exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 10

Speedup: 1.0×

• 49.2% 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

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return exp(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[Power[N[Exp[N[(x + 1.0), $MachinePrecision]], $MachinePrecision], 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. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return exp(1) ^ fma(x, x, -1.0)
end

Wolfram

code[x_] := N[Power[E, N[(x * 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. *-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. exp-1-e 100.0%

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

   4. fma-define 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{e}^{\left(\mathsf{fma}\left(x, x, -1\right)\right)}} \]
7. 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: 74.3% 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 75.9%

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

   1. exp-1-e 75.9%

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

9. Simplified 75.9%

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

Alternative 5: 74.3% 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 75.9%

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

   1. exp-1-e 75.9%

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

9. Simplified 75.9%

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

    1. unpow-prod-up 75.9%

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

    2. pow-to-exp 75.9%

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

    3. log-E 75.9%

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

    4. *-un-lft-identity 75.9%

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

    5. inv-pow 75.9%

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

    6. un-div-inv 75.9%

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

11. Applied egg-rr 75.9%

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

Alternative 6: 66.7% 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 75.9%

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

   1. exp-1-e 75.9%

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

9. Simplified 75.9%

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

    1. unpow-prod-up 75.9%

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

    2. pow-to-exp 75.9%

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

    3. log-E 75.9%

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

    4. *-un-lft-identity 75.9%

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

    5. inv-pow 75.9%

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

    6. un-div-inv 75.9%

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

11. Applied egg-rr 75.9%

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

12. Taylor expanded in x around 0 65.6%

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

13. Final simplification 65.6%

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

Alternative 7: 66.7% 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 75.9%

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

   1. exp-1-e 75.9%

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

9. Simplified 75.9%

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

    1. unpow-prod-up 75.9%

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

    2. pow-to-exp 75.9%

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

    3. log-E 75.9%

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

    4. *-un-lft-identity 75.9%

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

    5. inv-pow 75.9%

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

    6. un-div-inv 75.9%

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

11. Applied egg-rr 75.9%

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

12. Taylor expanded in x around 0 65.6%

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

    1. *-commutative 65.6%

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

14. Simplified 65.6%

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

Alternative 8: 75.6% 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 75.9%

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

   1. exp-1-e 75.9%

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

9. Simplified 75.9%

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

    1. unpow-prod-up 75.9%

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

    2. pow-to-exp 75.9%

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

    3. log-E 75.9%

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

    4. *-un-lft-identity 75.9%

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

    5. inv-pow 75.9%

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

    6. un-div-inv 75.9%

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

11. Applied egg-rr 75.9%

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

12. Taylor expanded in x around 0 73.4%

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

    1. *-commutative 73.4%

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

14. Simplified 73.4%

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

Alternative 9: 50.7% 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 75.9%

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

   1. exp-1-e 75.9%

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

9. Simplified 75.9%

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

10. Taylor expanded in x around 0 50.3%

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

    1. log-E 50.3%

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

    2. metadata-eval 50.3%

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

    3. log-E 50.3%

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

    4. log-E 50.3%

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

    5. metadata-eval 50.3%

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

    6. associate-/l* 50.3%

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

12. Simplified 50.3%

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

13. Taylor expanded in x around 0 50.3%

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

    1. metadata-eval 50.3%

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

    2. associate-/r* 50.3%

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

    3. neg-mul-1 50.3%

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

    4. remove-double-neg 50.3%

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

    5. distribute-neg-frac2 50.3%

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

    6. sub-neg 50.3%

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

    7. div-sub 50.3%

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

    8. distribute-frac-neg2 50.3%

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

    9. distribute-neg-frac 50.3%

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

    10. sub-neg 50.3%

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

    11. metadata-eval 50.3%

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

    12. distribute-neg-in 50.3%

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

    13. +-commutative 50.3%

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

    14. remove-double-neg 50.3%

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

15. Simplified 50.3%

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

Alternative 10: 51.5% 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. *-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. exp-1-e 100.0%

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

   4. fma-define 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 51.0%

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

Reproduce

herbie shell --seed 2024141 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))