exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 13.2s

Alternatives: 22

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\left(e^{x\_m + -1}\right)}^{\left(x\_m + 1\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (pow (exp (+ x_m -1.0)) (+ x_m 1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(exp((x_m + -1.0)), (x_m + 1.0));
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.exp((x_m + -1.0)), (x_m + 1.0));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.exp((x_m + -1.0)), (x_m + 1.0))

Julia

x_m = abs(x)
function code(x_m)
	return exp(Float64(x_m + -1.0)) ^ Float64(x_m + 1.0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = exp((x_m + -1.0)) ^ (x_m + 1.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[N[Exp[N[(x$95$m + -1.0), $MachinePrecision]], $MachinePrecision], N[(x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. sub-neg N/A

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

   3. distribute-neg-in N/A

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

   4. metadata-eval N/A

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

   5. remove-double-neg N/A

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

   6. exp-sum N/A

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

   7. metadata-eval N/A

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

   8. rec-exp N/A

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

   9. associate-/r/ N/A

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

   10. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. exp-1-e N/A

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

   14. lower-E.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. e-exp-1 N/A

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

   3. div-exp N/A

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

   4. lift-*.f64 N/A

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

   5. difference-of-sqr-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. pow-exp N/A

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

   4. *-commutative N/A

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

   5. exp-prod N/A

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

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{{\left(e^{x + -1}\right)}^{\left(x + 1\right)}} \]
9. Add Preprocessing

Alternative 2: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{if}\;e^{-1 + x\_m \cdot x\_m} \leq 0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot \mathsf{fma}\left(x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.5, x\_m\right), e, e\right)}{e \cdot e}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(t\_0, t\_0 \cdot 0.004629629629629629, 0.125\right) \cdot 4, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m x_m))))
   (if (<= (exp (+ -1.0 (* x_m x_m))) 0.5)
     (/ (fma (* x_m (fma x_m (* (* x_m x_m) 0.5) x_m)) E E) (* E E))
     (fma
      (* x_m x_m)
      (fma
       (* x_m x_m)
       (* (fma t_0 (* t_0 0.004629629629629629) 0.125) 4.0)
       1.0)
      1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = x_m * (x_m * x_m);
	double tmp;
	if (exp((-1.0 + (x_m * x_m))) <= 0.5) {
		tmp = fma((x_m * fma(x_m, ((x_m * x_m) * 0.5), x_m)), ((double) M_E), ((double) M_E)) / (((double) M_E) * ((double) M_E));
	} else {
		tmp = fma((x_m * x_m), fma((x_m * x_m), (fma(t_0, (t_0 * 0.004629629629629629), 0.125) * 4.0), 1.0), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(x_m * Float64(x_m * x_m))
	tmp = 0.0
	if (exp(Float64(-1.0 + Float64(x_m * x_m))) <= 0.5)
		tmp = Float64(fma(Float64(x_m * fma(x_m, Float64(Float64(x_m * x_m) * 0.5), x_m)), exp(1), exp(1)) / Float64(exp(1) * exp(1)));
	else
		tmp = fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), Float64(fma(t_0, Float64(t_0 * 0.004629629629629629), 0.125) * 4.0), 1.0), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[N[(-1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.5], N[(N[(N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision] * E + E), $MachinePrecision] / N[(E * E), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(t$95$0 * N[(t$95$0 * 0.004629629629629629), $MachinePrecision] + 0.125), $MachinePrecision] * 4.0), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 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. Step-by-step derivation

      1. lift-E.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. distribute-rgt-in N/A

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

      13. lift-/.f64 N/A

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

      14. un-div-inv N/A

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. flip3-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. lower-fma.f64 N/A

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

      7. cube-mult N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. lower-*.f64 N/A

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

      13. cube-mult N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{1}{216}}, {\frac{1}{2}}^{3}\right) \cdot \frac{1}{\left(x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{6}\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \frac{1}{2}\right)}, 1\right), 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{216}, \color{blue}{\frac{1}{8}}\right) \cdot \frac{1}{\left(x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{6}\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \frac{1}{2}\right)}, 1\right), 1\right) \]

   10. Applied rewrites 11.7%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot 0.004629629629629629, 0.125\right) \cdot \frac{1}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.027777777777777776, 0.25\right) - \left(x \cdot x\right) \cdot 0.08333333333333333}}, 1\right), 1\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{216}, \frac{1}{8}\right) \cdot \color{blue}{4}, 1\right), 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 90.8%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot 0.004629629629629629, 0.125\right) \cdot \color{blue}{4}, 1\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-1 + x \cdot x} \leq 0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), e, e\right)}{e \cdot e}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot 0.004629629629629629, 0.125\right) \cdot 4, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (exp (+ -1.0 (* x_m x_m))) 0.5)
   (/ (fma (* x_m (fma x_m (* (* x_m x_m) 0.5) x_m)) E E) (* E E))
   (fma
    (* x_m x_m)
    (fma (* x_m x_m) (fma x_m (* x_m 0.16666666666666666) 0.5) 1.0)
    1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Exp[N[(-1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.5], N[(N[(N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision] * E + E), $MachinePrecision] / N[(E * E), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 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. Step-by-step derivation

