exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 17.2s

Alternatives: 4

Speedup: 1.0×

• 49.7% 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

\[\begin{array}{l} \\ {e}^{\left(\left(-1 + \left(2 + x\right)\right) \cdot \left(-1 + x\right)\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow E (* (+ -1.0 (+ 2.0 x)) (+ -1.0 x))))

C

double code(double x) {
	return pow(((double) M_E), ((-1.0 + (2.0 + x)) * (-1.0 + x)));
}

Java

public static double code(double x) {
	return Math.pow(Math.E, ((-1.0 + (2.0 + x)) * (-1.0 + x)));
}

Python

def code(x):
	return math.pow(math.e, ((-1.0 + (2.0 + x)) * (-1.0 + x)))

Julia

function code(x)
	return exp(1) ^ Float64(Float64(-1.0 + Float64(2.0 + x)) * Float64(-1.0 + x))
end

MATLAB

function tmp = code(x)
	tmp = 2.71828182845904523536 ^ ((-1.0 + (2.0 + x)) * (-1.0 + x));
end

Wolfram

code[x_] := N[Power[E, N[(N[(-1.0 + N[(2.0 + x), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + 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. 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. Step-by-step derivation

   1. fma-undefine 100.0%

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

   2. difference-of-sqr--1 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 77.7%

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

   2. expm1-undefine 77.7%

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

10. Applied egg-rr 77.7%

    \[\leadsto {e}^{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(x + 1\right)} - 1\right)} \cdot \left(x + -1\right)\right)} \]
11. Step-by-step derivation

    1. sub-neg 77.7%

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

    2. metadata-eval 77.7%

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

    3. +-commutative 77.7%

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

    4. log1p-undefine 77.7%

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

    5. rem-exp-log 100.0%

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

    6. +-commutative 100.0%

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

    7. associate-+r+ 100.0%

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

    8. metadata-eval 100.0%

       \[\leadsto {e}^{\left(\left(-1 + \left(\color{blue}{2} + x\right)\right) \cdot \left(x + -1\right)\right)} \]

12. Simplified 100.0%

    \[\leadsto {e}^{\left(\color{blue}{\left(-1 + \left(2 + x\right)\right)} \cdot \left(x + -1\right)\right)} \]

13. Final simplification 100.0%

    \[\leadsto {e}^{\left(\left(-1 + \left(2 + x\right)\right) \cdot \left(-1 + x\right)\right)} \]
14. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Power[E, N[(N[(-1.0 + x), $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. *-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. Step-by-step derivation

   1. fma-undefine 100.0%

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

   2. difference-of-sqr--1 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

   \[\leadsto {e}^{\left(\left(-1 + x\right) \cdot \left(x + 1\right)\right)} \]
10. 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: 50.9% 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 57.1%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))