exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 10.4s

Alternatives: 17

Speedup: 1.0×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))

C

double code(double x) {
	return exp(-(1.0 - (x * x)));
}

Fortran

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

Java

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

Python

def code(x):
	return math.exp(-(1.0 - (x * x)))

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))

C

double code(double x) {
	return exp(-(1.0 - (x * x)));
}

Fortran

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

Java

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

Python

def code(x):
	return math.exp(-(1.0 - (x * x)))

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (exp (fma x x -1.0)))

C

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

Julia

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

Wolfram

code[x_] := N[Exp[N[(x * x + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
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 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 87.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp (+ -1.0 (* x x))) 0.5)
   (/ (fma x (* x E) E) (* E E))
   (/ (* x (fma x (* x (* x 0.5)) x)) E)))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Exp[N[(-1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.5], N[(N[(x * N[(x * E), $MachinePrecision] + E), $MachinePrecision] / N[(E * E), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / E), $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

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   4. 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

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. rec-exp N/A

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

      6. lower-/.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.9%

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

   9. Applied rewrites 98.9%

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

   if 0.5 < (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.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 81.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.9%

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

4. Final simplification 89.5%

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

Alternative 3: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right)\right), x \cdot x, \mathsf{fma}\left(x, x, 1\right)\right)}{e}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-5)
   (/
    (fma
     (* x (* x (fma x (* x 0.16666666666666666) 0.5)))
     (* x x)
     (fma x x 1.0))
    E)
   (exp (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-5) {
		tmp = fma((x * (x * fma(x, (x * 0.16666666666666666), 0.5))), (x * x), fma(x, x, 1.0)) / ((double) M_E);
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-5)
		tmp = Float64(fma(Float64(x * Float64(x * fma(x, Float64(x * 0.16666666666666666), 0.5))), Float64(x * x), fma(x, x, 1.0)) / exp(1));
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-5], N[(N[(N[(x * N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000024e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
   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 99.9%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 5.00000000000000024e-5 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 93.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.5\right), x\right)\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t\_0 \cdot \left(x \cdot t\_0\right), -1\right)}{e \cdot \mathsf{fma}\left(x, t\_0, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right)}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (* x (* x 0.5)) x)))
   (if (<= (* x x) 2e+152)
     (/ (fma x (* t_0 (* x t_0)) -1.0) (* E (fma x t_0 -1.0)))
     (/ (* x (* x (* (* x x) 0.5))) E))))

C

double code(double x) {
	double t_0 = fma(x, (x * (x * 0.5)), x);
	double tmp;
	if ((x * x) <= 2e+152) {
		tmp = fma(x, (t_0 * (x * t_0)), -1.0) / (((double) M_E) * fma(x, t_0, -1.0));
	} else {
		tmp = (x * (x * ((x * x) * 0.5))) / ((double) M_E);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, Float64(x * Float64(x * 0.5)), x)
	tmp = 0.0
	if (Float64(x * x) <= 2e+152)
		tmp = Float64(fma(x, Float64(t_0 * Float64(x * t_0)), -1.0) / Float64(exp(1) * fma(x, t_0, -1.0)));
	else
		tmp = Float64(Float64(x * Float64(x * Float64(Float64(x * x) * 0.5))) / exp(1));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e+152], N[(N[(x * N[(t$95$0 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(E * N[(x * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e152

   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 81.8%

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

   7. Applied rewrites 89.0%

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

   if 2.0000000000000001e152 < (*.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

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot {x}^{4}}{\mathsf{E}\left(\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.5\right), x\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.5\right), x\right)\right), -1\right)}{e \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.5\right), x\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right)}{e}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -2000000000:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right)}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \left(x \cdot 0.5\right), x \cdot x, \mathsf{fma}\left(x, x, 1\right)\right)}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -2000000000.0)
   (* (* x (* x (* x x))) (/ (fma (* x x) 0.16666666666666666 0.5) E))
   (/ (fma (* x (* x 0.5)) (* x x) (fma x x 1.0)) E)))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -2000000000.0) {
		tmp = (x * (x * (x * x))) * (fma((x * x), 0.16666666666666666, 0.5) / ((double) M_E));
	} else {
		tmp = fma((x * (x * 0.5)), (x * x), fma(x, x, 1.0)) / ((double) M_E);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -2000000000.0)
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) * Float64(fma(Float64(x * x), 0.16666666666666666, 0.5) / exp(1)));
	else
		tmp = Float64(fma(Float64(x * Float64(x * 0.5)), Float64(x * x), fma(x, x, 1.0)) / exp(1));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -2000000000.0], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -2e9