      1. lift-E.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. distribute-rgt-in N/A

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

      13. lift-/.f64 N/A

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

      14. un-div-inv N/A

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

4. Final simplification 91.2%

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

Alternative 4: 91.8% accurate, 0.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (exp (+ -1.0 (* x_m x_m))) 0.5)
   (/ (fma x_m (fma x_m (* (* x_m x_m) 0.5) x_m) 1.0) E)
   (fma
    (* x_m x_m)
    (fma (* x_m x_m) (fma x_m (* x_m 0.16666666666666666) 0.5) 1.0)
    1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Exp[N[(-1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.5], N[(N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] + x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / E), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 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. Step-by-step derivation

      1. lift-E.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-*l/ N/A

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

      9. neg-mul-1 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

4. Final simplification 91.2%

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

Alternative 5: 91.8% accurate, 0.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (exp (+ -1.0 (* x_m x_m))) 0.5)
   (/ (fma x_m (fma x_m (* (* x_m x_m) 0.5) x_m) 1.0) E)
   (fma
    (* x_m x_m)
    (* x_m (* x_m (fma (* x_m x_m) 0.16666666666666666 0.5)))
    1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Exp[N[(-1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.5], N[(N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] + x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / E), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 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. Step-by-step derivation

      1. lift-E.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-*l/ N/A

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

      9. neg-mul-1 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in x around inf

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

       1. distribute-rgt-in N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. associate-*l* N/A

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

       6. associate-*l/ N/A

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

       7. *-lft-identity N/A

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

       8. metadata-eval N/A

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

       9. pow-plus N/A

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

       10. associate-*l/ N/A

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

       11. *-lft-identity N/A

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

       12. associate-*l/ N/A

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

       13. unpow3 N/A

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. lft-mult-inverse N/A

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

       17. *-lft-identity N/A

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

       18. unpow2 N/A

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

   11. Applied rewrites 81.8%

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

4. Final simplification 91.2%

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

Alternative 6: 88.1% accurate, 0.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (exp (+ -1.0 (* x_m x_m))) 0.5)
   (/ (fma x_m x_m 1.0) E)
   (* x_m (* x_m (* x_m (* x_m 0.5))))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Exp[N[(-1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.5], N[(N[(x$95$m * x$95$m + 1.0), $MachinePrecision] / E), $MachinePrecision], N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

      5. remove-double-neg N/A

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

      6. exp-sum N/A

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

      7. metadata-eval N/A

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

      8. rec-exp N/A

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

      9. associate-/r/ N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. exp-1-e N/A

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

      14. lower-E.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. e-exp-1 N/A

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

      2. rec-exp N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. e-exp-1 N/A

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

      7. rec-exp N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. rec-exp N/A

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

      14. e-exp-1 N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

      17. lower-/.f64 N/A

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

      18. +-commutative N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-E.f64 99.8

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

   7. Applied rewrites 99.8%

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

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

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 71.8

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

   8. Applied rewrites 71.8%

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

   9. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. unpow2 N/A

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

       8. unpow3 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. unpow3 N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. associate-*r* N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 N/A

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

       20. *-commutative N/A

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

       21. lower-*.f64 71.8

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

   11. Applied rewrites 71.8%

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

4. Final simplification 86.3%

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

Alternative 7: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{x\_m \cdot x\_m}}{e} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (exp (* x_m x_m)) E))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Exp[N[(x$95$m * x$95$m), $MachinePrecision]], $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. sub-neg N/A

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

   3. distribute-neg-in N/A

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

   4. metadata-eval N/A

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

   5. remove-double-neg N/A

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

   6. exp-sum N/A

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

   7. metadata-eval N/A

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

   8. rec-exp N/A

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

   9. associate-/r/ N/A

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

   10. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. exp-1-e N/A

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

   14. lower-E.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 8: 100.0% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (exp (fma x_m x_m -1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Exp[N[(x$95$m * x$95$m + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. distribute-neg-in N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 9: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {e}^{\left(x\_m + -1\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (pow E (+ x_m -1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_E), (x_m + -1.0));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.E, (x_m + -1.0));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.e, (x_m + -1.0))

Julia

x_m = abs(x)
function code(x_m)
	return exp(1) ^ Float64(x_m + -1.0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 2.71828182845904523536 ^ (x_m + -1.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[E, N[(x$95$m + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. sub-neg N/A

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

   3. distribute-neg-in N/A

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

   4. metadata-eval N/A

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

   5. remove-double-neg N/A

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

   6. exp-sum N/A

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

   7. metadata-eval N/A

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

   8. rec-exp N/A

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

   9. associate-/r/ N/A

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

   10. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. exp-1-e N/A

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

   14. lower-E.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. e-exp-1 N/A

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

   3. div-exp N/A

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

   4. lift-*.f64 N/A

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

   5. difference-of-sqr-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. exp-1-e N/A

