exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 10

Speedup: 1.0×

• 49.5% 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

\[\begin{array}{l} x_m = \left|x\right| \\ {\left(e^{-1 - x\_m}\right)}^{\left(1 - x\_m\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (pow (exp (- -1.0 x_m)) (- 1.0 x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(exp((-1.0 - x_m)), (1.0 - x_m));
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.exp((-1.0 - x_m)), (1.0 - x_m));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.exp((-1.0 - x_m)), (1.0 - x_m))

Julia

x_m = abs(x)
function code(x_m)
	return exp(Float64(-1.0 - x_m)) ^ Float64(1.0 - x_m)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = exp((-1.0 - x_m)) ^ (1.0 - x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[N[Exp[N[(-1.0 - x$95$m), $MachinePrecision]], $MachinePrecision], N[(1.0 - x$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   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. *-un-lft-identity 99.9%

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

   2. exp-prod 99.9%

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

   3. exp-1-e 99.9%

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

   4. fma-define 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{e}^{\left(\mathsf{fma}\left(x, x, -1\right)\right)}} \]
7. Step-by-step derivation

   1. e-exp-1 99.9%

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

   2. pow-exp 99.9%

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

   3. *-un-lft-identity 99.9%

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

   4. metadata-eval 99.9%

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

   5. fmm-def 99.9%

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

   6. difference-of-sqr-1 99.9%

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

   7. add-log-exp 99.9%

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

   8. sub-neg 99.9%

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

   9. metadata-eval 99.9%

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

   10. pow-to-exp 100.0%

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

   11. add-sqr-sqrt 99.2%

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

   12. unpow-prod-down 100.0%

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

8. Applied egg-rr 100.0%

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

   1. pow-sqr 99.2%

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

   2. +-commutative 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in x around -inf 99.9%

    \[\leadsto \color{blue}{e^{-2 \cdot \left(\log \left(\sqrt{e^{1 - -1 \cdot x}}\right) \cdot \left(1 + -1 \cdot x\right)\right)}} \]
12. Step-by-step derivation

    1. associate-*r* 99.9%

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

    2. mul-1-neg 99.9%

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

    3. sub-neg 99.9%

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

    4. exp-prod 100.0%

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

13. Simplified 99.9%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (pow E (fma x_m x_m -1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[E, N[(x$95$m * x$95$m + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   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. *-un-lft-identity 99.9%

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

   2. exp-prod 99.9%

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

   3. exp-1-e 99.9%

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

   4. fma-define 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ e^{-1 + x\_m \cdot x\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (exp (+ -1.0 (* x_m x_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Exp[N[(-1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   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%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (pow E (+ -1.0 x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_E), (-1.0 + x_m));
}

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.e, (-1.0 + x_m))

Julia

x_m = abs(x)
function code(x_m)
	return exp(1) ^ Float64(-1.0 + x_m)
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[E, N[(-1.0 + x$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   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 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 72.8%

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

   1. exp-1-e 72.8%

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

9. Simplified 72.8%

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

10. Final simplification 72.8%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{x\_m}}{e} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (exp x_m) E))

C

x_m = fabs(x);
double code(double x_m) {
	return exp(x_m) / ((double) M_E);
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.exp(x_m) / Math.E;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.exp(x_m) / math.e

Julia

x_m = abs(x)
function code(x_m)
	return Float64(exp(x_m) / exp(1))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Exp[x$95$m], $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   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 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 72.8%

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

   1. exp-1-e 72.8%

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

9. Simplified 72.8%

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

    1. unpow-prod-up 72.8%

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

    2. pow-to-exp 72.8%

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

    3. log-E 72.8%

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

    4. *-un-lft-identity 72.8%

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

    5. inv-pow 72.8%

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

    6. un-div-inv 72.8%

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

11. Applied egg-rr 72.8%

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

Alternative 6: 82.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{e} + x\_m \cdot \left(\frac{1}{e} + x\_m \cdot \left(0.16666666666666666 \cdot \frac{x\_m}{e} + 0.5 \cdot \frac{1}{e}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (+
  (/ 1.0 E)
  (*
   x_m
   (+
    (/ 1.0 E)
    (* x_m (+ (* 0.16666666666666666 (/ x_m E)) (* 0.5 (/ 1.0 E))))))))