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
   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 85.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 85.9%

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

   if -2e9 < (-.f64 #s(literal 1 binary64) (*.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

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
   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 99.9%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \frac{1}{2}\right), x \cdot x, \mathsf{fma}\left(x, x, 1\right)\right)}{\mathsf{E}\left(\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.8%

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

Alternative 6: 91.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -2000000000:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \frac{x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right)}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \left(x \cdot 0.5\right), x \cdot x, \mathsf{fma}\left(x, x, 1\right)\right)}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -2000000000.0)
   (* (* x x) (* x (/ (* x (* x (* x 0.16666666666666666))) E)))
   (/ (fma (* x (* x 0.5)) (* x x) (fma x x 1.0)) E)))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -2000000000.0) {
		tmp = (x * x) * (x * ((x * (x * (x * 0.16666666666666666))) / ((double) M_E)));
	} else {
		tmp = fma((x * (x * 0.5)), (x * x), fma(x, x, 1.0)) / ((double) M_E);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -2000000000.0)
		tmp = Float64(Float64(x * x) * Float64(x * Float64(Float64(x * Float64(x * Float64(x * 0.16666666666666666))) / exp(1))));
	else
		tmp = Float64(fma(Float64(x * Float64(x * 0.5)), Float64(x * x), fma(x, x, 1.0)) / exp(1));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -2000000000.0], N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -2e9

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
   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 85.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 85.9%

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

   10. Applied rewrites 85.9%

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

   if -2e9 < (-.f64 #s(literal 1 binary64) (*.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

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
   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 99.9%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \frac{1}{2}\right), x \cdot x, \mathsf{fma}\left(x, x, 1\right)\right)}{\mathsf{E}\left(\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.8%

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

Alternative 7: 91.7% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma (* x (* x (fma x (* x 0.16666666666666666) 0.5))) (* x x) (fma x x 1.0))
  E))

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(x * N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
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 92.2%

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

7. Applied rewrites 92.2%

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

9. Applied rewrites 92.2%

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

Alternative 8: 91.7% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma x (fma (* x x) (* x (fma x (* x 0.16666666666666666) 0.5)) x) 1.0) E))

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
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 92.2%

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

7. Applied rewrites 92.2%

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

Alternative 9: 91.3% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma x (* x (* (* x x) (fma x (* x 0.16666666666666666) 0.5))) 1.0) E))

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
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 92.2%

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

7. Applied rewrites 92.2%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{\mathsf{fma}\left(x, {x}^{5} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right), 1\right)}{\mathsf{E}\left(\right)} \]

10. Applied rewrites 91.1%

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

Alternative 10: 87.6% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-5)
   (/ (fma x (* x E) E) (* E E))
   (/ (* x (* x (* (* x x) 0.5))) E)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-5) {
		tmp = fma(x, (x * ((double) M_E)), ((double) M_E)) / (((double) M_E) * ((double) M_E));
	} else {
		tmp = (x * (x * ((x * x) * 0.5))) / ((double) M_E);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-5)
		tmp = Float64(fma(x, Float64(x * exp(1)), exp(1)) / Float64(exp(1) * exp(1)));
	else
		tmp = Float64(Float64(x * Float64(x * Float64(Float64(x * x) * 0.5))) / exp(1));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-5], N[(N[(x * N[(x * E), $MachinePrecision] + E), $MachinePrecision] / N[(E * E), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000024e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   4. 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

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. rec-exp N/A

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

      6. lower-/.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.9%

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

   9. Applied rewrites 98.9%

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

   if 5.00000000000000024e-5 < (*.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 81.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot {x}^{4}}{\mathsf{E}\left(\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 81.9%