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

   2. lower-E.f64 74.4

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

9. Applied rewrites 74.4%

   \[\leadsto {\color{blue}{e}}^{\left(x + -1\right)} \]
10. Add Preprocessing

Alternative 10: 95.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \cdot x\_m \leq 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot \mathsf{fma}\left(x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.5, x\_m\right), e, e\right)}{e \cdot e}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(t\_0, t\_0 \cdot 0.004629629629629629, 0.125\right) \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 1.3333333333333333, 4\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m x_m))))
   (if (<= (* x_m x_m) 1e-6)
     (/ (fma (* x_m (fma x_m (* (* x_m x_m) 0.5) x_m)) E E) (* E E))
     (fma
      (* x_m x_m)
      (fma
       (* x_m x_m)
       (*
        (fma t_0 (* t_0 0.004629629629629629) 0.125)
        (fma x_m (* x_m 1.3333333333333333) 4.0))
       1.0)
      1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = x_m * (x_m * x_m);
	double tmp;
	if ((x_m * x_m) <= 1e-6) {
		tmp = fma((x_m * fma(x_m, ((x_m * x_m) * 0.5), x_m)), ((double) M_E), ((double) M_E)) / (((double) M_E) * ((double) M_E));
	} else {
		tmp = fma((x_m * x_m), fma((x_m * x_m), (fma(t_0, (t_0 * 0.004629629629629629), 0.125) * fma(x_m, (x_m * 1.3333333333333333), 4.0)), 1.0), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(x_m * Float64(x_m * x_m))
	tmp = 0.0
	if (Float64(x_m * x_m) <= 1e-6)
		tmp = Float64(fma(Float64(x_m * fma(x_m, Float64(Float64(x_m * x_m) * 0.5), x_m)), exp(1), exp(1)) / Float64(exp(1) * exp(1)));
	else
		tmp = fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), Float64(fma(t_0, Float64(t_0 * 0.004629629629629629), 0.125) * fma(x_m, Float64(x_m * 1.3333333333333333), 4.0)), 1.0), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 1e-6], N[(N[(N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision] * E + E), $MachinePrecision] / N[(E * E), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(t$95$0 * N[(t$95$0 * 0.004629629629629629), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 1.3333333333333333), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.99999999999999955e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 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. Step-by-step derivation

      1. lift-E.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. distribute-rgt-in N/A

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

      13. lift-/.f64 N/A

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

      14. un-div-inv N/A

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

   7. Applied rewrites 100.0%

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

   if 9.99999999999999955e-7 < (*.f64 x x)

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. flip3-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. lower-fma.f64 N/A

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

      7. cube-mult N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. lower-*.f64 N/A

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

      13. cube-mult N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{1}{216}}, {\frac{1}{2}}^{3}\right) \cdot \frac{1}{\left(x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{6}\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \frac{1}{2}\right)}, 1\right), 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{216}, \color{blue}{\frac{1}{8}}\right) \cdot \frac{1}{\left(x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{6}\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \frac{1}{2}\right)}, 1\right), 1\right) \]

   10. Applied rewrites 11.7%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot 0.004629629629629629, 0.125\right) \cdot \frac{1}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.027777777777777776, 0.25\right) - \left(x \cdot x\right) \cdot 0.08333333333333333}}, 1\right), 1\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{216}, \frac{1}{8}\right) \cdot \color{blue}{\left(4 + \frac{4}{3} \cdot {x}^{2}\right)}, 1\right), 1\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{216}, \frac{1}{8}\right) \cdot \color{blue}{\left(\frac{4}{3} \cdot {x}^{2} + 4\right)}, 1\right), 1\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{216}, \frac{1}{8}\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{4}{3}} + 4\right), 1\right), 1\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{216}, \frac{1}{8}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{4}{3} + 4\right), 1\right), 1\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{216}, \frac{1}{8}\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{4}{3}\right)} + 4\right), 1\right), 1\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{216}, \frac{1}{8}\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{4}{3}, 4\right)}, 1\right), 1\right) \]

       6. lower-*.f64 92.3

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot 0.004629629629629629, 0.125\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 1.3333333333333333}, 4\right), 1\right), 1\right) \]

   13. Applied rewrites 92.3%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot 0.004629629629629629, 0.125\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot 1.3333333333333333, 4\right)}, 1\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 91.9% accurate, 2.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (/ 1.0 E)
  (fma
   x_m
   (fma (fma (* x_m x_m) 0.16666666666666666 0.5) (* x_m (* x_m x_m)) x_m)
   1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 / E), $MachinePrecision] * N[(x$95$m * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[e^{-\left(1 - x \cdot x\right)} \]
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. Applied rewrites 91.2%