C

x_m = fabs(x);
double code(double x_m) {
	return (1.0 / ((double) M_E)) + (x_m * ((1.0 / ((double) M_E)) + (x_m * ((0.16666666666666666 * (x_m / ((double) M_E))) + (0.5 * (1.0 / ((double) M_E)))))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (1.0 / Math.E) + (x_m * ((1.0 / Math.E) + (x_m * ((0.16666666666666666 * (x_m / Math.E)) + (0.5 * (1.0 / Math.E))))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (1.0 / math.e) + (x_m * ((1.0 / math.e) + (x_m * ((0.16666666666666666 * (x_m / math.e)) + (0.5 * (1.0 / math.e))))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(1.0 / exp(1)) + Float64(x_m * Float64(Float64(1.0 / exp(1)) + Float64(x_m * Float64(Float64(0.16666666666666666 * Float64(x_m / exp(1))) + Float64(0.5 * Float64(1.0 / exp(1))))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (1.0 / 2.71828182845904523536) + (x_m * ((1.0 / 2.71828182845904523536) + (x_m * ((0.16666666666666666 * (x_m / 2.71828182845904523536)) + (0.5 * (1.0 / 2.71828182845904523536))))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 / E), $MachinePrecision] + N[(x$95$m * N[(N[(1.0 / E), $MachinePrecision] + N[(x$95$m * N[(N[(0.16666666666666666 * N[(x$95$m / E), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   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 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 72.8%

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

   1. exp-1-e 72.8%

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

9. Simplified 72.8%

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

    1. unpow-prod-up 72.8%

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

    2. pow-to-exp 72.8%

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

    3. log-E 72.8%

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

    4. *-un-lft-identity 72.8%

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

    5. inv-pow 72.8%

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

    6. un-div-inv 72.8%

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

11. Applied egg-rr 72.8%

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

12. Taylor expanded in x around 0 65.0%

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

13. Final simplification 65.0%

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

Alternative 7: 82.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1 + x\_m \cdot \left(1 + x\_m \cdot \left(0.5 + x\_m \cdot 0.16666666666666666\right)\right)}{e} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ (+ 1.0 (* x_m (+ 1.0 (* x_m (+ 0.5 (* x_m 0.16666666666666666)))))) E))

C

x_m = fabs(x);
double code(double x_m) {
	return (1.0 + (x_m * (1.0 + (x_m * (0.5 + (x_m * 0.16666666666666666)))))) / ((double) M_E);
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (1.0 + (x_m * (1.0 + (x_m * (0.5 + (x_m * 0.16666666666666666)))))) / Math.E;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (1.0 + (x_m * (1.0 + (x_m * (0.5 + (x_m * 0.16666666666666666)))))) / math.e

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(1.0 + Float64(x_m * Float64(1.0 + Float64(x_m * Float64(0.5 + Float64(x_m * 0.16666666666666666)))))) / exp(1))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (1.0 + (x_m * (1.0 + (x_m * (0.5 + (x_m * 0.16666666666666666)))))) / 2.71828182845904523536;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 + N[(x$95$m * N[(1.0 + N[(x$95$m * N[(0.5 + N[(x$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   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 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 72.8%

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

   1. exp-1-e 72.8%

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

9. Simplified 72.8%

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

    1. unpow-prod-up 72.8%

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

    2. pow-to-exp 72.8%

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

    3. log-E 72.8%

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

    4. *-un-lft-identity 72.8%

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

    5. inv-pow 72.8%

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

    6. un-div-inv 72.8%

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

11. Applied egg-rr 72.8%

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

12. Taylor expanded in x around 0 65.0%

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

    1. *-commutative 65.0%

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

14. Simplified 65.0%

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

Alternative 8: 75.2% accurate, 9.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (+ 1.0 (* x_m (+ 1.0 (* x_m 0.5)))) E))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 + N[(x$95$m * N[(1.0 + N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   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 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 72.8%

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

   1. exp-1-e 72.8%

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

9. Simplified 72.8%

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

    1. unpow-prod-up 72.8%

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

    2. pow-to-exp 72.8%

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

    3. log-E 72.8%

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

    4. *-un-lft-identity 72.8%

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

    5. inv-pow 72.8%

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

    6. un-div-inv 72.8%

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

11. Applied egg-rr 72.8%

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

12. Taylor expanded in x around 0 75.6%

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

    1. *-commutative 75.6%

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

14. Simplified 75.6%

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

Alternative 9: 51.9% accurate, 21.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{x\_m + 1}{e} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (+ x_m 1.0) E))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m + 1.0), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   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 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 72.8%

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

   1. exp-1-e 72.8%

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

9. Simplified 72.8%

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

    1. unpow-prod-up 72.8%

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

    2. pow-to-exp 72.8%

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

    3. log-E 72.8%

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

    4. *-un-lft-identity 72.8%

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

    5. inv-pow 72.8%

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

    6. un-div-inv 72.8%

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

11. Applied egg-rr 72.8%

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

12. Taylor expanded in x around 0 50.6%

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

13. Final simplification 50.6%

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

Alternative 10: 51.3% accurate, 35.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{e} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ 1.0 E))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 / ((double) M_E);
}

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0 / math.e

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 / exp(1))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / E), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. sqr-neg 99.9%

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

   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. *-un-lft-identity 99.9%

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

   2. exp-prod 99.9%

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

   3. exp-1-e 99.9%

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

   4. fma-define 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 51.5%

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

Reproduce

herbie shell --seed 2024172 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))