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

Alternative 11: 87.5% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-5)
   (/ (fma x (* x E) E) (* E E))
   (* x (/ (* x (* x (* x 0.5))) E))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-5) {
		tmp = fma(x, (x * ((double) M_E)), ((double) M_E)) / (((double) M_E) * ((double) M_E));
	} else {
		tmp = x * ((x * (x * (x * 0.5))) / ((double) M_E));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-5)
		tmp = Float64(fma(x, Float64(x * exp(1)), exp(1)) / Float64(exp(1) * exp(1)));
	else
		tmp = Float64(x * Float64(Float64(x * Float64(x * Float64(x * 0.5))) / exp(1)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-5], N[(N[(x * N[(x * E), $MachinePrecision] + E), $MachinePrecision] / N[(E * E), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000024e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   4. 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

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. rec-exp N/A

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

      6. lower-/.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.9%

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

   9. Applied rewrites 98.9%

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

   if 5.00000000000000024e-5 < (*.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 81.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.3%

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

Alternative 12: 87.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
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 92.2%

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

7. Applied rewrites 92.2%

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

9. Applied rewrites 92.2%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \frac{1}{2}\right), x \cdot x, \mathsf{fma}\left(x, x, 1\right)\right)}{\mathsf{E}\left(\right)} \]
11. Step-by-step derivation

12. Applied rewrites 89.9%

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

Alternative 13: 87.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
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 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. rec-exp N/A

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

   3. e-exp-1 N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. rec-exp N/A

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

   7. e-exp-1 N/A

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

   8. distribute-lft-in N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. rec-exp N/A

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

   12. e-exp-1 N/A

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

   13. associate-*r* N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

7. Applied rewrites 89.9%

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

Alternative 14: 75.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -2000000000:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -2000000000.0) (* x (/ x E)) (/ 1.0 E)))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -2000000000.0) {
		tmp = x * (x / ((double) M_E));
	} else {
		tmp = 1.0 / ((double) M_E);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -2000000000.0) {
		tmp = x * (x / Math.E);
	} else {
		tmp = 1.0 / Math.E;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - (x * x)) <= -2000000000.0:
		tmp = x * (x / math.e)
	else:
		tmp = 1.0 / math.e
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -2000000000.0)
		tmp = Float64(x * Float64(x / exp(1)));
	else
		tmp = Float64(1.0 / exp(1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - (x * x)) <= -2000000000.0)
		tmp = x * (x / 2.71828182845904523536);
	else
		tmp = 1.0 / 2.71828182845904523536;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -2000000000.0], N[(x * N[(x / E), $MachinePrecision]), $MachinePrecision], N[(1.0 / E), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -2e9

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   4. 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

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. rec-exp N/A

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

      6. lower-/.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.9%

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

   10. Applied rewrites 56.9%

       \[\leadsto \frac{x}{e} \cdot x \]

   if -2e9 < (-.f64 #s(literal 1 binary64) (*.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

      \[\leadsto \color{blue}{e^{-1}} \]
   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 97.5

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

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{\frac{1}{e}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -2000000000:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, x \cdot e, e\right)}{e \cdot e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fma x (* x E) E) (* E E)))

C

double code(double x) {
	return fma(x, (x * ((double) M_E)), ((double) M_E)) / (((double) M_E) * ((double) M_E));
}

Julia

function code(x)
	return Float64(fma(x, Float64(x * exp(1)), exp(1)) / Float64(exp(1) * exp(1)))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
4. 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

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. rec-exp N/A

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

   6. lower-/.f64 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 75.8

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

5. Applied rewrites 75.8%

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

7. Applied rewrites 75.8%

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

9. Applied rewrites 75.8%

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

Alternative 16: 75.7% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
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 100.0

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

4. Applied rewrites 100.0%

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

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. metadata-eval N/A

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

   3. rec-exp N/A

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

   4. e-exp-1 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-E.f64 75.8

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

7. Applied rewrites 75.8%

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

Alternative 17: 50.8% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. 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 45.5

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

5. Applied rewrites 45.5%

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

Reproduce

herbie shell --seed 2024226 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))