exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 9.8s

Alternatives: 13

Speedup: 1.0×

• 50.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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 2: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{e} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.5\right), x\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-7)
   (* (/ 1.0 E) (fma x (fma x (* x (* x 0.5)) x) 1.0))
   (exp (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-7) {
		tmp = (1.0 / ((double) M_E)) * fma(x, fma(x, (x * (x * 0.5)), x), 1.0);
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-7)
		tmp = Float64(Float64(1.0 / exp(1)) * fma(x, fma(x, Float64(x * Float64(x * 0.5)), x), 1.0));
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-7], N[(N[(1.0 / E), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999977e-7

   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. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r* N/A

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

      4. *-lft-identity N/A

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

      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 100.0%

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

   if 4.99999999999999977e-7 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

Alternative 3: 91.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{e} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.5\right), x\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-7)
   (* (/ 1.0 E) (fma x (fma x (* x (* x 0.5)) x) 1.0))
   (* x (* (* x (* x x)) (fma x (* x 0.16666666666666666) 0.5)))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-7) {
		tmp = (1.0 / ((double) M_E)) * fma(x, fma(x, (x * (x * 0.5)), x), 1.0);
	} else {
		tmp = x * ((x * (x * x)) * fma(x, (x * 0.16666666666666666), 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-7)
		tmp = Float64(Float64(1.0 / exp(1)) * fma(x, fma(x, Float64(x * Float64(x * 0.5)), x), 1.0));
	else
		tmp = Float64(x * Float64(Float64(x * Float64(x * x)) * fma(x, Float64(x * 0.16666666666666666), 0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-7], N[(N[(1.0 / E), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999977e-7

   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. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r* N/A

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

      4. *-lft-identity N/A

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

      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 100.0%

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

   if 4.99999999999999977e-7 < (*.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%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. cube-prod N/A

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

       4. unpow2 N/A

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

       5. unpow3 N/A

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

       6. pow-sqr N/A

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

       7. metadata-eval N/A

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

       8. associate-*r* N/A

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

       9. metadata-eval N/A

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

       10. pow-sqr N/A

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

       11. +-commutative N/A

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

       12. distribute-rgt-in N/A

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

       13. associate-*l* N/A

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

       14. lft-mult-inverse N/A

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

       15. metadata-eval N/A

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

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

       18. associate-*l* N/A

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

       19. lower-*.f64 N/A

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

   11. Simplified 88.1%

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

Alternative 4: 92.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-+r+ N/A

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

   4. distribute-rgt1-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

5. Simplified 94.1%

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

Alternative 5: 91.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{e} \cdot \mathsf{fma}\left(x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-7)
   (* (/ 1.0 E) (fma x x 1.0))
   (* x (* (* x (* x x)) (fma x (* x 0.16666666666666666) 0.5)))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-7) {
		tmp = (1.0 / ((double) M_E)) * fma(x, x, 1.0);
	} else {
		tmp = x * ((x * (x * x)) * fma(x, (x * 0.16666666666666666), 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-7)
		tmp = Float64(Float64(1.0 / exp(1)) * fma(x, x, 1.0));
	else
		tmp = Float64(x * Float64(Float64(x * Float64(x * x)) * fma(x, Float64(x * 0.16666666666666666), 0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-7], N[(N[(1.0 / E), $MachinePrecision] * N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999977e-7

   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 99.9

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

   5. Simplified 99.9%

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

   if 4.99999999999999977e-7 < (*.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%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. cube-prod N/A

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

       4. unpow2 N/A

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

       5. unpow3 N/A

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

       6. pow-sqr N/A

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

       7. metadata-eval N/A

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

       8. associate-*r* N/A

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

       9. metadata-eval N/A

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

       10. pow-sqr N/A

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

       11. +-commutative N/A

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

       12. distribute-rgt-in N/A

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

       13. associate-*l* N/A

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

       14. lft-mult-inverse N/A

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

       15. metadata-eval N/A

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

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

       18. associate-*l* N/A

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

       19. lower-*.f64 N/A

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

   11. Simplified 88.1%

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

Alternative 6: 91.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{e} \cdot \mathsf{fma}\left(x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-7)
   (* (/ 1.0 E) (fma x x 1.0))
   (* 0.16666666666666666 (* (* x x) (* (* x x) (* x x))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-7) {
		tmp = (1.0 / ((double) M_E)) * fma(x, x, 1.0);
	} else {
		tmp = 0.16666666666666666 * ((x * x) * ((x * x) * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-7)
		tmp = Float64(Float64(1.0 / exp(1)) * fma(x, x, 1.0));
	else
		tmp = Float64(0.16666666666666666 * Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-7], N[(N[(1.0 / E), $MachinePrecision] * N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999977e-7

   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 99.9

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

   5. Simplified 99.9%

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

   if 4.99999999999999977e-7 < (*.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%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{6} \cdot {x}^{4}}, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 88.1

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

   11. Simplified 88.1%

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

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{6}} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{6}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{6} \cdot {x}^{\color{blue}{\left(5 + 1\right)}} \]

       3. pow-plus N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left({x}^{5} \cdot x\right)} \]

       4. metadata-eval N/A

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

       5. pow-plus N/A

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

       6. associate-*r* N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. metadata-eval N/A

