exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 7

Speedup: 1.0×

• 49.6% 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. 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. 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. 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 3: 98.7% 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. 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. 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 72.4%

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

   1. exp-1-e 72.4%

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

9. Simplified 72.4%

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

    1. metadata-eval 72.4%

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

    2. sub-neg 72.4%

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

    3. pow-sub 72.4%

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

    4. pow-to-exp 72.4%

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

    5. log-E 72.4%

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

    6. *-un-lft-identity 72.4%

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

    7. pow1 72.4%

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

11. Applied egg-rr 72.4%

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

12. Final simplification 72.4%

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

Alternative 4: 83.4% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{e} + x\_m \cdot \left(\frac{1}{e} + x\_m \cdot \left(0.16666666666666666 \cdot \frac{x\_m}{e} + 0.5 \cdot \frac{1}{e}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (+
  (/ 1.0 E)
  (*
   x_m
   (+
    (/ 1.0 E)
    (* x_m (+ (* 0.16666666666666666 (/ x_m E)) (* 0.5 (/ 1.0 E))))))))

C

x_m = fabs(x);
double code(double x_m) {
	return (1.0 / ((double) M_E)) + (x_m * ((1.0 / ((double) M_E)) + (x_m * ((0.16666666666666666 * (x_m / ((double) M_E))) + (0.5 * (1.0 / ((double) M_E)))))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (1.0 / Math.E) + (x_m * ((1.0 / Math.E) + (x_m * ((0.16666666666666666 * (x_m / Math.E)) + (0.5 * (1.0 / Math.E))))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (1.0 / math.e) + (x_m * ((1.0 / math.e) + (x_m * ((0.16666666666666666 * (x_m / math.e)) + (0.5 * (1.0 / math.e))))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(1.0 / exp(1)) + Float64(x_m * Float64(Float64(1.0 / exp(1)) + Float64(x_m * Float64(Float64(0.16666666666666666 * Float64(x_m / exp(1))) + Float64(0.5 * Float64(1.0 / exp(1))))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (1.0 / 2.71828182845904523536) + (x_m * ((1.0 / 2.71828182845904523536) + (x_m * ((0.16666666666666666 * (x_m / 2.71828182845904523536)) + (0.5 * (1.0 / 2.71828182845904523536))))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 / E), $MachinePrecision] + N[(x$95$m * N[(N[(1.0 / E), $MachinePrecision] + N[(x$95$m * N[(N[(0.16666666666666666 * N[(x$95$m / E), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. 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 72.4%

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

   1. exp-1-e 72.4%

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

9. Simplified 72.4%

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

    1. metadata-eval 72.4%

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

    2. sub-neg 72.4%

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

    3. pow-sub 72.4%

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

    4. pow-to-exp 72.4%

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

    5. log-E 72.4%

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

    6. *-un-lft-identity 72.4%

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

    7. pow1 72.4%

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

11. Applied egg-rr 72.4%

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

12. Taylor expanded in x around 0 63.5%

    \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \frac{x}{e} + 0.5 \cdot \frac{1}{e}\right) + \frac{1}{e}\right) + \frac{1}{e}} \]

13. Final simplification 63.5%

    \[\leadsto \frac{1}{e} + x \cdot \left(\frac{1}{e} + x \cdot \left(0.16666666666666666 \cdot \frac{x}{e} + 0.5 \cdot \frac{1}{e}\right)\right) \]
14. Add Preprocessing

Alternative 5: 75.8% accurate, 7.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (+ (/ 1.0 E) (* x_m (+ (/ 1.0 E) (* (/ x_m E) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	return (1.0 / ((double) M_E)) + (x_m * ((1.0 / ((double) M_E)) + ((x_m / ((double) M_E)) * 0.5)));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (1.0 / Math.E) + (x_m * ((1.0 / Math.E) + ((x_m / Math.E) * 0.5)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (1.0 / math.e) + (x_m * ((1.0 / math.e) + ((x_m / math.e) * 0.5)))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(1.0 / exp(1)) + Float64(x_m * Float64(Float64(1.0 / exp(1)) + Float64(Float64(x_m / exp(1)) * 0.5))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (1.0 / 2.71828182845904523536) + (x_m * ((1.0 / 2.71828182845904523536) + ((x_m / 2.71828182845904523536) * 0.5)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 / E), $MachinePrecision] + N[(x$95$m * N[(N[(1.0 / E), $MachinePrecision] + N[(N[(x$95$m / E), $MachinePrecision] * 0.5), $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. 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 72.4%

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

   1. exp-1-e 72.4%

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

9. Simplified 72.4%

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

    1. metadata-eval 72.4%

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

    2. sub-neg 72.4%

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

    3. pow-sub 72.4%

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

    4. pow-to-exp 72.4%

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

    5. log-E 72.4%

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

    6. *-un-lft-identity 72.4%

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

    7. pow1 72.4%

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

11. Applied egg-rr 72.4%

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

12. Taylor expanded in x around 0 72.6%

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

13. Final simplification 72.6%

    \[\leadsto \frac{1}{e} + x \cdot \left(\frac{1}{e} + \frac{x}{e} \cdot 0.5\right) \]
14. Add Preprocessing

Alternative 6: 52.1% accurate, 21.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m + 1.0), $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. 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 72.4%

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

   1. exp-1-e 72.4%

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

9. Simplified 72.4%

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

10. Taylor expanded in x around 0 47.3%

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

    1. e-exp-1 47.3%

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

    2. rec-exp 47.3%

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

    3. metadata-eval 47.3%

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

    4. log-E 47.3%

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

    5. metadata-eval 47.3%

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

    6. log-E 47.3%

       \[\leadsto e^{-1} + \frac{x \cdot {\color{blue}{\log e}}^{2}}{e} \]

    7. associate-/l* 47.3%

       \[\leadsto e^{-1} + \color{blue}{x \cdot \frac{{\log e}^{2}}{e}} \]

    8. log-E 47.3%

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

    9. metadata-eval 47.3%

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

    10. e-exp-1 47.3%

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

    11. rec-exp 47.3%

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

    12. metadata-eval 47.3%

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

    13. distribute-rgt1-in 47.3%

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

    14. +-commutative 47.3%

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

    15. metadata-eval 47.3%

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

    16. rec-exp 47.3%

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

    17. e-exp-1 47.3%

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

    18. associate-*r/ 47.3%

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

    19. *-rgt-identity 47.3%

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

    20. +-commutative 47.3%

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

12. Simplified 47.3%

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

13. Final simplification 47.3%

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

Alternative 7: 51.5% 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. 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 72.4%

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

   1. exp-1-e 72.4%

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

9. Simplified 72.4%

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

10. Taylor expanded in x around 0 48.0%

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

11. Final simplification 48.0%

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

Reproduce

herbie shell --seed 2024085 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))