exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 10.8s

Alternatives: 16

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 N/A

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

   2. associate--r- N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{e}, \frac{1}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-11) (fma x (/ x E) (/ 1.0 E)) (exp (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-11) {
		tmp = fma(x, (x / ((double) M_E)), (1.0 / ((double) M_E)));
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-11)
		tmp = fma(x, Float64(x / exp(1)), Float64(1.0 / exp(1)));
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-11], N[(x * N[(x / E), $MachinePrecision] + N[(1.0 / E), $MachinePrecision]), $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000018e-11

   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. *-lowering-*.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. /-lowering-/.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. E-lowering-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. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. un-div-inv N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. E-lowering-E.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. E-lowering-E.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if 5.00000000000000018e-11 < (*.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. *-lowering-*.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: 92.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 0.5\\ t_1 := \mathsf{fma}\left(x, t\_0, x\right)\\ t_2 := x \cdot t\_1\\ \mathbf{if}\;x \cdot x \leq 10^{+143}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_2, -1\right)}{e \cdot \mathsf{fma}\left(x, t\_1, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{t\_0}{e}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) 0.5)) (t_1 (fma x t_0 x)) (t_2 (* x t_1)))
   (if (<= (* x x) 1e+143)
     (/ (fma t_2 t_2 -1.0) (* E (fma x t_1 -1.0)))
     (* x (* x (/ t_0 E))))))

C

double code(double x) {
	double t_0 = (x * x) * 0.5;
	double t_1 = fma(x, t_0, x);
	double t_2 = x * t_1;
	double tmp;
	if ((x * x) <= 1e+143) {
		tmp = fma(t_2, t_2, -1.0) / (((double) M_E) * fma(x, t_1, -1.0));
	} else {
		tmp = x * (x * (t_0 / ((double) M_E)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * 0.5)
	t_1 = fma(x, t_0, x)
	t_2 = Float64(x * t_1)
	tmp = 0.0
	if (Float64(x * x) <= 1e+143)
		tmp = Float64(fma(t_2, t_2, -1.0) / Float64(exp(1) * fma(x, t_1, -1.0)));
	else
		tmp = Float64(x * Float64(x * Float64(t_0 / exp(1))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0 + x), $MachinePrecision]}, Block[{t$95$2 = N[(x * t$95$1), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e+143], N[(N[(t$95$2 * t$95$2 + -1.0), $MachinePrecision] / N[(E * N[(x * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(t$95$0 / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e143

   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. *-lowering-*.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 80.0%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. frac-times N/A

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

      4. *-rgt-identity N/A

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

      5. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 92.1%

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

   if 1e143 < (*.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. *-lowering-*.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%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{x}^{3}}{\mathsf{E}\left(\right)}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. metadata-eval N/A

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

       7. associate-*r/ N/A

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

       8. associate-*l* N/A

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

       9. *-lowering-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. associate-*r/ N/A

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

       13. *-commutative N/A

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

       14. /-lowering-/.f64 N/A

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

       15. *-commutative N/A

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

       16. *-lowering-*.f64 N/A

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 N/A

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

       19. E-lowering-E.f64 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+143}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), -1\right)}{e \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{\left(x \cdot x\right) \cdot 0.5}{e}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-+r+ N/A

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

   4. distribute-rgt1-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

5. Simplified 93.3%

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

   1. distribute-rgt-in N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. un-div-inv N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

7. Applied egg-rr 93.4%

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

Alternative 5: 91.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-+r+ N/A

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

   4. distribute-rgt1-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

5. Simplified 93.3%

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

   1. distribute-rgt-in N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. un-div-inv N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