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

Alternative 12: 91.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;1 - x\_m \cdot x\_m \leq -500:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, x\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.16666666666666666, 0.5\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{e}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (* x_m x_m)) -500.0)
   (fma
    (* x_m x_m)
    (* x_m (* x_m (fma (* x_m x_m) 0.16666666666666666 0.5)))
    1.0)
   (/ 1.0 (/ E (fma x_m x_m 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((1.0 - (x_m * x_m)) <= -500.0) {
		tmp = fma((x_m * x_m), (x_m * (x_m * fma((x_m * x_m), 0.16666666666666666, 0.5))), 1.0);
	} else {
		tmp = 1.0 / (((double) M_E) / fma(x_m, x_m, 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(1.0 - Float64(x_m * x_m)) <= -500.0)
		tmp = fma(Float64(x_m * x_m), Float64(x_m * Float64(x_m * fma(Float64(x_m * x_m), 0.16666666666666666, 0.5))), 1.0);
	else
		tmp = Float64(1.0 / Float64(exp(1) / fma(x_m, x_m, 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -500.0], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(E / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -500

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in x around inf

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

       1. distribute-rgt-in N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. associate-*l* N/A

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

       6. associate-*l/ N/A

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

       7. *-lft-identity N/A

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

       8. metadata-eval N/A

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

       9. pow-plus N/A

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

       10. associate-*l/ N/A

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

       11. *-lft-identity N/A

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

       12. associate-*l/ N/A

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

       13. unpow3 N/A

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. lft-mult-inverse N/A

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

       17. *-lft-identity N/A

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

       18. unpow2 N/A

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

   11. Applied rewrites 81.8%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

      5. remove-double-neg N/A

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

      6. exp-sum N/A

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

      7. metadata-eval N/A

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

      8. rec-exp N/A

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

      9. associate-/r/ N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. exp-1-e N/A

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

      14. lower-E.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. e-exp-1 N/A

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

      2. rec-exp N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. e-exp-1 N/A

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

      7. rec-exp N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. rec-exp N/A

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

      14. e-exp-1 N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

      17. lower-/.f64 N/A

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

      18. +-commutative N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-E.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-E.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

Alternative 13: 91.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;1 - x\_m \cdot x\_m \leq -500:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{e}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (* x_m x_m)) -500.0)
   (fma (* x_m x_m) (* (* x_m x_m) (* (* x_m x_m) 0.16666666666666666)) 1.0)
   (/ 1.0 (/ E (fma x_m x_m 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((1.0 - (x_m * x_m)) <= -500.0) {
		tmp = fma((x_m * x_m), ((x_m * x_m) * ((x_m * x_m) * 0.16666666666666666)), 1.0);
	} else {
		tmp = 1.0 / (((double) M_E) / fma(x_m, x_m, 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(1.0 - Float64(x_m * x_m)) <= -500.0)
		tmp = fma(Float64(x_m * x_m), Float64(Float64(x_m * x_m) * Float64(Float64(x_m * x_m) * 0.16666666666666666)), 1.0);
	else
		tmp = Float64(1.0 / Float64(exp(1) / fma(x_m, x_m, 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -500.0], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(E / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -500

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 81.8

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

   11. Applied rewrites 81.8%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

      5. remove-double-neg N/A

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

      6. exp-sum N/A

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

      7. metadata-eval N/A

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

      8. rec-exp N/A

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

      9. associate-/r/ N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. exp-1-e N/A

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

      14. lower-E.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. e-exp-1 N/A

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

      2. rec-exp N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. e-exp-1 N/A

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

      7. rec-exp N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. rec-exp N/A

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

      14. e-exp-1 N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

      17. lower-/.f64 N/A

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

      18. +-commutative N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-E.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-E.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

Alternative 14: 91.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{if}\;1 - x\_m \cdot x\_m \leq -500:\\ \;\;\;\;0.16666666666666666 \cdot \left(t\_0 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{e}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m x_m))))
   (if (<= (- 1.0 (* x_m x_m)) -500.0)
     (* 0.16666666666666666 (* t_0 t_0))
     (/ 1.0 (/ E (fma x_m x_m 1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = x_m * (x_m * x_m);
	double tmp;
	if ((1.0 - (x_m * x_m)) <= -500.0) {
		tmp = 0.16666666666666666 * (t_0 * t_0);
	} else {
		tmp = 1.0 / (((double) M_E) / fma(x_m, x_m, 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(x_m * Float64(x_m * x_m))
	tmp = 0.0
	if (Float64(1.0 - Float64(x_m * x_m)) <= -500.0)
		tmp = Float64(0.16666666666666666 * Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / Float64(exp(1) / fma(x_m, x_m, 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -500.0], N[(0.16666666666666666 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(E / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -500

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. cube-mult N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 81.8

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

   11. Applied rewrites 81.8%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

      5. remove-double-neg N/A

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

      6. exp-sum N/A

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

      7. metadata-eval N/A

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

      8. rec-exp N/A

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

      9. associate-/r/ N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. exp-1-e N/A

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

      14. lower-E.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. e-exp-1 N/A

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

      2. rec-exp N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. e-exp-1 N/A

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

      7. rec-exp N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. rec-exp N/A

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

      14. e-exp-1 N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

      17. lower-/.f64 N/A

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

      18. +-commutative N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-E.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-E.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

Alternative 15: 88.1% accurate, 2.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (* x_m x_m)) -500.0)
   (fma x_m (* x_m (fma (* x_m x_m) 0.5 1.0)) 1.0)
   (/ 1.0 (/ E (fma x_m x_m 1.0)))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -500.0], N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(E / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -500

