exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 6

Speedup: 1.0×

• 48.9% 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

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	return pow(exp((x_m + 1.0)), (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)) ** (x_m + (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[N[Exp[N[(x$95$m + 1.0), $MachinePrecision]], $MachinePrecision], 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--1 100.0%

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

   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%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto {\left(e^{x + 1}\right)}^{\left(x + -1\right)} \]
8. 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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Exp[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--1 100.0%

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

   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%

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

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 70.8%

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

   1. exp-1-e 70.8%

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

9. Simplified 70.8%

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

    1. e-exp-1 70.8%

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

    2. pow-exp 70.8%

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

    3. *-un-lft-identity 70.8%

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

    4. expm1-log1p-u 69.5%

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

    5. expm1-udef 69.5%

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

    6. exp-diff 69.5%

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

11. Applied egg-rr 70.8%

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

12. Final simplification 70.8%

    \[\leadsto \frac{e^{x}}{e} \]
13. Add Preprocessing

Alternative 4: 52.4% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(E + N[(x$95$m * E), $MachinePrecision]), $MachinePrecision] / N[(E * E), $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--1 100.0%

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

   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%

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

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 70.8%

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

   1. exp-1-e 70.8%

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

9. Simplified 70.8%

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

10. Taylor expanded in x around 0 50.9%

    \[\leadsto \color{blue}{\frac{1}{e} + \frac{x \cdot \log e}{e}} \]
11. Step-by-step derivation

    1. log-E 50.9%

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

    2. associate-/l* 50.9%

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

12. Simplified 50.9%

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

    1. frac-2neg 50.9%

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

    2. metadata-eval 50.9%

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

    3. frac-2neg 50.9%

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

    4. /-rgt-identity 50.9%

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

    5. frac-add 50.9%

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

    6. metadata-eval 50.9%

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

    7. associate-/r/ 50.9%

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

    8. metadata-eval 50.9%

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

    9. frac-2neg 50.9%

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

    10. remove-double-div 50.9%

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

14. Applied egg-rr 50.9%

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

15. Final simplification 50.9%

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

Alternative 5: 52.4% accurate, 15.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m / E), $MachinePrecision] + N[(1.0 / E), $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--1 100.0%

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

   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%

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

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 70.8%

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

   1. exp-1-e 70.8%

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

9. Simplified 70.8%

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

10. Taylor expanded in x around 0 50.9%

    \[\leadsto \color{blue}{\frac{1}{e} + \frac{x \cdot \log e}{e}} \]
11. Step-by-step derivation

    1. log-E 50.9%

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

    2. associate-/l* 50.9%

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

12. Simplified 50.9%

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

13. Taylor expanded in x around 0 50.9%

    \[\leadsto \color{blue}{\frac{1}{e} + \frac{x}{e}} \]
14. Step-by-step derivation

    1. +-commutative 50.9%

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

15. Simplified 50.9%

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

16. Final simplification 50.9%

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

Alternative 6: 51.8% 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. 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%

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

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 70.8%

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

   1. exp-1-e 70.8%

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

9. Simplified 70.8%

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

10. Taylor expanded in x around 0 51.5%

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

11. Final simplification 51.5%

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

Reproduce

herbie shell --seed 2024023 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))