exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 12.3s

Alternatives: 19

Speedup: 1.0×

• 48.6% 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, 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-*.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. distribute-neg-in N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.05:\\ \;\;\;\;\frac{1}{\frac{e \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right), x \cdot \left(x \cdot x\right), x\right), x \cdot e, e\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.05)
   (/
    1.0
    (/
     (* E E)
     (fma
      (fma (fma (* x x) 0.16666666666666666 0.5) (* x (* x x)) x)
      (* x E)
      E)))
   (exp (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.05) {
		tmp = 1.0 / ((((double) M_E) * ((double) M_E)) / fma(fma(fma((x * x), 0.16666666666666666, 0.5), (x * (x * x)), x), (x * ((double) M_E)), ((double) M_E)));
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.05)
		tmp = Float64(1.0 / Float64(Float64(exp(1) * exp(1)) / fma(fma(fma(Float64(x * x), 0.16666666666666666, 0.5), Float64(x * Float64(x * x)), x), Float64(x * exp(1)), exp(1))));
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.05], N[(1.0 / N[(N[(E * E), $MachinePrecision] / N[(N[(N[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(x * E), $MachinePrecision] + E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.050000000000000003

   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. Simplified 99.6%

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

      1. lift-E.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-*l/ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

   7. Applied egg-rr 99.7%

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

   9. Applied egg-rr 99.7%

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

   if 0.050000000000000003 < (*.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. Simplified 100.0%

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

Alternative 3: 91.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
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. Simplified 92.0%

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

   1. lift-E.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. associate-*l/ N/A

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

   9. clear-num N/A

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

   10. lower-/.f64 N/A

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

7. Applied egg-rr 92.0%

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

9. Applied egg-rr 92.0%

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

Alternative 4: 91.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -1000:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right)}{e}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), e, e\right)}{e \cdot e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -1000.0)
   (* x (* (* x x) (* x (/ (fma (* x x) 0.16666666666666666 0.5) E))))
   (/ (fma (* x (fma x (* (* x x) 0.5) x)) E E) (* E E))))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -1000.0) {
		tmp = x * ((x * x) * (x * (fma((x * x), 0.16666666666666666, 0.5) / ((double) M_E))));
	} else {
		tmp = fma((x * fma(x, ((x * x) * 0.5), x)), ((double) M_E), ((double) M_E)) / (((double) M_E) * ((double) M_E));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -1000.0)
		tmp = Float64(x * Float64(Float64(x * x) * Float64(x * Float64(fma(Float64(x * x), 0.16666666666666666, 0.5) / exp(1)))));
	else
		tmp = Float64(fma(Float64(x * fma(x, Float64(Float64(x * x) * 0.5), x)), exp(1), exp(1)) / Float64(exp(1) * exp(1)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -1000.0], N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * E + E), $MachinePrecision] / N[(E * E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -1e3

   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. Simplified 82.6%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 82.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-E.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 82.6

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

   10. Applied egg-rr 82.6%

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

   if -1e3 < (-.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. Simplified 99.5%

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

      1. lift-E.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. frac-add N/A

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

      12. lift-E.f64 N/A

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

      13. lift-E.f64 N/A

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

   7. Applied egg-rr 99.5%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -1000:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right)}{e}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), e, e\right)}{e \cdot e}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -1000:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right)}{e}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot e, \mathsf{fma}\left(x \cdot x, 0.5, -1\right), e\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -1000.0)
   (* x (* (* x x) (* x (/ (fma (* x x) 0.16666666666666666 0.5) E))))
   (/ 1.0 (fma (* (* x x) E) (fma (* x x) 0.5 -1.0) E))))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -1000.0) {
		tmp = x * ((x * x) * (x * (fma((x * x), 0.16666666666666666, 0.5) / ((double) M_E))));
	} else {
		tmp = 1.0 / fma(((x * x) * ((double) M_E)), fma((x * x), 0.5, -1.0), ((double) M_E));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -1000.0)
		tmp = Float64(x * Float64(Float64(x * x) * Float64(x * Float64(fma(Float64(x * x), 0.16666666666666666, 0.5) / exp(1)))));
	else
		tmp = Float64(1.0 / fma(Float64(Float64(x * x) * exp(1)), fma(Float64(x * x), 0.5, -1.0), exp(1)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -1000.0], N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] * E), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.5 + -1.0), $MachinePrecision] + E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -1e3

   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. Simplified 82.6%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 82.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-E.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 82.6

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

   10. Applied egg-rr 82.6%

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

   if -1e3 < (-.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. Simplified 99.6%

