exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 6.7s

Alternatives: 7

Speedup: 1.0×

• 49.8% 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

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

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 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   3. associate--r- 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. sqr-neg 99.9%

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

3. Simplified 99.9%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-undefine 99.9%

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

   3. exp-diff 99.9%

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

   4. fma-define 99.9%

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

   5. metadata-eval 99.9%

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

   6. fma-neg 99.9%

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

   7. pow1 99.9%

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

   8. pow-to-exp 99.9%

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

   9. expm1-define 99.9%

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

   10. log1p-expm1-u 99.9%

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

   11. pow-to-exp 99.9%

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

   12. pow1 99.9%

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

   13. pow2 99.9%

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

   14. exp-1-e 99.9%

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

6. Applied egg-rr 99.9%

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

   1. unpow2 99.9%

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

   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. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{{x}^{2}}}{e} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   3. associate--r- 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. sqr-neg 99.9%

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

3. Simplified 99.9%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-undefine 99.9%

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

   3. exp-diff 99.9%

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

   4. fma-define 99.9%

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

   5. metadata-eval 99.9%

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

   6. fma-neg 99.9%

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

   7. pow1 99.9%

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

   8. pow-to-exp 99.9%

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

   9. expm1-define 99.9%

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

   10. log1p-expm1-u 99.9%

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

   11. pow-to-exp 99.9%

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

   12. pow1 99.9%

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

   13. pow2 99.9%

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

   14. exp-1-e 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \frac{e^{{x}^{2}}}{e} \]
8. 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 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   3. associate--r- 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. sqr-neg 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

   \[\leadsto e^{x \cdot x + -1} \]
6. Add Preprocessing

Alternative 4: 74.7% 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 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   3. associate--r- 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. sqr-neg 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
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 78.4%

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

   1. exp-1-e 78.4%

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

9. Simplified 78.4%

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

10. Taylor expanded in x around inf 78.4%

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

    1. log-E 78.4%

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

    2. sub-neg 78.4%

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

    3. metadata-eval 78.4%

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

    4. *-lft-identity 78.4%

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

12. Simplified 78.4%

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

13. Final simplification 78.4%

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

Alternative 5: 50.5% accurate, 15.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   3. associate--r- 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. sqr-neg 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
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 78.4%

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

   1. exp-1-e 78.4%

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

9. Simplified 78.4%

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

10. Taylor expanded in x around 0 49.7%

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

    1. exp-1-e 49.7%

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

    2. rec-exp 49.7%

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

    3. metadata-eval 49.7%

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

    4. log-E 49.7%

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

    5. associate-*r/ 49.7%

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

    6. exp-1-e 49.7%

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

    7. rec-exp 49.7%

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

    8. metadata-eval 49.7%

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

    9. distribute-rgt1-in 49.7%

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

    10. metadata-eval 49.7%

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

    11. rec-exp 49.7%

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

    12. exp-1-e 49.7%

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

    13. associate-*r/ 49.7%

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

    14. distribute-rgt1-in 49.7%

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

    15. *-rgt-identity 49.7%

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

12. Simplified 49.7%

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

13. Taylor expanded in x around 0 49.7%

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

14. Final simplification 49.7%

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

Alternative 6: 50.5% 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 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   3. associate--r- 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. sqr-neg 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
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 78.4%

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

   1. exp-1-e 78.4%

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

9. Simplified 78.4%

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

10. Taylor expanded in x around 0 49.7%

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

    1. exp-1-e 49.7%

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

    2. rec-exp 49.7%

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

    3. metadata-eval 49.7%

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

    4. log-E 49.7%

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

    5. associate-*r/ 49.7%

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

    6. exp-1-e 49.7%

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

    7. rec-exp 49.7%

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

    8. metadata-eval 49.7%

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

    9. distribute-rgt1-in 49.7%

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

    10. metadata-eval 49.7%

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

    11. rec-exp 49.7%

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

    12. exp-1-e 49.7%

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

    13. associate-*r/ 49.7%

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

    14. distribute-rgt1-in 49.7%

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

    15. *-rgt-identity 49.7%

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

12. Simplified 49.7%

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

13. Final simplification 49.7%

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

Alternative 7: 51.2% 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 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   3. associate--r- 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. sqr-neg 99.9%

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

3. Simplified 99.9%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-undefine 99.9%

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

   3. exp-diff 99.9%

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

   4. fma-define 99.9%

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

   5. metadata-eval 99.9%

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

   6. fma-neg 99.9%

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

   7. pow1 99.9%

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

   8. pow-to-exp 99.9%

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

   9. expm1-define 99.9%

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

   10. log1p-expm1-u 99.9%

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

   11. pow-to-exp 99.9%

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

   12. pow1 99.9%

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

   13. pow2 99.9%

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

   14. exp-1-e 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 49.8%

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

8. Final simplification 49.8%

   \[\leadsto \frac{1}{e} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))