exp neg sub

Percentage Accurate: 100.0% → 99.9%

Time: 5.6s

Alternatives: 11

Speedup: 1.0×

• 50.3% 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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(e^{3 \cdot \mathsf{fma}\left(x, x, -1\right)}\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (exp (* 3.0 (fma x x -1.0))) 0.3333333333333333))

C

double code(double x) {
	return pow(exp((3.0 * fma(x, x, -1.0))), 0.3333333333333333);
}

Julia

function code(x)
	return exp(Float64(3.0 * fma(x, x, -1.0))) ^ 0.3333333333333333
end

Wolfram

code[x_] := N[Power[N[Exp[N[(3.0 * N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.3333333333333333], $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. add-cbrt-cube 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(e^{x \cdot x + -1} \cdot e^{x \cdot x + -1}\right) \cdot e^{x \cdot x + -1}}} \]

   2. pow1/3 99.2%

      \[\leadsto \color{blue}{{\left(\left(e^{x \cdot x + -1} \cdot e^{x \cdot x + -1}\right) \cdot e^{x \cdot x + -1}\right)}^{0.3333333333333333}} \]

   3. add-exp-log 100.0%

      \[\leadsto {\color{blue}{\left(e^{\log \left(\left(e^{x \cdot x + -1} \cdot e^{x \cdot x + -1}\right) \cdot e^{x \cdot x + -1}\right)}\right)}}^{0.3333333333333333} \]

   4. pow3 100.0%

      \[\leadsto {\left(e^{\log \color{blue}{\left({\left(e^{x \cdot x + -1}\right)}^{3}\right)}}\right)}^{0.3333333333333333} \]

   5. log-pow 100.0%

      \[\leadsto {\left(e^{\color{blue}{3 \cdot \log \left(e^{x \cdot x + -1}\right)}}\right)}^{0.3333333333333333} \]

   6. add-log-exp 100.0%

      \[\leadsto {\left(e^{3 \cdot \color{blue}{\left(x \cdot x + -1\right)}}\right)}^{0.3333333333333333} \]

   7. fma-define 100.0%

      \[\leadsto {\left(e^{3 \cdot \color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{0.3333333333333333} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(e^{3 \cdot \mathsf{fma}\left(x, x, -1\right)}\right)}^{0.3333333333333333}} \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (exp (* (+ x -1.0) (+ x 1.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Exp[N[(N[(x + -1.0), $MachinePrecision] * N[(x + 1.0), $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. sub-neg 100.0%

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

   3. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-1 + x \cdot x} \end{array} \]

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(-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]

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 4: 74.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (pow E (+ x -1.0)))

C

double code(double x) {
	return pow(((double) M_E), (x + -1.0));
}

Java

public static double code(double x) {
	return Math.pow(Math.E, (x + -1.0));
}

Python

def code(x):
	return math.pow(math.e, (x + -1.0))

Julia

function code(x)
	return exp(1) ^ Float64(x + -1.0)
end

MATLAB

function tmp = code(x)
	tmp = 2.71828182845904523536 ^ (x + -1.0);
end

Wolfram

code[x_] := N[Power[E, N[(x + -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. Taylor expanded in x around 0 74.9%

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

   1. exp-1-e 74.9%

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

9. Simplified 74.9%

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

Alternative 5: 74.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Exp[x], $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 74.9%

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

   1. exp-1-e 74.9%

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

9. Simplified 74.9%

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

    1. e-exp-1 74.9%

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

    2. pow-exp 74.9%

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

    3. *-un-lft-identity 74.9%

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

    4. expm1-log1p-u 74.0%

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

    5. expm1-undefine 74.0%

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

    6. exp-diff 74.0%

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

11. Applied egg-rr 74.9%

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

Alternative 6: 66.1% accurate, 4.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (+
  (/ 1.0 E)
  (*
   x
   (+ (/ 1.0 E) (* x (+ (* 0.16666666666666666 (/ x E)) (* 0.5 (/ 1.0 E))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / E), $MachinePrecision] + N[(x * N[(N[(1.0 / E), $MachinePrecision] + N[(x * N[(N[(0.16666666666666666 * N[(x / 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 74.9%