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

      1. lift-E.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-*l/ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   10. Simplified 99.5%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -1000:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right)}{e}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot e, \mathsf{fma}\left(x \cdot x, 0.5, -1\right), e\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -1000:\\ \;\;\;\;x \cdot \left(x \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right)}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot e, \mathsf{fma}\left(x \cdot x, 0.5, -1\right), e\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -1000.0)
   (* x (* x (/ (* (* x x) (* x (* x 0.16666666666666666))) E)))
   (/ 1.0 (fma (* (* x x) E) (fma (* x x) 0.5 -1.0) E))))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -1000.0) {
		tmp = x * (x * (((x * x) * (x * (x * 0.16666666666666666))) / ((double) M_E)));
	} else {
		tmp = 1.0 / fma(((x * x) * ((double) M_E)), fma((x * x), 0.5, -1.0), ((double) M_E));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -1000.0)
		tmp = Float64(x * Float64(x * Float64(Float64(Float64(x * x) * Float64(x * Float64(x * 0.16666666666666666))) / exp(1))));
	else
		tmp = Float64(1.0 / fma(Float64(Float64(x * x) * exp(1)), fma(Float64(x * x), 0.5, -1.0), exp(1)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -1000.0], N[(x * N[(x * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] * E), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.5 + -1.0), $MachinePrecision] + E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -1e3

   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. Simplified 82.6%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{6} \cdot {x}^{\color{blue}{\left(2 \cdot 3\right)}}}{\mathsf{E}\left(\right)} \]

      3. pow-sqr N/A

         \[\leadsto \frac{\frac{1}{6} \cdot \color{blue}{\left({x}^{3} \cdot {x}^{3}\right)}}{\mathsf{E}\left(\right)} \]

      4. cube-prod N/A

         \[\leadsto \frac{\frac{1}{6} \cdot \color{blue}{{\left(x \cdot x\right)}^{3}}}{\mathsf{E}\left(\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{6} \cdot {\color{blue}{\left({x}^{2}\right)}}^{3}}{\mathsf{E}\left(\right)} \]

      6. unpow3 N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{6} \cdot \left({x}^{\color{blue}{4}} \cdot {x}^{2}\right)}{\mathsf{E}\left(\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{6} \cdot {x}^{4}\right) \cdot {x}^{2}}}{\mathsf{E}\left(\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left({x}^{4} \cdot \frac{1}{6}\right)} \cdot {x}^{2}}{\mathsf{E}\left(\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{{x}^{4} \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)}}{\mathsf{E}\left(\right)} \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*r* N/A

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

      15. associate-*r/ N/A

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

      16. associate-*r/ N/A

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

      17. associate-*r/ N/A

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

      18. associate-*r* N/A

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

   8. Simplified 82.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-E.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. *-commutative N/A

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

      16. associate-*r/ N/A

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

   10. Applied egg-rr 82.6%

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

   if -1e3 < (-.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. Simplified 99.6%

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

      1. lift-E.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-*l/ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   10. Simplified 99.5%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -1000:\\ \;\;\;\;x \cdot \left(x \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right)}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot e, \mathsf{fma}\left(x \cdot x, 0.5, -1\right), e\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
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. Simplified 92.0%

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

   1. lift-E.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. distribute-rgt-in N/A

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

7. Applied egg-rr 92.0%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

10. Simplified 92.0%

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

11. Final simplification 92.0%

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

Alternative 8: 91.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
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. Simplified 92.0%

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

Alternative 9: 91.7% accurate, 2.5× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 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. Simplified 92.0%

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

   1. lift-E.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-fma.f64 N/A

      \[\leadsto \frac{-1}{\mathsf{neg}\left(\mathsf{E}\left(\right)\right)} \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{6}, \frac{1}{2}\right), x \cdot \left(x \cdot x\right), x\right), 1\right)} \]

   10. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{-1 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{6}, \frac{1}{2}\right), x \cdot \left(x \cdot x\right), x\right), 1\right)}{\mathsf{neg}\left(\mathsf{E}\left(\right)\right)}} \]

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

7. Applied egg-rr 92.0%

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

Alternative 10: 87.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{e}, \frac{1}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{x}{e} \cdot \mathsf{fma}\left(x, x \cdot 0.5, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.05)
   (fma x (/ x E) (/ 1.0 E))
   (* x (* (/ x E) (fma x (* x 0.5) 1.0)))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.05) {
		tmp = fma(x, (x / ((double) M_E)), (1.0 / ((double) M_E)));
	} else {
		tmp = x * ((x / ((double) M_E)) * fma(x, (x * 0.5), 1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.05)
		tmp = fma(x, Float64(x / exp(1)), Float64(1.0 / exp(1)));
	else
		tmp = Float64(x * Float64(Float64(x / exp(1)) * fma(x, Float64(x * 0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.05], N[(x * N[(x / E), $MachinePrecision] + N[(1.0 / E), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x / E), $MachinePrecision] * N[(x * N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.050000000000000003

