exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 10.3s

Alternatives: 14

Speedup: 1.0×

• 50.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. lift--.f64 N/A

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

   4. associate--r- N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-exp.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. difference-of-sqr--1 N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 91.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;e^{-1 + x\_m \cdot x\_m} \leq 0.5:\\ \;\;\;\;\frac{1}{\frac{e}{\mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.5, x\_m\right), 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x\_m \cdot \frac{x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{e}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (exp (+ -1.0 (* x_m x_m))) 0.5)
   (/ 1.0 (/ E (fma x_m (fma x_m (* (* x_m x_m) 0.5) x_m) 1.0)))
   (* 0.16666666666666666 (* x_m (/ (* x_m (* x_m (* x_m (* x_m x_m)))) E)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (exp((-1.0 + (x_m * x_m))) <= 0.5) {
		tmp = 1.0 / (((double) M_E) / fma(x_m, fma(x_m, ((x_m * x_m) * 0.5), x_m), 1.0));
	} else {
		tmp = 0.16666666666666666 * (x_m * ((x_m * (x_m * (x_m * (x_m * x_m)))) / ((double) M_E)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (exp(Float64(-1.0 + Float64(x_m * x_m))) <= 0.5)
		tmp = Float64(1.0 / Float64(exp(1) / fma(x_m, fma(x_m, Float64(Float64(x_m * x_m) * 0.5), x_m), 1.0)));
	else
		tmp = Float64(0.16666666666666666 * Float64(x_m * Float64(Float64(x_m * Float64(x_m * Float64(x_m * Float64(x_m * x_m)))) / exp(1))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.f64 x x)))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.4%

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

   if 0.5 < (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.f64 x x))))

   1. Initial program 99.9%

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

   4. Applied rewrites 59.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(-1 \cdot \left(e^{-1} \cdot \log \left(\frac{1}{\mathsf{E}\left(\right)}\right)\right) + \left(\frac{-1}{2} \cdot e^{-1} + {x}^{2} \cdot \left(-1 \cdot \left(e^{-1} \cdot \log \left(\frac{1}{\mathsf{E}\left(\right)}\right)\right) + \left(\frac{1}{6} \cdot e^{-1} + e^{-1} \cdot \log \left(\frac{1}{\mathsf{E}\left(\right)}\right)\right)\right)\right)\right)\right)} \]

   7. Applied rewrites 81.7%

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

   9. Applied rewrites 41.1%

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot x}{e}, \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.027777777777777776, -0.25\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.16666666666666666, -0.5\right)}}, 1\right), \frac{1}{e}\right) \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{1}{6} \cdot \color{blue}{\frac{{x}^{6}}{\mathsf{E}\left(\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 81.7%

       \[\leadsto 0.16666666666666666 \cdot \color{blue}{\left(\frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x}{e} \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-1 + x \cdot x} \leq 0.5:\\ \;\;\;\;\frac{1}{\frac{e}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{e}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;e^{-1 + x\_m \cdot x\_m} \leq 0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.5, x\_m\right), 1\right)}{e}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x\_m \cdot \frac{x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{e}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (exp (+ -1.0 (* x_m x_m))) 0.5)
   (/ (fma x_m (fma x_m (* (* x_m x_m) 0.5) x_m) 1.0) E)
   (* 0.16666666666666666 (* x_m (/ (* x_m (* x_m (* x_m (* x_m x_m)))) E)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (exp((-1.0 + (x_m * x_m))) <= 0.5) {
		tmp = fma(x_m, fma(x_m, ((x_m * x_m) * 0.5), x_m), 1.0) / ((double) M_E);
	} else {
		tmp = 0.16666666666666666 * (x_m * ((x_m * (x_m * (x_m * (x_m * x_m)))) / ((double) M_E)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (exp(Float64(-1.0 + Float64(x_m * x_m))) <= 0.5)
		tmp = Float64(fma(x_m, fma(x_m, Float64(Float64(x_m * x_m) * 0.5), x_m), 1.0) / exp(1));
	else
		tmp = Float64(0.16666666666666666 * Float64(x_m * Float64(Float64(x_m * Float64(x_m * Float64(x_m * Float64(x_m * x_m)))) / exp(1))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.f64 x x)))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.4%

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

   if 0.5 < (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.f64 x x))))

   1. Initial program 99.9%

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

   4. Applied rewrites 59.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(-1 \cdot \left(e^{-1} \cdot \log \left(\frac{1}{\mathsf{E}\left(\right)}\right)\right) + \left(\frac{-1}{2} \cdot e^{-1} + {x}^{2} \cdot \left(-1 \cdot \left(e^{-1} \cdot \log \left(\frac{1}{\mathsf{E}\left(\right)}\right)\right) + \left(\frac{1}{6} \cdot e^{-1} + e^{-1} \cdot \log \left(\frac{1}{\mathsf{E}\left(\right)}\right)\right)\right)\right)\right)\right)} \]

