exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 8

Speedup: 1.0×

• 50.1% 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

\[\begin{array}{l} \\ \frac{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot 0.5\right)}}{e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow (exp (* x 2.0)) (* x 0.5)) E))

C

double code(double x) {
	return pow(exp((x * 2.0)), (x * 0.5)) / ((double) M_E);
}

Java

public static double code(double x) {
	return Math.pow(Math.exp((x * 2.0)), (x * 0.5)) / Math.E;
}

Python

def code(x):
	return math.pow(math.exp((x * 2.0)), (x * 0.5)) / math.e

Julia

function code(x)
	return Float64((exp(Float64(x * 2.0)) ^ Float64(x * 0.5)) / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * 2.0)) ^ (x * 0.5)) / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(N[Power[N[Exp[N[(x * 2.0), $MachinePrecision]], $MachinePrecision], N[(x * 0.5), $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. *-un-lft-identity 100.0%

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

   2. exp-prod 99.9%

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

   3. expm1-log1p-u 99.9%

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

   4. expm1-undefine 99.9%

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

   5. pow-sub 99.9%

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

   6. fma-define 99.9%

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

   7. metadata-eval 99.9%

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

   8. fma-neg 99.9%

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

   9. pow1 99.9%

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

   10. pow-to-exp 99.9%

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

   11. expm1-define 99.9%

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

   12. log1p-expm1-u 99.9%

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

   13. pow-to-exp 100.0%

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

   14. pow1 100.0%

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

   15. exp-prod 100.0%

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

   16. *-un-lft-identity 100.0%

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

   17. pow2 100.0%

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

   18. pow1 100.0%

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

   19. exp-1-e 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{e}} \]
7. Step-by-step derivation

   1. unpow2 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{\color{blue}{x \cdot x}}}{e} \]
9. Step-by-step derivation

   1. pow-exp 100.0%

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

   2. sqr-pow 100.0%

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

   3. pow-prod-down 100.0%

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

   4. pow2 100.0%

      \[\leadsto \frac{{\color{blue}{\left({\left(e^{x}\right)}^{2}\right)}}^{\left(\frac{x}{2}\right)}}{e} \]

   5. div-inv 100.0%

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

   6. metadata-eval 100.0%

      \[\leadsto \frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(x \cdot \color{blue}{0.5}\right)}}{e} \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(x \cdot 0.5\right)}}}{e} \]
11. Step-by-step derivation

    1. unpow2 100.0%

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

12. Applied egg-rr 100.0%

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

    1. exp-lft-sqr 100.0%

       \[\leadsto \frac{{\color{blue}{\left(e^{x \cdot 2}\right)}}^{\left(x \cdot 0.5\right)}}{e} \]

14. Simplified 100.0%

    \[\leadsto \frac{{\color{blue}{\left(e^{x \cdot 2}\right)}}^{\left(x \cdot 0.5\right)}}{e} \]
15. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[N[Exp[x], $MachinePrecision], 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. *-un-lft-identity 100.0%

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

   2. exp-prod 99.9%

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

   3. expm1-log1p-u 99.9%

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

   4. expm1-undefine 99.9%

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

   5. pow-sub 99.9%

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

   6. fma-define 99.9%

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

   7. metadata-eval 99.9%

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

   8. fma-neg 99.9%

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

   9. pow1 99.9%

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

   10. pow-to-exp 99.9%

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

   11. expm1-define 99.9%

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

   12. log1p-expm1-u 99.9%

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

   13. pow-to-exp 100.0%

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

   14. pow1 100.0%

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

   15. exp-prod 100.0%

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

   16. *-un-lft-identity 100.0%

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

   17. pow2 100.0%

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

   18. pow1 100.0%

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

   19. exp-1-e 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{e}} \]
7. Step-by-step derivation

   1. unpow2 100.0%

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

   2. exp-prod 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e} \]
9. Add Preprocessing

Alternative 3: 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 4: 74.1% 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 99.9%

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

   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 71.8%

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

   1. exp-1-e 71.8%

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

9. Simplified 71.8%

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

    1. unpow-prod-up 71.8%

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

    2. pow-to-exp 71.8%

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

    3. log-E 71.8%

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

    4. *-un-lft-identity 71.8%

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

    5. inv-pow 71.8%

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

    6. un-div-inv 71.8%

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

11. Applied egg-rr 71.8%

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

Alternative 5: 66.4% 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 99.9%

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

   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 71.8%

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

   1. exp-1-e 71.8%

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

9. Simplified 71.8%

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

    1. unpow-prod-up 71.8%

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

    2. pow-to-exp 71.8%

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

    3. log-E 71.8%

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

    4. *-un-lft-identity 71.8%

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

    5. inv-pow 71.8%

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

    6. un-div-inv 71.8%

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

11. Applied egg-rr 71.8%

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

12. Taylor expanded in x around 0 63.3%

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

    1. *-commutative 63.3%

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

14. Simplified 63.3%

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

Alternative 6: 75.7% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + 1.0), $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 99.9%

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

   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 71.8%

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

   1. exp-1-e 71.8%

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

9. Simplified 71.8%

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

    1. unpow-prod-up 71.8%

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

    2. pow-to-exp 71.8%

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

    3. log-E 71.8%

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

    4. *-un-lft-identity 71.8%

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

    5. inv-pow 71.8%

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

    6. un-div-inv 71.8%

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

11. Applied egg-rr 71.8%

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

12. Taylor expanded in x around 0 74.3%

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

    1. *-commutative 74.3%

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

14. Simplified 74.3%

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

15. Final simplification 74.3%

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

Alternative 7: 50.1% 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 99.9%

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

   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 71.8%

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

   1. exp-1-e 71.8%

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

9. Simplified 71.8%

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

10. Taylor expanded in x around 0 46.0%

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

    1. log-E 46.0%

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

    2. metadata-eval 46.0%

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

    3. log-E 46.0%

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

    4. log-E 46.0%

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

    5. metadata-eval 46.0%

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

    6. *-commutative 46.0%

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

    7. *-commutative 46.0%

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

    8. associate-/l* 46.0%

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

    9. distribute-rgt1-in 46.0%

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

    10. +-commutative 46.0%

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

    11. associate-*r/ 46.0%

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

    12. +-commutative 46.0%

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

    13. distribute-rgt1-in 46.0%

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

    14. *-rgt-identity 46.0%

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

12. Simplified 46.0%

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

13. Final simplification 46.0%

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

Alternative 8: 50.8% 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 99.9%

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

   3. expm1-log1p-u 99.9%

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

   4. expm1-undefine 99.9%

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

   5. pow-sub 99.9%

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

   6. fma-define 99.9%

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

   7. metadata-eval 99.9%

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

   8. fma-neg 99.9%

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

   9. pow1 99.9%

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

   10. pow-to-exp 99.9%

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

   11. expm1-define 99.9%

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

   12. log1p-expm1-u 99.9%

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

   13. pow-to-exp 100.0%

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

   14. pow1 100.0%

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

   15. exp-prod 100.0%

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

   16. *-un-lft-identity 100.0%

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

   17. pow2 100.0%

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

   18. pow1 100.0%

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

   19. exp-1-e 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 46.7%

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

Reproduce

herbie shell --seed 2024144 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))