exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 12.2s

Alternatives: 14

Speedup: 1.0×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return math.exp((x * x)) / math.e

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.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. distribute-neg-in N/A

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

   4. metadata-eval N/A

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

   5. remove-double-neg N/A

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

   6. exp-sum N/A

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

   7. metadata-eval N/A

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

   8. rec-exp N/A

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

   9. associate-/r/ N/A

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

   10. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. exp-1-e N/A

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

   14. lower-E.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 92.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.5\right), x\right), 1\right)} \]
   6. 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. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. frac-2neg N/A

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

      13. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
   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 83.7%

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

   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 83.7%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\frac{x \cdot x}{e} \cdot \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

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

Alternative 3: 92.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.5\right), x\right), 1\right)} \]
   6. 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. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. frac-2neg N/A

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

      13. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]
   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 83.7%

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

   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. cube-prod N/A

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

      6. unpow2 N/A

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

      7. cube-unmult N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. associate-*l/ N/A

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

      17. associate-*r/ N/A

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

      18. associate-*r/ N/A

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

      19. associate-*r* N/A

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

   8. Simplified 83.7%

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

4. Final simplification 92.2%

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

Alternative 4: 88.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   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 99.7

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

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \mathsf{fma}\left(x, x, 1\right)} \]
   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 99.7

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

   7. Applied egg-rr 99.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Simplified 79.0%

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

   6. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. associate-*r* N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

   8. Simplified 79.0%

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

   9. Taylor expanded in x around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. *-lft-identity N/A

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

       6. associate-*l/ N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. distribute-lft1-in N/A

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

       12. +-commutative N/A

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

       13. associate-*r/ N/A

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

       14. *-rgt-identity N/A

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

       15. lower-/.f64 N/A

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

   11. Simplified 79.0%

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

4. Final simplification 89.8%

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

Alternative 5: 88.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   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 99.7

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

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \mathsf{fma}\left(x, x, 1\right)} \]
   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 99.7

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

   7. Applied egg-rr 99.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Simplified 79.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. lower-/.f64 N/A

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

   8. Simplified 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-E.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 79.0

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

   10. Applied egg-rr 79.0%

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

4. Final simplification 89.8%

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

Alternative 6: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x \cdot x + -1} \leq 0.5:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (exp(((x * x) + -1.0)) <= 0.5) {
		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 (Math.exp(((x * x) + -1.0)) <= 0.5) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = x * (x / Math.E);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.exp(((x * x) + -1.0)) <= 0.5:
		tmp = 1.0 / math.e
	else:
		tmp = x * (x / math.e)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (exp(Float64(Float64(x * x) + -1.0)) <= 0.5)
		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 (exp(((x * x) + -1.0)) <= 0.5)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = x * (x / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1}} \]
   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 99.2

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

   5. Simplified 99.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   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 54.9

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

   5. Simplified 54.9%

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

   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 54.9

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

   8. Simplified 54.9%

      \[\leadsto \color{blue}{\frac{x \cdot x}{e}} \]
   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 54.9

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

   10. Applied egg-rr 54.9%

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

4. Final simplification 78.1%

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

Alternative 7: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing
   3. 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. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

      5. remove-double-neg N/A

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

      6. exp-sum N/A

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

      7. metadata-eval N/A

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

      8. rec-exp N/A

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

      9. associate-/r/ N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. exp-1-e N/A

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

      14. lower-E.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   7. Simplified 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

Alternative 8: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x + -1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 9: 92.4% accurate, 2.5× speedup

Math

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

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

Wolfram

code[x_] := N[(N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $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. 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. distribute-neg-in N/A

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

   4. metadata-eval N/A

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

   5. remove-double-neg N/A

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

   6. exp-sum N/A

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

   7. metadata-eval N/A

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

   8. rec-exp N/A

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

   9. associate-/r/ N/A

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

   10. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. exp-1-e N/A

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

   14. lower-E.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

7. Simplified 92.2%

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

Alternative 10: 92.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $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. 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. distribute-neg-in N/A

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

   4. metadata-eval N/A

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

   5. remove-double-neg N/A

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

   6. exp-sum N/A

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

   7. metadata-eval N/A

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

   8. rec-exp N/A

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

   9. associate-/r/ N/A

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

   10. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. exp-1-e N/A

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

   14. lower-E.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

7. Simplified 92.2%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{6} \cdot {x}^{3}\right)}, x\right), 1\right)}{\mathsf{E}\left(\right)} \]
9. Step-by-step derivation

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 92.1

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

10. Simplified 92.1%

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

Alternative 11: 92.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-+r+ N/A

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

   4. distribute-rgt1-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

5. Simplified 92.2%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot {x}^{5}}, 1\right) \]
7. Step-by-step derivation

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. unpow2 N/A

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

   19. lower-*.f64 91.8

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

8. Simplified 91.8%

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

   1. lift-E.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. lower-/.f64 91.8

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

10. Applied egg-rr 91.8%

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

Alternative 12: 88.9% 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 89.9%

   \[\leadsto \color{blue}{\frac{1}{e} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.5\right), x\right), 1\right)} \]
6. 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. *-rgt-identity N/A

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

   8. *-rgt-identity N/A

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

   9. lift-fma.f64 N/A

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

   10. associate-*l/ N/A

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

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

   13. lower-/.f64 89.9

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

7. Applied egg-rr 89.9%

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

Alternative 13: 76.9% 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 78.4

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

5. Simplified 78.4%

   \[\leadsto \color{blue}{\frac{1}{e} \cdot \mathsf{fma}\left(x, x, 1\right)} \]
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 78.4

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

7. Applied egg-rr 78.4%

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

Alternative 14: 51.5% 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 53.4

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

5. Simplified 53.4%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))