^{\left(x + 0.5\right)}} \cdot \sqrt{e^{-1 + x}} \]
13. Add Preprocessing

Alternative 2: 100.0% accurate, 0.3× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (pow (exp (+ -1.0 x_m)) (+ x_m 0.5)) (exp (* (+ -1.0 x_m) 0.5))))

C

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

Fortran

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

Java

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

Python

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

Julia

x_m = abs(x)
function code(x_m)
	return Float64((exp(Float64(-1.0 + x_m)) ^ Float64(x_m + 0.5)) * exp(Float64(Float64(-1.0 + x_m) * 0.5)))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[N[Exp[N[(-1.0 + x$95$m), $MachinePrecision]], $MachinePrecision], N[(x$95$m + 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(-1.0 + x$95$m), $MachinePrecision] * 0.5), $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. 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 99.9%

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

   3. expm1-log1p-u 99.9%

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

   4. expm1-undefine 99.9%

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

   5. pow-sub 99.9%

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

   6. fma-define 99.9%

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

   7. metadata-eval 99.9%

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

   8. fma-neg 99.9%

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

   9. pow1 99.9%

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

   10. pow-to-exp 99.9%

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

   11. expm1-define 99.9%

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

   12. log1p-expm1-u 99.9%

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

   13. pow-to-exp 99.9%

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

   14. pow1 99.9%

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

   15. exp-prod 99.9%

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

   16. *-un-lft-identity 99.9%

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

   17. pow2 99.9%

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

   18. pow1 99.9%

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

   19. exp-1-e 99.9%

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

6. Applied egg-rr 99.9%

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

   1. e-exp-1 99.9%

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

   2. div-exp 100.0%

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

   3. unpow2 100.0%

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

   4. difference-of-sqr-1 99.9%

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

   5. sub-neg 99.9%

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

   6. metadata-eval 99.9%

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

   7. add-log-exp 99.9%

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

   8. log-pow 99.9%

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

   9. add-exp-log 100.0%

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

   10. pow-plus 76.9%

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

   11. add-sqr-sqrt 76.9%

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

   12. associate-*r* 76.9%

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

   13. +-commutative 76.9%

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

   14. +-commutative 76.9%

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

   15. +-commutative 76.9%

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

8. Applied egg-rr 76.9%

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

   1. add-log-exp 76.5%

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

   2. *-un-lft-identity 76.5%

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

   3. log-prod 76.5%

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

   4. metadata-eval 76.5%

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

   5. add-log-exp 76.9%

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

   6. pow1/2 76.9%

      \[\leadsto \left(0 + {\left(e^{-1 + x}\right)}^{x} \cdot \color{blue}{{\left(e^{-1 + x}\right)}^{0.5}}\right) \cdot \sqrt{e^{-1 + x}} \]

   7. pow-prod-up 76.9%

      \[\leadsto \left(0 + \color{blue}{{\left(e^{-1 + x}\right)}^{\left(x + 0.5\right)}}\right) \cdot \sqrt{e^{-1 + x}} \]

10. Applied egg-rr 76.9%

    \[\leadsto \color{blue}{\left(0 + {\left(e^{-1 + x}\right)}^{\left(x + 0.5\right)}\right)} \cdot \sqrt{e^{-1 + x}} \]
11. Step-by-step derivation

    1. +-lft-identity 76.9%

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

12. Simplified 76.9%

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

    1. pow1/2 76.9%

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

    2. pow-exp 76.9%

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

14. Applied egg-rr 76.9%

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

Alternative 3: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{{\left(e^{x\_m}\right)}^{x\_m}}{e} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (pow (exp x_m) x_m) E))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(exp(x_m), x_m) / ((double) M_E);
}

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.exp(x_m), x_m) / math.e

Julia

x_m = abs(x)
function code(x_m)
	return Float64((exp(x_m) ^ x_m) / exp(1))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[N[Exp[x$95$m], $MachinePrecision], x$95$m], $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 99.9%

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

   3. expm1-log1p-u 99.9%

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

   4. expm1-undefine 99.9%

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

   5. pow-sub 99.9%

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

   6. fma-define 99.9%

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

   7. metadata-eval 99.9%

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

   8. fma-neg 99.9%

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

   9. pow1 99.9%

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

   10. pow-to-exp 99.9%

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

   11. expm1-define 99.9%

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

   12. log1p-expm1-u 99.9%

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

   13. pow-to-exp 99.9%

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

   14. pow1 99.9%

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

   15. exp-prod 99.9%

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

   16. *-un-lft-identity 99.9%

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

   17. pow2 99.9%

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

   18. pow1 99.9%

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

   19. exp-1-e 99.9%

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

6. Applied egg-rr 99.9%

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

   1. unpow2 99.9%

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

   2. exp-prod 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e} \]
9. Add Preprocessing

Alternative 4: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ e^{-1 + x\_m \cdot x\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (exp (+ -1.0 (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	return exp((-1.0 + (x_m * x_m)));
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.exp((-1.0 + (x_m * x_m)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.exp((-1.0 + (x_m * x_m)))

Julia

x_m = abs(x)
function code(x_m)
	return exp(Float64(-1.0 + Float64(x_m * x_m)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = exp((-1.0 + (x_m * x_m)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Exp[N[(-1.0 + N[(x$95$m * x$95$m), $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. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto e^{-1 + x \cdot x} \]
6. Add Preprocessing

Alternative 5: 50.7% accurate, 35.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{e} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ 1.0 E))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 / ((double) M_E);
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0 / Math.E;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0 / math.e

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 / exp(1))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0 / 2.71828182845904523536;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / 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 99.9%

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

   3. expm1-log1p-u 99.9%

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

   4. expm1-undefine 99.9%

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

   5. pow-sub 99.9%

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

   6. fma-define 99.9%

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

   7. metadata-eval 99.9%

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

   8. fma-neg 99.9%

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

   9. pow1 99.9%

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

   10. pow-to-exp 99.9%

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

   11. expm1-define 99.9%

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

   12. log1p-expm1-u 99.9%

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

   13. pow-to-exp 99.9%

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

   14. pow1 99.9%

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

   15. exp-prod 99.9%

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

   16. *-un-lft-identity 99.9%

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

   17. pow2 99.9%

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

   18. pow1 99.9%

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

   19. exp-1-e 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 54.5%

   \[\leadsto \frac{\color{blue}{1}}{e} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024128 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))