exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 4

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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Exp[N[(N[(x * x), $MachinePrecision] + -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. Add Preprocessing

Alternative 2: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Exp[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 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 70.6%

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

   1. exp-1-e 70.6%

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

9. Simplified 70.6%

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

10. Taylor expanded in x around inf 70.6%

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

    1. log-E 70.6%

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

    2. sub-neg 70.6%

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

    3. metadata-eval 70.6%

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

    4. *-lft-identity 70.6%

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

12. Simplified 70.6%

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

Alternative 3: 49.9% 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 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 70.6%

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

   1. exp-1-e 70.6%

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

9. Simplified 70.6%

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

10. Taylor expanded in x around 0 48.8%

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

    1. exp-1-e 48.8%

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

    2. log-E 48.8%

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

    3. metadata-eval 48.8%

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

    4. log-E 48.8%

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

    5. associate-/l* 48.8%

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

    6. log-E 48.8%

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

    7. metadata-eval 48.8%

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

    8. exp-1-e 48.8%

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

    9. /-rgt-identity 48.8%

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

    10. associate-*r/ 48.8%

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

    11. associate-*l/ 48.8%

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

    12. /-rgt-identity 48.8%

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

    13. distribute-rgt1-in 48.8%

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

    14. exp-1-e 48.8%

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

    15. associate-*r/ 48.8%

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

    16. *-rgt-identity 48.8%

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

    17. +-commutative 48.8%

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

12. Simplified 48.8%

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

13. Final simplification 48.8%

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

Alternative 4: 50.6% 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 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 70.6%

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

   1. exp-1-e 70.6%

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

9. Simplified 70.6%

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

10. Taylor expanded in x around 0 49.9%

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

Reproduce

herbie shell --seed 2024103 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))