7. Applied egg-rr 93.4%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 93.4

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

10. Simplified 93.4%

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

Alternative 6: 91.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{e}, \frac{1}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right)}{e}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-11)
   (fma x (/ x E) (/ 1.0 E))
   (* (* x x) (* x (* x (/ (fma x (* x 0.16666666666666666) 0.5) E))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-11) {
		tmp = fma(x, (x / ((double) M_E)), (1.0 / ((double) M_E)));
	} else {
		tmp = (x * x) * (x * (x * (fma(x, (x * 0.16666666666666666), 0.5) / ((double) M_E))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-11)
		tmp = fma(x, Float64(x / exp(1)), Float64(1.0 / exp(1)));
	else
		tmp = Float64(Float64(x * x) * Float64(x * Float64(x * Float64(fma(x, Float64(x * 0.16666666666666666), 0.5) / exp(1)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-11], N[(x * N[(x / E), $MachinePrecision] + N[(1.0 / E), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000018e-11

   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. *-lowering-*.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. /-lowering-/.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. E-lowering-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. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. un-div-inv N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. E-lowering-E.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. E-lowering-E.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if 5.00000000000000018e-11 < (*.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 87.9%

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

   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 87.9%

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

Alternative 7: 91.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{e}, \frac{1}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 \cdot \frac{x \cdot x}{e}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-11)
   (fma x (/ x E) (/ 1.0 E))
   (* (* (* x x) (* x x)) (* 0.16666666666666666 (/ (* x x) E)))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-11) {
		tmp = fma(x, (x / ((double) M_E)), (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 (Float64(x * x) <= 5e-11)
		tmp = fma(x, Float64(x / exp(1)), Float64(1.0 / exp(1)));
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(0.16666666666666666 * Float64(Float64(x * x) / exp(1))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-11], N[(x * N[(x / E), $MachinePrecision] + N[(1.0 / E), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 * N[(N[(x * x), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000018e-11

   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. *-lowering-*.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. /-lowering-/.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. E-lowering-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. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. un-div-inv N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. E-lowering-E.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. E-lowering-E.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if 5.00000000000000018e-11 < (*.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 87.9%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. cube-prod N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*r* N/A

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

      15. associate-*r/ N/A

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

      16. associate-*r/ N/A

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

      17. associate-*r/ N/A

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

      18. associate-*r* N/A

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

   8. Simplified 87.9%

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

Alternative 8: 87.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -100000:\\ \;\;\;\;x \cdot \left(\frac{x}{e} \cdot \mathsf{fma}\left(x, x \cdot 0.5, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{e}, \frac{1}{e}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -100000.0)
   (* x (* (/ x E) (fma x (* x 0.5) 1.0)))
   (fma x (/ x E) (/ 1.0 E))))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -100000.0) {
		tmp = x * ((x / ((double) M_E)) * fma(x, (x * 0.5), 1.0));
	} else {
		tmp = fma(x, (x / ((double) M_E)), (1.0 / ((double) M_E)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -100000.0)
		tmp = Float64(x * Float64(Float64(x / exp(1)) * fma(x, Float64(x * 0.5), 1.0)));
	else
		tmp = fma(x, Float64(x / exp(1)), Float64(1.0 / exp(1)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -100000.0], N[(x * N[(N[(x / E), $MachinePrecision] * N[(x * N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x / E), $MachinePrecision] + N[(1.0 / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -1e5

   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. *-lowering-*.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 79.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 79.1%

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

   if -1e5 < (-.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. *-lowering-*.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. /-lowering-/.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. E-lowering-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. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. un-div-inv N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. E-lowering-E.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. E-lowering-E.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -100000:\\ \;\;\;\;x \cdot \left(\frac{x}{e} \cdot \mathsf{fma}\left(x, x \cdot 0.5, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{e}, \frac{1}{e}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -100000:\\ \;\;\;\;x \cdot \left(x \cdot \frac{\left(x \cdot x\right) \cdot 0.5}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{e}, \frac{1}{e}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -100000.0)
   (* x (* x (/ (* (* x x) 0.5) E)))
   (fma x (/ x E) (/ 1.0 E))))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -100000.0) {
		tmp = x * (x * (((x * x) * 0.5) / ((double) M_E)));
	} else {
		tmp = fma(x, (x / ((double) M_E)), (1.0 / ((double) M_E)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -100000.0)
		tmp = Float64(x * Float64(x * Float64(Float64(Float64(x * x) * 0.5) / exp(1))));
	else
		tmp = fma(x, Float64(x / exp(1)), Float64(1.0 / exp(1)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -100000.0], N[(x * N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x / E), $MachinePrecision] + N[(1.0 / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -1e5