   7. Applied rewrites 81.7%

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

   9. Applied rewrites 41.1%

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot x}{e}, \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.027777777777777776, -0.25\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.16666666666666666, -0.5\right)}}, 1\right), \frac{1}{e}\right) \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{1}{6} \cdot \color{blue}{\frac{{x}^{6}}{\mathsf{E}\left(\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 81.7%

       \[\leadsto 0.16666666666666666 \cdot \color{blue}{\left(\frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x}{e} \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-1 + x \cdot x} \leq 0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), 1\right)}{e}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{e}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Exp[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. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. lift--.f64 N/A

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

   4. associate--r- N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 5: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. lift--.f64 N/A

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

   4. associate--r- N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-exp.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. difference-of-sqr--1 N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. exp-1-e N/A

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

   2. lower-E.f64 67.8

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

9. Applied rewrites 67.8%

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

Alternative 6: 93.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.5, x\_m\right)\\ \mathbf{if}\;x\_m \cdot x\_m \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, t\_0 \cdot t\_0, -1\right)}{e \cdot \mathsf{fma}\left(x\_m, t\_0, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(0.5 \cdot \left(x\_m \cdot \frac{x\_m \cdot x\_m}{e}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma x_m (* (* x_m x_m) 0.5) x_m)))
   (if (<= (* x_m x_m) 2e+151)
     (/ (fma (* x_m x_m) (* t_0 t_0) -1.0) (* E (fma x_m t_0 -1.0)))
     (* x_m (* 0.5 (* x_m (/ (* x_m x_m) E)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(x_m, ((x_m * x_m) * 0.5), x_m);
	double tmp;
	if ((x_m * x_m) <= 2e+151) {
		tmp = fma((x_m * x_m), (t_0 * t_0), -1.0) / (((double) M_E) * fma(x_m, t_0, -1.0));
	} else {
		tmp = x_m * (0.5 * (x_m * ((x_m * x_m) / ((double) M_E))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(x_m, Float64(Float64(x_m * x_m) * 0.5), x_m)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 2e+151)
		tmp = Float64(fma(Float64(x_m * x_m), Float64(t_0 * t_0), -1.0) / Float64(exp(1) * fma(x_m, t_0, -1.0)));
	else
		tmp = Float64(x_m * Float64(0.5 * Float64(x_m * Float64(Float64(x_m * x_m) / exp(1)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] + x$95$m), $MachinePrecision]}, If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 2e+151], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(E * N[(x$95$m * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(0.5 * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000003e151

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 88.7%

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.5 \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(x, 0.5 \cdot \left(x \cdot x\right), x\right), -1\right)}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5 \cdot \left(x \cdot x\right), x\right), -1\right) \cdot e}} \]

   if 2.00000000000000003e151 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{{x}^{4}}{\mathsf{E}\left(\right)}} \]

   8. Applied rewrites 100.0%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\frac{x \cdot x}{e} \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right) \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), -1\right)}{e \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \frac{x \cdot x}{e}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto e^{-1} + \color{blue}{\left(e^{-1} \cdot {x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right) \cdot {x}^{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto e^{-1} + \left(\color{blue}{{x}^{2} \cdot e^{-1}} + \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right) \cdot {x}^{2}\right) \]

   3. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(e^{-1} + {x}^{2} \cdot e^{-1}\right) + \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right) \cdot {x}^{2}} \]

   4. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left({x}^{2} + 1\right) \cdot e^{-1}} + \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right) \cdot {x}^{2} \]

   5. *-commutative N/A

      \[\leadsto \left({x}^{2} + 1\right) \cdot e^{-1} + \color{blue}{\left(\left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right) \cdot {x}^{2}\right)} \cdot {x}^{2} \]

   6. associate-*l* N/A

      \[\leadsto \left({x}^{2} + 1\right) \cdot e^{-1} + \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right) \cdot \left({x}^{2} \cdot {x}^{2}\right)} \]

   7. +-commutative N/A

      \[\leadsto \left({x}^{2} + 1\right) \cdot e^{-1} + \color{blue}{\left(\frac{1}{2} \cdot e^{-1} + \frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   8. associate-*r* N/A

      \[\leadsto \left({x}^{2} + 1\right) \cdot e^{-1} + \left(\frac{1}{2} \cdot e^{-1} + \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot e^{-1}}\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   9. distribute-rgt-out N/A

      \[\leadsto \left({x}^{2} + 1\right) \cdot e^{-1} + \color{blue}{\left(e^{-1} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

5. Applied rewrites 89.5%

   \[\leadsto \color{blue}{\frac{1}{e} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), x \cdot \left(x \cdot x\right), x\right), 1\right)} \]
6. Add Preprocessing