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

       10. pow-sqr N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 88.1

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

   14. Simplified 88.1%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{e} \cdot \mathsf{fma}\left(x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -200000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e} \cdot \mathsf{fma}\left(x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -200000000.0)
   (* x (fma x (* (* x x) 0.5) x))
   (* (/ 1.0 E) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -200000000.0) {
		tmp = x * fma(x, ((x * x) * 0.5), x);
	} else {
		tmp = (1.0 / ((double) M_E)) * fma(x, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -200000000.0)
		tmp = Float64(x * fma(x, Float64(Float64(x * x) * 0.5), x));
	else
		tmp = Float64(Float64(1.0 / exp(1)) * fma(x, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -200000000.0], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / E), $MachinePrecision] * N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -2e8

   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%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {x}^{3}, x\right)}, 1\right) \]

      12. cube-mult N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 78.6

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

   8. Simplified 78.6%

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

   9. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. lft-mult-inverse N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. distribute-lft-in N/A

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

       14. *-rgt-identity N/A

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

       15. lower-fma.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 78.6

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

   11. Simplified 78.6%

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

   if -2e8 < (-.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 99.9

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

   5. Simplified 99.9%

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

Alternative 8: 87.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -200000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -200000000.0)
   (* x (fma x (* (* x x) 0.5) x))
   (/ 1.0 E)))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -200000000.0) {
		tmp = x * fma(x, ((x * x) * 0.5), x);
	} else {
		tmp = 1.0 / ((double) M_E);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -200000000.0)
		tmp = Float64(x * fma(x, Float64(Float64(x * x) * 0.5), x));
	else
		tmp = Float64(1.0 / exp(1));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -200000000.0], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(1.0 / E), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -2e8

   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%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {x}^{3}, x\right)}, 1\right) \]

      12. cube-mult N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 78.6

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

   8. Simplified 78.6%

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

   9. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. lft-mult-inverse N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. distribute-lft-in N/A

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

       14. *-rgt-identity N/A

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

       15. lower-fma.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 78.6

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

   11. Simplified 78.6%

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

   if -2e8 < (-.f64 #s(literal 1 binary64) (*.f64 x x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. rec-exp N/A

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

      3. lower-/.f64 N/A

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

      4. exp-1-e N/A

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

      5. lower-E.f64 99.3

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

   5. Simplified 99.3%

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

Alternative 9: 87.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -200000000:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -200000000.0) (* x (* x (* (* x x) 0.5))) (/ 1.0 E)))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -200000000.0) {
		tmp = x * (x * ((x * x) * 0.5));
	} else {
		tmp = 1.0 / ((double) M_E);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -200000000.0) {
		tmp = x * (x * ((x * x) * 0.5));
	} else {
		tmp = 1.0 / Math.E;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - (x * x)) <= -200000000.0:
		tmp = x * (x * ((x * x) * 0.5))
	else:
		tmp = 1.0 / math.e
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -200000000.0)
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.5)));
	else
		tmp = Float64(1.0 / exp(1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - (x * x)) <= -200000000.0)
		tmp = x * (x * ((x * x) * 0.5));
	else
		tmp = 1.0 / 2.71828182845904523536;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -200000000.0], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / E), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -2e8

   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%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {x}^{3}, x\right)}, 1\right) \]

      12. cube-mult N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 78.6

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

   8. Simplified 78.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow3 N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 78.6

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

   11. Simplified 78.6%

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

   if -2e8 < (-.f64 #s(literal 1 binary64) (*.f64 x x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. rec-exp N/A

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

      3. lower-/.f64 N/A

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

      4. exp-1-e N/A

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

      5. lower-E.f64 99.3

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

   5. Simplified 99.3%

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

Alternative 10: 76.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 5e-7) (/ 1.0 E) (fma x x 1.0)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-7) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = fma(x, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-7)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = fma(x, x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-7], N[(1.0 / E), $MachinePrecision], N[(x * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999977e-7

   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 99.3

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

   5. Simplified 99.3%

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

   if 4.99999999999999977e-7 < (*.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%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} + 1} \]

      2. unpow2 N/A

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

      3. lower-fma.f64 49.0

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

   8. Simplified 49.0%

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

Alternative 11: 36.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -200000000:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -200000000.0) (* x x) 1.0))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -200000000.0) {
		tmp = x * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((1.0d0 - (x * x)) <= (-200000000.0d0)) then
        tmp = x * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -200000000.0) {
		tmp = x * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - (x * x)) <= -200000000.0:
		tmp = x * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -200000000.0)
		tmp = Float64(x * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - (x * x)) <= -200000000.0)
		tmp = x * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -200000000.0], N[(x * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -2e8

   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%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} + 1} \]

      2. unpow2 N/A

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

      3. lower-fma.f64 49.0

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

   8. Simplified 49.0%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 49.0

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

   11. Simplified 49.0%

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

   if -2e8 < (-.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 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 17.8

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

   5. Simplified 17.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 17.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 36.0% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 58.9

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

5. Simplified 58.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} + 1} \]

   2. unpow2 N/A

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

   3. lower-fma.f64 33.4

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

8. Simplified 33.4%

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

Alternative 13: 10.4% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

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 58.9

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

5. Simplified 58.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 10.5%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))