   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 99.2

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

   5. Simplified 99.2%

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

      1. lift-E.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. un-div-inv N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.2

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

   7. Applied egg-rr 99.2%

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

   if 0.050000000000000003 < (*.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. Simplified 75.2%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 75.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{e}, \frac{1}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{x}{e} \cdot \mathsf{fma}\left(x, x \cdot 0.5, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{e}, \frac{1}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.05)
   (fma x (/ x E) (/ 1.0 E))
   (* x (* x (* 0.5 (/ (* x x) E))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.05) {
		tmp = fma(x, (x / ((double) M_E)), (1.0 / ((double) M_E)));
	} else {
		tmp = x * (x * (0.5 * ((x * x) / ((double) M_E))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.05)
		tmp = fma(x, Float64(x / exp(1)), Float64(1.0 / exp(1)));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * Float64(Float64(x * x) / exp(1)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.05], N[(x * N[(x / E), $MachinePrecision] + N[(1.0 / E), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(N[(x * x), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.050000000000000003

   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 99.2

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

   5. Simplified 99.2%

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

      1. lift-E.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. un-div-inv N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.2

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

   7. Applied egg-rr 99.2%

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

   if 0.050000000000000003 < (*.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. Simplified 75.2%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-E.f64 75.2

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

   8. Simplified 75.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-E.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 75.2

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

   10. Applied egg-rr 75.2%

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

Alternative 12: 87.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
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. Simplified 88.6%

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

Alternative 13: 87.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5), $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

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
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. Simplified 88.6%

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

   1. lift-E.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. associate-*l/ N/A

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

   9. neg-mul-1 N/A

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

   10. frac-2neg N/A

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

   11. lower-/.f64 88.6

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

7. Applied egg-rr 88.6%

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

8. Final simplification 88.6%

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

Alternative 14: 75.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.05:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 0.05) (/ 1.0 E) (/ (* x x) E)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.05) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = (x * x) / ((double) M_E);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.05) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = (x * x) / Math.E;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.05:
		tmp = 1.0 / math.e
	else:
		tmp = (x * x) / math.e
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.05)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(Float64(x * x) / exp(1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.05)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = (x * x) / 2.71828182845904523536;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.05], N[(1.0 / E), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / E), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.050000000000000003

   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 98.5

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

   5. Simplified 98.5%

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

   if 0.050000000000000003 < (*.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

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   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 48.3

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

   5. Simplified 48.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-E.f64 48.3

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

   8. Simplified 48.3%

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

Alternative 15: 75.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.05:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 0.05) (/ 1.0 E) (* x (/ x E))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.05) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = x * (x / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.05) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = x * (x / Math.E);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.05:
		tmp = 1.0 / math.e
	else:
		tmp = x * (x / math.e)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.05)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(x * Float64(x / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.05)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = x * (x / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.05], N[(1.0 / E), $MachinePrecision], N[(x * N[(x / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.050000000000000003

   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 98.5

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

   5. Simplified 98.5%

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

   if 0.050000000000000003 < (*.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

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   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 48.3

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

   5. Simplified 48.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-E.f64 48.3

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

   8. Simplified 48.3%

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

      1. lift-E.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 47.6

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

   10. Applied egg-rr 47.6%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.05:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 76.3% accurate, 4.0× speedup

Math

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

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(Float64(x * x), exp(1), exp(1)) / Float64(exp(1) * exp(1)))
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * E + 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 76.3

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

5. Simplified 76.3%

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

   1. lift-E.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. lift-/.f64 N/A

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

   6. un-div-inv N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. frac-add N/A

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

   10. lift-E.f64 N/A

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

   11. lift-E.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-fma.f64 N/A

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

   15. lift-E.f64 N/A

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

   16. lift-E.f64 N/A

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

   17. lower-*.f64 76.3

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

7. Applied egg-rr 76.3%

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

Alternative 17: 76.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 / E), $MachinePrecision] * N[(x * x + 1.0), $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 76.3

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

5. Simplified 76.3%

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

Alternative 18: 76.2% 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. 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 76.3

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

5. Simplified 76.3%

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

   1. lift-E.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-fma.f64 N/A

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

   5. associate-*l/ N/A

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

   6. neg-mul-1 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 76.3

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

7. Applied egg-rr 76.3%

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

Alternative 19: 52.0% 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 55.6

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

5. Simplified 55.6%

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

Reproduce

herbie shell --seed 2024212 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))