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

   1. exp-1-e 74.9%

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

9. Simplified 74.9%

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

    1. e-exp-1 74.9%

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

    2. pow-exp 74.9%

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

    3. *-un-lft-identity 74.9%

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

    4. expm1-log1p-u 74.0%

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

    5. expm1-undefine 74.0%

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

    6. exp-diff 74.0%

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

11. Applied egg-rr 74.9%

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

12. Taylor expanded in x around 0 66.9%

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

13. Final simplification 66.9%

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

Alternative 7: 66.1% accurate, 7.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))) E))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

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

   2. sqr-neg 100.0%

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

   3. associate--r- 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

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

   1. 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 74.9%

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

   1. exp-1-e 74.9%

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

9. Simplified 74.9%

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

    1. e-exp-1 74.9%

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

    2. pow-exp 74.9%

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

    3. *-un-lft-identity 74.9%

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

    4. expm1-log1p-u 74.0%

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

    5. expm1-undefine 74.0%

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

    6. exp-diff 74.0%

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

11. Applied egg-rr 74.9%

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

12. Taylor expanded in x around 0 66.9%

    \[\leadsto \frac{\color{blue}{1 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)}}{e} \]
13. Step-by-step derivation

    1. *-commutative 66.9%

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

14. Simplified 66.9%

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

Alternative 8: 75.5% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (+ 1.0 (* x (+ 1.0 (* x 0.5)))) E))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

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

   2. sqr-neg 100.0%

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

   3. associate--r- 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

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

   1. 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 74.9%

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

   1. exp-1-e 74.9%

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

9. Simplified 74.9%

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

    1. e-exp-1 74.9%

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

    2. pow-exp 74.9%

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

    3. *-un-lft-identity 74.9%

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

    4. expm1-log1p-u 74.0%

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

    5. expm1-undefine 74.0%

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

    6. exp-diff 74.0%

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

11. Applied egg-rr 74.9%

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

12. Taylor expanded in x around 0 76.5%

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

    1. *-commutative 76.5%

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

14. Simplified 76.5%

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

Alternative 9: 49.8% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \frac{x}{e} + \frac{1}{e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (/ x E) (/ 1.0 E)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x / 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. 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 74.9%

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

   1. exp-1-e 74.9%

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

9. Simplified 74.9%

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

10. Taylor expanded in x around 0 50.6%

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

    1. log-E 50.6%

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

    2. metadata-eval 50.6%

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

    3. log-E 50.6%

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

    4. associate-/l* 50.6%

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

    5. log-E 50.6%

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

    6. metadata-eval 50.6%

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

    7. /-rgt-identity 50.6%

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

    8. associate-*r/ 50.6%

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

    9. associate-*l/ 50.6%

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

    10. /-rgt-identity 50.6%

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

12. Simplified 50.6%

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

13. Taylor expanded in x around 0 50.6%

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

14. Final simplification 50.6%

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

Alternative 10: 49.8% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \frac{x + 1}{e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ x 1.0) E))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x + 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 74.9%

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

   1. exp-1-e 74.9%

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

9. Simplified 74.9%

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

10. Taylor expanded in x around 0 50.6%

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

    1. log-E 50.6%

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

    2. metadata-eval 50.6%

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

    3. log-E 50.6%

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

    4. associate-/l* 50.6%

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

    5. log-E 50.6%

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

    6. metadata-eval 50.6%

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

    7. /-rgt-identity 50.6%

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

    8. associate-*r/ 50.6%

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

    9. associate-*l/ 50.6%

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

    10. /-rgt-identity 50.6%

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

    11. distribute-rgt1-in 50.6%

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

    12. associate-*r/ 50.6%

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

    13. *-rgt-identity 50.6%

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

    14. +-commutative 50.6%

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

12. Simplified 50.6%

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

13. Final simplification 50.6%

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

Alternative 11: 50.5% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 E))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := 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 74.9%

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

   1. exp-1-e 74.9%

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

9. Simplified 74.9%

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

10. Taylor expanded in x around 0 51.2%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))