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 71.8

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

   8. Applied rewrites 71.8%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

      5. remove-double-neg N/A

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

      6. exp-sum N/A

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

      7. metadata-eval N/A

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

      8. rec-exp N/A

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

      9. associate-/r/ N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. exp-1-e N/A

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

      14. lower-E.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. e-exp-1 N/A

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

      2. rec-exp N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. e-exp-1 N/A

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

      7. rec-exp N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. rec-exp N/A

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

      14. e-exp-1 N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

      17. lower-/.f64 N/A

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

      18. +-commutative N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-E.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-E.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

Alternative 16: 88.1% accurate, 3.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (* x_m x_m)) -500.0)
   (fma x_m (* x_m (fma (* x_m x_m) 0.5 1.0)) 1.0)
   (/ (fma x_m x_m 1.0) E)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -500.0], N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x$95$m * x$95$m + 1.0), $MachinePrecision] / E), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -500

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

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

   5. Applied rewrites 99.3%

      \[\leadsto e^{\color{blue}{x \cdot x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + 1} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + 1 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)} + 1 \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right), 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}, 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right), 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right), 1\right) \]

      10. lower-*.f64 71.8

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right), 1\right) \]

   8. Applied rewrites 71.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right), 1\right)} \]

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(\left(1 - \color{blue}{x \cdot x}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto e^{\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(x \cdot x\right)\right)\right)}\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto e^{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right)}} \]

      4. metadata-eval N/A

         \[\leadsto e^{\color{blue}{-1} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right)} \]

      5. remove-double-neg N/A

         \[\leadsto e^{-1 + \color{blue}{x \cdot x}} \]

      6. exp-sum N/A

         \[\leadsto \color{blue}{e^{-1} \cdot e^{x \cdot x}} \]

      7. metadata-eval N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(1\right)}} \cdot e^{x \cdot x} \]

      8. rec-exp N/A

         \[\leadsto \color{blue}{\frac{1}{e^{1}}} \cdot e^{x \cdot x} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{1}}{e^{x \cdot x}}}} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{x \cdot x}}}{e^{1}} \]

      13. exp-1-e N/A

          \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

      14. lower-E.f64 100.0

          \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{e}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{\mathsf{E}\left(\right)} + \frac{{x}^{2}}{\mathsf{E}\left(\right)}} \]
   6. Step-by-step derivation

      1. e-exp-1 N/A

         \[\leadsto \frac{1}{\color{blue}{e^{1}}} + \frac{{x}^{2}}{\mathsf{E}\left(\right)} \]

      2. rec-exp N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(1\right)}} + \frac{{x}^{2}}{\mathsf{E}\left(\right)} \]

      3. metadata-eval N/A

         \[\leadsto e^{\color{blue}{-1}} + \frac{{x}^{2}}{\mathsf{E}\left(\right)} \]

      4. *-rgt-identity N/A

         \[\leadsto e^{-1} + \frac{\color{blue}{{x}^{2} \cdot 1}}{\mathsf{E}\left(\right)} \]

      5. associate-*r/ N/A

         \[\leadsto e^{-1} + \color{blue}{{x}^{2} \cdot \frac{1}{\mathsf{E}\left(\right)}} \]

      6. e-exp-1 N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \frac{1}{\color{blue}{e^{1}}} \]

      7. rec-exp N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \color{blue}{e^{\mathsf{neg}\left(1\right)}} \]

      8. metadata-eval N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{\color{blue}{-1}} \]

      9. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} + 1\right) \cdot e^{-1}} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(1 + {x}^{2}\right)} \cdot e^{-1} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{e^{-1} \cdot \left(1 + {x}^{2}\right)} \]

      12. metadata-eval N/A

          \[\leadsto e^{\color{blue}{\mathsf{neg}\left(1\right)}} \cdot \left(1 + {x}^{2}\right) \]

      13. rec-exp N/A

          \[\leadsto \color{blue}{\frac{1}{e^{1}}} \cdot \left(1 + {x}^{2}\right) \]

      14. e-exp-1 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \cdot \left(1 + {x}^{2}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + {x}^{2}\right)}{\mathsf{E}\left(\right)}} \]

      16. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{1 + {x}^{2}}}{\mathsf{E}\left(\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 + {x}^{2}}{\mathsf{E}\left(\right)}} \]

      18. +-commutative N/A

          \[\leadsto \frac{\color{blue}{{x}^{2} + 1}}{\mathsf{E}\left(\right)} \]

      19. unpow2 N/A

          \[\leadsto \frac{\color{blue}{x \cdot x} + 1}{\mathsf{E}\left(\right)} \]

