exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 9.0s

Alternatives: 15

Speedup: 1.0×

• 50.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. lift--.f64 N/A

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

   4. associate--r- N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 92.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 82.6%

      \[\leadsto \frac{x \cdot x}{e} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

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

Alternative 3: 92.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 82.6%

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

4. Final simplification 91.5%

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

Alternative 4: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 0.5\right), 1\right) \cdot x, \frac{1}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 2e-5)
   (fma
    (/ x E)
    (* (fma (* x x) (fma (* 0.16666666666666666 x) x 0.5) 1.0) x)
    (/ 1.0 E))
   (exp (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-5) {
		tmp = fma((x / ((double) M_E)), (fma((x * x), fma((0.16666666666666666 * x), x, 0.5), 1.0) * x), (1.0 / ((double) M_E)));
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-5)
		tmp = fma(Float64(x / exp(1)), Float64(fma(Float64(x * x), fma(Float64(0.16666666666666666 * x), x, 0.5), 1.0) * x), Float64(1.0 / exp(1)));
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-5], N[(N[(x / E), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(0.16666666666666666 * x), $MachinePrecision] * x + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * x), $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) < 2.00000000000000016e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 2.00000000000000016e-5 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 0.5\right), 1\right) \cdot x, \frac{1}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 91.5%

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

7. Applied rewrites 91.5%

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

9. Applied rewrites 91.5%

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

10. Final simplification 91.5%

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

Alternative 6: 91.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.5, 1\right), x \cdot x, 1\right)}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right)}{e} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 2e-5)
   (/ (fma (fma (* x x) 0.5 1.0) (* x x) 1.0) E)
   (* (/ (fma 0.16666666666666666 (* x x) 0.5) E) (* (* (* x x) x) x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-5) {
		tmp = fma(fma((x * x), 0.5, 1.0), (x * x), 1.0) / ((double) M_E);
	} else {
		tmp = (fma(0.16666666666666666, (x * x), 0.5) / ((double) M_E)) * (((x * x) * x) * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-5)
		tmp = Float64(fma(fma(Float64(x * x), 0.5, 1.0), Float64(x * x), 1.0) / exp(1));
	else
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(x * x), 0.5) / exp(1)) * Float64(Float64(Float64(x * x) * x) * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-5], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / E), $MachinePrecision], N[(N[(N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] / E), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000016e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if 2.00000000000000016e-5 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 82.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.5, 1\right), x \cdot x, 1\right)}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right)}{e} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -500:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \frac{x}{e}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{e} - \frac{-1}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -500.0)
   (* (* (fma 0.5 (* x x) 1.0) (/ x E)) x)
   (- (/ (* x x) E) (/ -1.0 E))))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -500.0) {
		tmp = (fma(0.5, (x * x), 1.0) * (x / ((double) M_E))) * x;
	} else {
		tmp = ((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)) <= -500.0)
		tmp = Float64(Float64(fma(0.5, Float64(x * x), 1.0) * Float64(x / exp(1))) * x);
	else
		tmp = Float64(Float64(Float64(x * 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], -500.0], N[(N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / E), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] / 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)) < -500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 76.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 76.6%

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

   10. Applied rewrites 76.6%

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

   12. Applied rewrites 76.6%

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

   if -500 < (-.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. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. rec-exp N/A

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

      7. lower-/.f64 N/A

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

      8. exp-1-e N/A

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

      9. lower-E.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

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

Alternative 8: 88.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -500:\\ \;\;\;\;\left(\frac{0.5}{e} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{e} - \frac{-1}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -500.0)
   (* (* (/ 0.5 E) x) (* (* x x) x))
   (- (/ (* x x) E) (/ -1.0 E))))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -500.0) {
		tmp = ((0.5 / ((double) M_E)) * x) * ((x * x) * x);
	} else {
		tmp = ((x * x) / ((double) M_E)) - (-1.0 / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -500.0) {
		tmp = ((0.5 / Math.E) * x) * ((x * x) * x);
	} else {
		tmp = ((x * x) / Math.E) - (-1.0 / Math.E);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - (x * x)) <= -500.0:
		tmp = ((0.5 / math.e) * x) * ((x * x) * x)
	else:
		tmp = ((x * x) / math.e) - (-1.0 / math.e)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -500.0)
		tmp = Float64(Float64(Float64(0.5 / exp(1)) * x) * Float64(Float64(x * x) * x));
	else
		tmp = Float64(Float64(Float64(x * x) / exp(1)) - Float64(-1.0 / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - (x * x)) <= -500.0)
		tmp = ((0.5 / 2.71828182845904523536) * x) * ((x * x) * x);
	else
		tmp = ((x * x) / 2.71828182845904523536) - (-1.0 / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -500.0], N[(N[(N[(0.5 / E), $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] / 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)) < -500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 76.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 76.6%

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

   if -500 < (-.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. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. rec-exp N/A

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

      7. lower-/.f64 N/A

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

      8. exp-1-e N/A

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

      9. lower-E.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -500:\\ \;\;\;\;\left(\frac{0.5}{e} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{e} - \frac{-1}{e}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 91.5%

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

7. Applied rewrites 91.5%

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

Alternative 10: 88.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(x * 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. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt1-in N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. associate-+r+ N/A

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

   8. +-commutative N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 88.5%

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

7. Applied rewrites 88.5%

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

Alternative 11: 75.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(x * 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

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

   1. distribute-rgt1-in N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 N/A

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

   5. metadata-eval N/A

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

   6. rec-exp N/A

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

   7. lower-/.f64 N/A

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

   8. exp-1-e N/A

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

   9. lower-E.f64 77.6

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

5. Applied rewrites 77.6%

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

7. Applied rewrites 77.6%

   \[\leadsto \frac{x \cdot x}{e} - \color{blue}{\frac{-1}{e}} \]
8. Add Preprocessing

Alternative 12: 75.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. distribute-rgt1-in N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 N/A

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

   5. metadata-eval N/A

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

   6. rec-exp N/A

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

   7. lower-/.f64 N/A

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

   8. exp-1-e N/A

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

   9. lower-E.f64 77.6

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

5. Applied rewrites 77.6%

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

7. Applied rewrites 77.6%

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

Alternative 13: 75.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 2e-5) (/ 1.0 E) (* (/ x E) x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-5) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = (x / ((double) M_E)) * x;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 2e-5) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = (x / Math.E) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 2e-5:
		tmp = 1.0 / math.e
	else:
		tmp = (x / math.e) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-5)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(Float64(x / exp(1)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 2e-5)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = (x / 2.71828182845904523536) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-5], N[(1.0 / E), $MachinePrecision], N[(N[(x / E), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000016e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. rec-exp N/A

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

      3. lower-/.f64 N/A

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

      4. exp-1-e N/A

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

      5. lower-E.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if 2.00000000000000016e-5 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. rec-exp N/A

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

      7. lower-/.f64 N/A

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

      8. exp-1-e N/A

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

      9. lower-E.f64 54.5

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

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.5%

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

   10. Applied rewrites 54.5%

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

Alternative 14: 75.4% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. lift--.f64 N/A

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

   4. associate--r- N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. rec-exp N/A

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

   4. e-exp-1 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-E.f64 77.6

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

7. Applied rewrites 77.6%

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

Alternative 15: 50.3% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. rec-exp N/A

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

   3. lower-/.f64 N/A

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

   4. exp-1-e N/A

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

   5. lower-E.f64 51.7

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

5. Applied rewrites 51.7%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))