   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. *-lowering-*.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 79.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 79.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{x}^{3}}{\mathsf{E}\left(\right)}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. metadata-eval N/A

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

       7. associate-*r/ N/A

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

       8. associate-*l* N/A

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

       9. *-lowering-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. associate-*r/ N/A

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

       13. *-commutative N/A

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

       14. /-lowering-/.f64 N/A

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

       15. *-commutative N/A

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

       16. *-lowering-*.f64 N/A

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 N/A

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

       19. E-lowering-E.f64 79.1

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

   11. Simplified 79.1%

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

   if -1e5 < (-.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. *-lowering-*.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. /-lowering-/.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. E-lowering-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. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. un-div-inv N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. E-lowering-E.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. E-lowering-E.f64 100.0

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

   7. Applied egg-rr 100.0%

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

Alternative 10: 91.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-+r+ N/A

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

   4. distribute-rgt1-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

5. Simplified 93.3%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. associate-*l* N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. E-lowering-E.f64 93.3

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

7. Applied egg-rr 93.3%

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

Alternative 11: 87.5% 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. *-lowering-*.f64 N/A

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

5. Simplified 88.5%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. E-lowering-E.f64 88.5

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

7. Applied egg-rr 88.5%

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

Alternative 12: 75.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -100000:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -100000.0) (* x (/ x E)) (/ 1.0 E)))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -100000.0) {
		tmp = x * (x / ((double) M_E));
	} else {
		tmp = 1.0 / ((double) M_E);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -100000.0) {
		tmp = x * (x / Math.E);
	} else {
		tmp = 1.0 / Math.E;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - (x * x)) <= -100000.0:
		tmp = x * (x / math.e)
	else:
		tmp = 1.0 / math.e
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -100000.0)
		tmp = Float64(x * Float64(x / exp(1)));
	else
		tmp = Float64(1.0 / exp(1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - (x * x)) <= -100000.0)
		tmp = x * (x / 2.71828182845904523536);
	else
		tmp = 1.0 / 2.71828182845904523536;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -100000.0], N[(x * N[(x / E), $MachinePrecision]), $MachinePrecision], N[(1.0 / E), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -1e5

   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. *-lowering-*.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. /-lowering-/.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. E-lowering-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. accelerator-lowering-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. /-lowering-/.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. *-lowering-*.f64 N/A

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

      4. E-lowering-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. associate-*l/ N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. E-lowering-E.f64 54.9

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

   10. Applied egg-rr 54.9%

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

   if -1e5 < (-.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. /-lowering-/.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. E-lowering-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. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -100000:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. distribute-rgt1-in N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.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. /-lowering-/.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. E-lowering-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. accelerator-lowering-fma.f64 75.3

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

5. Simplified 75.3%

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

   1. distribute-rgt-in N/A

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

   2. associate-*l* N/A

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

   3. un-div-inv N/A

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

   4. div-inv N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. E-lowering-E.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. E-lowering-E.f64 75.3

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

7. Applied egg-rr 75.3%

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

Alternative 14: 75.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. distribute-rgt1-in N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.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. /-lowering-/.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. E-lowering-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. accelerator-lowering-fma.f64 75.3

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

5. Simplified 75.3%

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

   1. distribute-rgt-in N/A

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

   2. un-div-inv N/A

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

   3. div-inv N/A

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

   4. frac-add N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. E-lowering-E.f64 N/A

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

   10. E-lowering-E.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. E-lowering-E.f64 N/A

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

   13. E-lowering-E.f64 75.3

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

7. Applied egg-rr 75.3%

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

Alternative 15: 75.8% 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. *-lowering-*.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. /-lowering-/.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. E-lowering-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. accelerator-lowering-fma.f64 75.3

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

5. Simplified 75.3%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. E-lowering-E.f64 75.3

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

7. Applied egg-rr 75.3%

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

Alternative 16: 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. /-lowering-/.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. E-lowering-E.f64 46.7

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

5. Simplified 46.7%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))