      20. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

      21. lower-E.f64 99.8

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right)}{\color{blue}{e}} \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{e}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 88.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;1 - x\_m \cdot x\_m \leq -500:\\ \;\;\;\;x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \left(x\_m \cdot 0.5\right), x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, x\_m, 1\right)}{e}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (* x_m x_m)) -500.0)
   (* x_m (fma x_m (* x_m (* x_m 0.5)) x_m))
   (/ (fma x_m x_m 1.0) E)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((1.0 - (x_m * x_m)) <= -500.0) {
		tmp = x_m * fma(x_m, (x_m * (x_m * 0.5)), x_m);
	} else {
		tmp = fma(x_m, x_m, 1.0) / ((double) M_E);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(1.0 - Float64(x_m * x_m)) <= -500.0)
		tmp = Float64(x_m * fma(x_m, Float64(x_m * Float64(x_m * 0.5)), x_m));
	else
		tmp = Float64(fma(x_m, x_m, 1.0) / exp(1));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -500.0], N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m + 1.0), $MachinePrecision] / E), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -500

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

         \[\leadsto e^{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto e^{\color{blue}{x \cdot x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + 1} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + 1 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)} + 1 \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right), 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}, 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right), 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right), 1\right) \]

      10. lower-*.f64 71.8

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right), 1\right) \]

   8. Applied rewrites 71.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{4} + \frac{1}{{x}^{2}} \cdot {x}^{4}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{{x}^{2}} \cdot {x}^{4} + \frac{1}{2} \cdot {x}^{4}} \]

       3. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot {x}^{4}}{{x}^{2}}} + \frac{1}{2} \cdot {x}^{4} \]

       4. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{{x}^{4}}}{{x}^{2}} + \frac{1}{2} \cdot {x}^{4} \]

       5. metadata-eval N/A

          \[\leadsto \frac{{x}^{\color{blue}{\left(3 + 1\right)}}}{{x}^{2}} + \frac{1}{2} \cdot {x}^{4} \]

       6. pow-plus N/A

          \[\leadsto \frac{\color{blue}{{x}^{3} \cdot x}}{{x}^{2}} + \frac{1}{2} \cdot {x}^{4} \]

       7. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{{x}^{3}}{{x}^{2}} \cdot x} + \frac{1}{2} \cdot {x}^{4} \]

       8. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{1 \cdot {x}^{3}}}{{x}^{2}} \cdot x + \frac{1}{2} \cdot {x}^{4} \]

       9. associate-*l/ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{{x}^{2}} \cdot {x}^{3}\right)} \cdot x + \frac{1}{2} \cdot {x}^{4} \]

       10. unpow3 N/A

           \[\leadsto \left(\frac{1}{{x}^{2}} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right) \cdot x + \frac{1}{2} \cdot {x}^{4} \]

       11. unpow2 N/A

           \[\leadsto \left(\frac{1}{{x}^{2}} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right)\right) \cdot x + \frac{1}{2} \cdot {x}^{4} \]

       12. associate-*r* N/A

           \[\leadsto \color{blue}{\left(\left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right) \cdot x\right)} \cdot x + \frac{1}{2} \cdot {x}^{4} \]

       13. lft-mult-inverse N/A

           \[\leadsto \left(\color{blue}{1} \cdot x\right) \cdot x + \frac{1}{2} \cdot {x}^{4} \]

       14. associate-*r* N/A

           \[\leadsto \color{blue}{1 \cdot \left(x \cdot x\right)} + \frac{1}{2} \cdot {x}^{4} \]

       15. unpow2 N/A

           \[\leadsto 1 \cdot \color{blue}{{x}^{2}} + \frac{1}{2} \cdot {x}^{4} \]

       16. metadata-eval N/A

           \[\leadsto 1 \cdot {x}^{2} + \frac{1}{2} \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}} \]

       17. pow-sqr N/A

           \[\leadsto 1 \cdot {x}^{2} + \frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \]

       18. associate-*l* N/A

           \[\leadsto 1 \cdot {x}^{2} + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

       19. distribute-rgt-in N/A

           \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]

       20. unpow2 N/A

           \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) \]

   11. Applied rewrites 71.8%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.5\right), x\right)} \]

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(\left(1 - \color{blue}{x \cdot x}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto e^{\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(x \cdot x\right)\right)\right)}\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto e^{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right)}} \]

      4. metadata-eval N/A

         \[\leadsto e^{\color{blue}{-1} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right)} \]

      5. remove-double-neg N/A

         \[\leadsto e^{-1 + \color{blue}{x \cdot x}} \]

      6. exp-sum N/A

         \[\leadsto \color{blue}{e^{-1} \cdot e^{x \cdot x}} \]

      7. metadata-eval N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(1\right)}} \cdot e^{x \cdot x} \]

      8. rec-exp N/A

         \[\leadsto \color{blue}{\frac{1}{e^{1}}} \cdot e^{x \cdot x} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{1}}{e^{x \cdot x}}}} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{x \cdot x}}}{e^{1}} \]

      13. exp-1-e N/A

          \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

      14. lower-E.f64 100.0

          \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{e}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{\mathsf{E}\left(\right)} + \frac{{x}^{2}}{\mathsf{E}\left(\right)}} \]
   6. Step-by-step derivation

      1. e-exp-1 N/A

         \[\leadsto \frac{1}{\color{blue}{e^{1}}} + \frac{{x}^{2}}{\mathsf{E}\left(\right)} \]

