exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 5

Speedup: 1.0×

• 49.4% 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, 0.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (pow (exp (* 2.0 (+ 1.0 x_m))) (* 0.5 (+ x_m -1.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[N[Exp[N[(2.0 * N[(1.0 + x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. sqr-neg 100.0%

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

   2. neg-sub0 100.0%

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

   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. flip-+ 76.1%

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

   4. metadata-eval 76.1%

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

   5. div-sub 76.1%

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

   6. pow-sub 76.1%

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

   7. exp-1-e 76.1%

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

   8. pow2 76.1%

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

   9. pow2 76.1%

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

   10. pow-prod-up 76.2%

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

   11. metadata-eval 76.2%

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

   12. fma-neg 76.2%

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

   13. metadata-eval 76.2%

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

   14. exp-1-e 76.2%

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

   15. fma-neg 76.2%

       \[\leadsto \frac{{e}^{\left(\frac{{x}^{4}}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{{e}^{\left(\frac{1}{\color{blue}{\mathsf{fma}\left(x, x, --1\right)}}\right)}} \]

   16. metadata-eval 76.2%

       \[\leadsto \frac{{e}^{\left(\frac{{x}^{4}}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{{e}^{\left(\frac{1}{\mathsf{fma}\left(x, x, \color{blue}{1}\right)}\right)}} \]

6. Applied egg-rr 76.2%

   \[\leadsto \color{blue}{\frac{{e}^{\left(\frac{{x}^{4}}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{{e}^{\left(\frac{1}{\mathsf{fma}\left(x, x, 1\right)}\right)}}} \]
7. Step-by-step derivation

   1. pow-div 76.1%

      \[\leadsto \color{blue}{{e}^{\left(\frac{{x}^{4}}{\mathsf{fma}\left(x, x, 1\right)} - \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\right)}} \]

   2. pow-to-exp 76.1%

      \[\leadsto \color{blue}{e^{\log e \cdot \left(\frac{{x}^{4}}{\mathsf{fma}\left(x, x, 1\right)} - \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\right)}} \]

   3. log-E 76.1%

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

   4. *-un-lft-identity 76.1%

      \[\leadsto e^{\color{blue}{\frac{{x}^{4}}{\mathsf{fma}\left(x, x, 1\right)} - \frac{1}{\mathsf{fma}\left(x, x, 1\right)}}} \]

   5. sub-div 76.1%

      \[\leadsto e^{\color{blue}{\frac{{x}^{4} - 1}{\mathsf{fma}\left(x, x, 1\right)}}} \]

   6. metadata-eval 76.1%

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

   7. pow-prod-up 76.1%

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

   8. pow-prod-down 76.1%

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

   9. pow2 76.1%

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

   10. metadata-eval 76.1%

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

   11. fma-udef 76.1%

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

   12. flip-- 100.0%

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

   13. difference-of-sqr-1 99.9%

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

   14. sub-neg 99.9%

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

   15. metadata-eval 99.9%

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

   16. pow-exp 100.0%

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

   17. exp-sum 100.0%

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

   18. e-exp-1 100.0%

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

   19. unpow-prod-down 76.5%

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

8. Applied egg-rr 76.6%

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

   1. *-un-lft-identity 76.6%

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

   2. pow-exp 76.5%

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

   3. e-exp-1 76.5%

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

   4. sqr-pow 76.5%

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

   5. sqr-pow 76.5%

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

   6. pow-prod-down 76.5%

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

   7. pow-prod-down 76.5%

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

   8. pow-prod-down 100.0%

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

10. Applied egg-rr 100.0%

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

    1. pow-plus 100.0%

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

    2. *-commutative 100.0%

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

    3. +-commutative 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around inf 100.0%

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

14. Final simplification 100.0%

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

Alternative 2: 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. sqr-neg 100.0%

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

   2. neg-sub0 100.0%

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

   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 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sqr-neg 100.0%

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

   2. neg-sub0 100.0%

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

   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. difference-of-sqr--1 99.9%

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

   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%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 75.3%

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

   1. exp-1-e 75.3%

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

9. Simplified 75.3%

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

    1. e-exp-1 75.3%

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

    2. pow-exp 75.3%

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

    3. *-un-lft-identity 75.3%

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

11. Applied egg-rr 75.3%

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

12. Final simplification 75.3%

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

Alternative 4: 52.0% accurate, 21.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 + 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. sqr-neg 100.0%

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

   2. neg-sub0 100.0%

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

   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. difference-of-sqr--1 99.9%

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

   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%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 75.3%

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

   1. exp-1-e 75.3%

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

9. Simplified 75.3%

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

    1. e-exp-1 75.3%

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

    2. pow-exp 75.3%

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

    3. *-un-lft-identity 75.3%

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

11. Applied egg-rr 75.3%

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

12. Taylor expanded in x around 0 52.0%

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

    1. distribute-rgt1-in 52.0%

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

    2. metadata-eval 52.0%

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

    3. rec-exp 52.0%

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

    4. e-exp-1 52.0%

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

    5. associate-*r/ 52.0%

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

    6. distribute-rgt1-in 52.0%

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

    7. *-rgt-identity 52.0%

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

14. Simplified 52.0%

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

15. Final simplification 52.0%

    \[\leadsto \frac{1 + x}{e} \]
16. Add Preprocessing

Alternative 5: 51.4% 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. sqr-neg 100.0%

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

   2. neg-sub0 100.0%

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

   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. flip-+ 76.1%

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

   4. metadata-eval 76.1%

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

   5. div-sub 76.1%

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

   6. pow-sub 76.1%

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

   7. exp-1-e 76.1%

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

   8. pow2 76.1%

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

   9. pow2 76.1%

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

   10. pow-prod-up 76.2%

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

   11. metadata-eval 76.2%

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

   12. fma-neg 76.2%

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

   13. metadata-eval 76.2%

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

   14. exp-1-e 76.2%

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

   15. fma-neg 76.2%

       \[\leadsto \frac{{e}^{\left(\frac{{x}^{4}}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{{e}^{\left(\frac{1}{\color{blue}{\mathsf{fma}\left(x, x, --1\right)}}\right)}} \]

   16. metadata-eval 76.2%

       \[\leadsto \frac{{e}^{\left(\frac{{x}^{4}}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{{e}^{\left(\frac{1}{\mathsf{fma}\left(x, x, \color{blue}{1}\right)}\right)}} \]

6. Applied egg-rr 76.2%

   \[\leadsto \color{blue}{\frac{{e}^{\left(\frac{{x}^{4}}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{{e}^{\left(\frac{1}{\mathsf{fma}\left(x, x, 1\right)}\right)}}} \]

7. Taylor expanded in x around 0 52.6%

   \[\leadsto \color{blue}{\frac{1}{e}} \]

8. Final simplification 52.6%

   \[\leadsto \frac{1}{e} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))