Alternative 8: 87.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;1 - x\_m \cdot x\_m \leq -10:\\ \;\;\;\;x\_m \cdot \left(0.5 \cdot \left(x\_m \cdot \frac{x\_m \cdot x\_m}{e}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{e}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (* x_m x_m)) -10.0)
   (* x_m (* 0.5 (* x_m (/ (* x_m x_m) E))))
   (/ 1.0 (/ E (fma x_m x_m 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((1.0 - (x_m * x_m)) <= -10.0) {
		tmp = x_m * (0.5 * (x_m * ((x_m * x_m) / ((double) M_E))));
	} else {
		tmp = 1.0 / (((double) M_E) / fma(x_m, x_m, 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(1.0 - Float64(x_m * x_m)) <= -10.0)
		tmp = Float64(x_m * Float64(0.5 * Float64(x_m * Float64(Float64(x_m * x_m) / exp(1)))));
	else
		tmp = Float64(1.0 / Float64(exp(1) / fma(x_m, x_m, 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -10.0], N[(x$95$m * N[(0.5 * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(E / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{{x}^{4}}{\mathsf{E}\left(\right)}} \]

   8. Applied rewrites 76.2%

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

   if -10 < (-.f64 #s(literal 1 binary64) (*.f64 x x))

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   6. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. rec-exp N/A

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

      6. e-exp-1 N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

      13. lower-E.f64 99.3

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{e}{\mathsf{fma}\left(x, x, 1\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -10:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \frac{x \cdot x}{e}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{e}{\mathsf{fma}\left(x, x, 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 0.4:\\ \;\;\;\;\frac{1}{\frac{e}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{e} \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (* x_m x_m) 0.4)
   (/ 1.0 (/ E (fma x_m x_m 1.0)))
   (* (/ (* x_m x_m) E) (fma x_m (* x_m 0.5) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((x_m * x_m) <= 0.4) {
		tmp = 1.0 / (((double) M_E) / fma(x_m, x_m, 1.0));
	} else {
		tmp = ((x_m * x_m) / ((double) M_E)) * fma(x_m, (x_m * 0.5), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 0.4)
		tmp = Float64(1.0 / Float64(exp(1) / fma(x_m, x_m, 1.0)));
	else
		tmp = Float64(Float64(Float64(x_m * x_m) / exp(1)) * fma(x_m, Float64(x_m * 0.5), 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 0.4], N[(1.0 / N[(E / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] / E), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.40000000000000002

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   6. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. rec-exp N/A

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

      6. e-exp-1 N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

      13. lower-E.f64 99.3

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.3%

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

   if 0.40000000000000002 < (*.f64 x x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 76.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)} + \frac{1}{{x}^{2} \cdot \mathsf{E}\left(\right)}\right)} \]

   10. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(x, x \cdot 0.5, 1\right) \cdot \color{blue}{\frac{x \cdot x}{e}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.4:\\ \;\;\;\;\frac{1}{\frac{e}{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{e} \cdot \mathsf{fma}\left(x, x \cdot 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. *-rgt-identity N/A

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

   9. associate-+l+ N/A

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

   10. +-commutative N/A

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

   11. lower-*.f64 N/A

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

5. Applied rewrites 86.4%

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

7. Applied rewrites 86.4%

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

8. Final simplification 86.4%

   \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), 1\right)}{e} \]
9. Add Preprocessing

Alternative 11: 75.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 77.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
6. Step-by-step derivation

   1. distribute-rgt1-in N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. rec-exp N/A

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

   6. e-exp-1 N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

   13. lower-E.f64 68.5

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

7. Applied rewrites 68.5%

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

9. Applied rewrites 68.5%

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

Alternative 12: 75.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 1:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{e}\\ \end{array} \end{array} \]

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((x_m * x_m) <= 1.0) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = (x_m * x_m) / ((double) M_E);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if ((x_m * x_m) <= 1.0) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = (x_m * x_m) / Math.E;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if (x_m * x_m) <= 1.0:
		tmp = 1.0 / math.e
	else:
		tmp = (x_m * x_m) / math.e
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 1.0)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(Float64(x_m * x_m) / exp(1));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if ((x_m * x_m) <= 1.0)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = (x_m * x_m) / 2.71828182845904523536;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(1\right)}} \]

      2. rec-exp N/A

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

      3. lower-/.f64 N/A

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

      4. exp-1-e N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \]

      5. lower-E.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 1 < (*.f64 x x)

   1. Initial program 99.9%

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

   4. Applied rewrites 59.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   6. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. rec-exp N/A

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

      6. e-exp-1 N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

      13. lower-E.f64 44.1

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

   7. Applied rewrites 44.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{{x}^{2}}{\mathsf{E}\left(\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.1%

       \[\leadsto \frac{x \cdot x}{e} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 75.7% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 77.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
6. Step-by-step derivation

   1. distribute-rgt1-in N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. rec-exp N/A

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

   6. e-exp-1 N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

   13. lower-E.f64 68.5

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

7. Applied rewrites 68.5%

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

Alternative 14: 50.3% accurate, 9.3× 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. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1}} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(1\right)}} \]

   2. rec-exp N/A

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

   3. lower-/.f64 N/A

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

   4. exp-1-e N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \]

   5. lower-E.f64 45.3

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

5. Applied rewrites 45.3%

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

Reproduce

herbie shell --seed 2024223 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))