      2. rec-exp N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(1\right)}} + \frac{{x}^{2}}{\mathsf{E}\left(\right)} \]

      3. metadata-eval N/A

         \[\leadsto e^{\color{blue}{-1}} + \frac{{x}^{2}}{\mathsf{E}\left(\right)} \]

      4. *-rgt-identity N/A

         \[\leadsto e^{-1} + \frac{\color{blue}{{x}^{2} \cdot 1}}{\mathsf{E}\left(\right)} \]

      5. associate-*r/ N/A

         \[\leadsto e^{-1} + \color{blue}{{x}^{2} \cdot \frac{1}{\mathsf{E}\left(\right)}} \]

      6. e-exp-1 N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \frac{1}{\color{blue}{e^{1}}} \]

      7. rec-exp N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \color{blue}{e^{\mathsf{neg}\left(1\right)}} \]

      8. metadata-eval N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{\color{blue}{-1}} \]

      9. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} + 1\right) \cdot e^{-1}} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(1 + {x}^{2}\right)} \cdot e^{-1} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{e^{-1} \cdot \left(1 + {x}^{2}\right)} \]

      12. metadata-eval N/A

          \[\leadsto e^{\color{blue}{\mathsf{neg}\left(1\right)}} \cdot \left(1 + {x}^{2}\right) \]

      13. rec-exp N/A

          \[\leadsto \color{blue}{\frac{1}{e^{1}}} \cdot \left(1 + {x}^{2}\right) \]

      14. e-exp-1 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \cdot \left(1 + {x}^{2}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + {x}^{2}\right)}{\mathsf{E}\left(\right)}} \]

      16. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{1 + {x}^{2}}}{\mathsf{E}\left(\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 + {x}^{2}}{\mathsf{E}\left(\right)}} \]

      18. +-commutative N/A

          \[\leadsto \frac{\color{blue}{{x}^{2} + 1}}{\mathsf{E}\left(\right)} \]

      19. unpow2 N/A

          \[\leadsto \frac{\color{blue}{x \cdot x} + 1}{\mathsf{E}\left(\right)} \]

      20. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

      21. lower-E.f64 99.8

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right)}{\color{blue}{e}} \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{e}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 76.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 10^{-6}:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (* x_m x_m) 1e-6) (/ 1.0 E) (fma x_m x_m 1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((x_m * x_m) <= 1e-6) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = fma(x_m, x_m, 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 1e-6)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = fma(x_m, x_m, 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 1e-6], N[(1.0 / E), $MachinePrecision], N[(x$95$m * x$95$m + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.99999999999999955e-7

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(1\right)}} \]

      2. rec-exp N/A

         \[\leadsto \color{blue}{\frac{1}{e^{1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{e^{1}}} \]

      4. exp-1-e N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \]

      5. lower-E.f64 99.3

         \[\leadsto \frac{1}{\color{blue}{e}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{1}{e}} \]

   if 9.99999999999999955e-7 < (*.f64 x x)

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

         \[\leadsto e^{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto e^{\color{blue}{x \cdot x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} + 1} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} + 1 \]

      3. lower-fma.f64 47.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \]

   8. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 35.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;1 - x\_m \cdot x\_m \leq -500:\\ \;\;\;\;x\_m \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (* x_m x_m)) -500.0) (* x_m x_m) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((1.0 - (x_m * x_m)) <= -500.0) {
		tmp = x_m * x_m;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if ((1.0d0 - (x_m * x_m)) <= (-500.0d0)) then
        tmp = x_m * x_m
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if ((1.0 - (x_m * x_m)) <= -500.0) {
		tmp = x_m * x_m;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if (1.0 - (x_m * x_m)) <= -500.0:
		tmp = x_m * x_m
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(1.0 - Float64(x_m * x_m)) <= -500.0)
		tmp = Float64(x_m * x_m);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if ((1.0 - (x_m * x_m)) <= -500.0)
		tmp = x_m * x_m;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -500.0], N[(x$95$m * x$95$m), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -500

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.3

         \[\leadsto e^{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto e^{\color{blue}{x \cdot x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} + 1} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} + 1 \]

      3. lower-fma.f64 47.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \]

   8. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{x \cdot x} \]

       2. lower-*.f64 47.2

          \[\leadsto \color{blue}{x \cdot x} \]

   11. Applied rewrites 47.2%

       \[\leadsto \color{blue}{x \cdot x} \]

   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 inf

      \[\leadsto e^{\color{blue}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto e^{\color{blue}{x \cdot x}} \]

      2. lower-*.f64 17.8

         \[\leadsto e^{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 17.8%

      \[\leadsto e^{\color{blue}{x \cdot x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 76.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\mathsf{fma}\left(x\_m, x\_m, 1\right)}{e} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (fma x_m x_m 1.0) E))

C

x_m = fabs(x);
double code(double x_m) {
	return fma(x_m, x_m, 1.0) / ((double) M_E);
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(fma(x_m, x_m, 1.0) / exp(1))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * x$95$m + 1.0), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto e^{\mathsf{neg}\left(\left(1 - \color{blue}{x \cdot x}\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto e^{\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(x \cdot x\right)\right)\right)}\right)} \]

   3. distribute-neg-in N/A

      \[\leadsto e^{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right)}} \]

   4. metadata-eval N/A

      \[\leadsto e^{\color{blue}{-1} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right)} \]

   5. remove-double-neg N/A

      \[\leadsto e^{-1 + \color{blue}{x \cdot x}} \]

   6. exp-sum N/A

      \[\leadsto \color{blue}{e^{-1} \cdot e^{x \cdot x}} \]

   7. metadata-eval N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(1\right)}} \cdot e^{x \cdot x} \]

   8. rec-exp N/A

      \[\leadsto \color{blue}{\frac{1}{e^{1}}} \cdot e^{x \cdot x} \]

   9. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{1}}{e^{x \cdot x}}}} \]

   10. clear-num N/A

       \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

   11. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

   12. lower-exp.f64 N/A

       \[\leadsto \frac{\color{blue}{e^{x \cdot x}}}{e^{1}} \]

   13. exp-1-e N/A

       \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   14. lower-E.f64 99.9

       \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{e}} \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{\mathsf{E}\left(\right)} + \frac{{x}^{2}}{\mathsf{E}\left(\right)}} \]
6. Step-by-step derivation

   1. e-exp-1 N/A

      \[\leadsto \frac{1}{\color{blue}{e^{1}}} + \frac{{x}^{2}}{\mathsf{E}\left(\right)} \]

   2. rec-exp N/A

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(1\right)}} + \frac{{x}^{2}}{\mathsf{E}\left(\right)} \]

   3. metadata-eval N/A

      \[\leadsto e^{\color{blue}{-1}} + \frac{{x}^{2}}{\mathsf{E}\left(\right)} \]

   4. *-rgt-identity N/A

      \[\leadsto e^{-1} + \frac{\color{blue}{{x}^{2} \cdot 1}}{\mathsf{E}\left(\right)} \]

   5. associate-*r/ N/A

      \[\leadsto e^{-1} + \color{blue}{{x}^{2} \cdot \frac{1}{\mathsf{E}\left(\right)}} \]

   6. e-exp-1 N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot \frac{1}{\color{blue}{e^{1}}} \]

   7. rec-exp N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot \color{blue}{e^{\mathsf{neg}\left(1\right)}} \]

   8. metadata-eval N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{\color{blue}{-1}} \]

   9. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left({x}^{2} + 1\right) \cdot e^{-1}} \]

   10. +-commutative N/A

       \[\leadsto \color{blue}{\left(1 + {x}^{2}\right)} \cdot e^{-1} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{e^{-1} \cdot \left(1 + {x}^{2}\right)} \]

   12. metadata-eval N/A

       \[\leadsto e^{\color{blue}{\mathsf{neg}\left(1\right)}} \cdot \left(1 + {x}^{2}\right) \]

   13. rec-exp N/A

       \[\leadsto \color{blue}{\frac{1}{e^{1}}} \cdot \left(1 + {x}^{2}\right) \]

   14. e-exp-1 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \cdot \left(1 + {x}^{2}\right) \]

   15. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + {x}^{2}\right)}{\mathsf{E}\left(\right)}} \]

   16. *-lft-identity N/A

       \[\leadsto \frac{\color{blue}{1 + {x}^{2}}}{\mathsf{E}\left(\right)} \]

   17. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1 + {x}^{2}}{\mathsf{E}\left(\right)}} \]

   18. +-commutative N/A

       \[\leadsto \frac{\color{blue}{{x}^{2} + 1}}{\mathsf{E}\left(\right)} \]

   19. unpow2 N/A

       \[\leadsto \frac{\color{blue}{x \cdot x} + 1}{\mathsf{E}\left(\right)} \]

   20. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

   21. lower-E.f64 74.4

       \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right)}{\color{blue}{e}} \]

7. Applied rewrites 74.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{e}} \]
8. Add Preprocessing

Alternative 21: 35.1% accurate, 15.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(x\_m, x\_m, 1\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (fma x_m x_m 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	return fma(x_m, x_m, 1.0);
}

Julia

x_m = abs(x)
function code(x_m)
	return fma(x_m, x_m, 1.0)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * x$95$m + 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[e^{-\left(1 - x \cdot x\right)} \]
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 57.3

      \[\leadsto e^{\color{blue}{x \cdot x}} \]

5. Applied rewrites 57.3%

   \[\leadsto e^{\color{blue}{x \cdot x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} + 1} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{x \cdot x} + 1 \]

   3. lower-fma.f64 32.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \]

8. Applied rewrites 32.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \]
9. Add Preprocessing

Alternative 22: 10.5% accurate, 111.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0

Julia

x_m = abs(x)
function code(x_m)
	return 1.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1.0

Derivation

1. Initial program 99.9%

   \[e^{-\left(1 - x \cdot x\right)} \]
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 57.3

      \[\leadsto e^{\color{blue}{x \cdot x}} \]

5. Applied rewrites 57.3%

   \[\leadsto e^{\color{blue}{x \cdot x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 10.7%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))