NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.9% → 99.9%

Time: 15.6s

Alternatives: 15

Speedup: 9.7×

• 38.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.699057133428455e+54
  2. eps = -5.705967244562707e-226

Specification

Math

\[\begin{array}{l} \\ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \left(e^{-x} \cdot \left(x + 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot eps\_m} + e^{x \cdot \left(-1 - eps\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (* 0.5 (* 2.0 (* (exp (- x)) (+ x 1.0))))
   (* 0.5 (+ (exp (* x eps_m)) (exp (* x (- -1.0 eps_m)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = 0.5 * (2.0 * (exp(-x) * (x + 1.0)));
	} else {
		tmp = 0.5 * (exp((x * eps_m)) + exp((x * (-1.0 - eps_m))));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.0d0) then
        tmp = 0.5d0 * (2.0d0 * (exp(-x) * (x + 1.0d0)))
    else
        tmp = 0.5d0 * (exp((x * eps_m)) + exp((x * ((-1.0d0) - eps_m))))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = 0.5 * (2.0 * (Math.exp(-x) * (x + 1.0)));
	} else {
		tmp = 0.5 * (Math.exp((x * eps_m)) + Math.exp((x * (-1.0 - eps_m))));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.0:
		tmp = 0.5 * (2.0 * (math.exp(-x) * (x + 1.0)))
	else:
		tmp = 0.5 * (math.exp((x * eps_m)) + math.exp((x * (-1.0 - eps_m))))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.0)
		tmp = Float64(0.5 * Float64(2.0 * Float64(exp(Float64(-x)) * Float64(x + 1.0))));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(x * eps_m)) + exp(Float64(x * Float64(-1.0 - eps_m)))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.0)
		tmp = 0.5 * (2.0 * (exp(-x) * (x + 1.0)));
	else
		tmp = 0.5 * (exp((x * eps_m)) + exp((x * (-1.0 - eps_m))));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.0], N[(0.5 * N[(2.0 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 70.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-out N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-+.f64 69.1

          \[\leadsto 0.5 \cdot \left(2 \cdot \left(e^{-x} \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 \cdot \left(e^{-x} \cdot \left(x + 1\right)\right)\right)} \]

   if 1 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

         \[\leadsto 0.5 \cdot \left(e^{\color{blue}{x \cdot \varepsilon}} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto 0.5 \cdot \left(e^{\color{blue}{x \cdot \varepsilon}} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 2.0000004:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \left(e^{-x} \cdot \left(x + 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{1 + \left(x \cdot -0.5 + \mathsf{fma}\left(eps\_m, \mathsf{fma}\left(x, -0.5, \mathsf{fma}\left(eps\_m, \mathsf{fma}\left(x, -0.5, \mathsf{fma}\left(x, eps\_m, 0.5 \cdot x\right)\right), x \cdot -0.5\right)\right), \mathsf{fma}\left(0.5, x, -1\right)\right)\right)}{eps\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0000004)
   (* 0.5 (* 2.0 (* (exp (- x)) (+ x 1.0))))
   (fma
    0.5
    (*
     x
     (/
      (+
       1.0
       (+
        (* x -0.5)
        (fma
         eps_m
         (fma
          x
          -0.5
          (fma eps_m (fma x -0.5 (fma x eps_m (* 0.5 x))) (* x -0.5)))
         (fma 0.5 x -1.0))))
      eps_m))
    1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 2.0000004) {
		tmp = 0.5 * (2.0 * (exp(-x) * (x + 1.0)));
	} else {
		tmp = fma(0.5, (x * ((1.0 + ((x * -0.5) + fma(eps_m, fma(x, -0.5, fma(eps_m, fma(x, -0.5, fma(x, eps_m, (0.5 * x))), (x * -0.5))), fma(0.5, x, -1.0)))) / eps_m)), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) <= 2.0000004)
		tmp = Float64(0.5 * Float64(2.0 * Float64(exp(Float64(-x)) * Float64(x + 1.0))));
	else
		tmp = fma(0.5, Float64(x * Float64(Float64(1.0 + Float64(Float64(x * -0.5) + fma(eps_m, fma(x, -0.5, fma(eps_m, fma(x, -0.5, fma(x, eps_m, Float64(0.5 * x))), Float64(x * -0.5))), fma(0.5, x, -1.0)))) / eps_m)), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0000004], N[(0.5 * N[(2.0 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(N[(1.0 + N[(N[(x * -0.5), $MachinePrecision] + N[(eps$95$m * N[(x * -0.5 + N[(eps$95$m * N[(x * -0.5 + N[(x * eps$95$m + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 2.00000039999999979

   1. Initial program 63.6%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-out N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-+.f64 99.6

          \[\leadsto 0.5 \cdot \left(2 \cdot \left(e^{-x} \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 \cdot \left(e^{-x} \cdot \left(x + 1\right)\right)\right)} \]

   if 2.00000039999999979 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 89.9%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.0000004:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \left(e^{-x} \cdot \left(x + 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{1 + \left(x \cdot -0.5 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, -0.5, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, -0.5, \mathsf{fma}\left(x, \varepsilon, 0.5 \cdot x\right)\right), x \cdot -0.5\right)\right), \mathsf{fma}\left(0.5, x, -1\right)\right)\right)}{\varepsilon}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 2.0000004:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{1 + \left(x \cdot -0.5 + \mathsf{fma}\left(eps\_m, \mathsf{fma}\left(x, -0.5, \mathsf{fma}\left(eps\_m, \mathsf{fma}\left(x, -0.5, \mathsf{fma}\left(x, eps\_m, 0.5 \cdot x\right)\right), x \cdot -0.5\right)\right), \mathsf{fma}\left(0.5, x, -1\right)\right)\right)}{eps\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0000004)
   (exp (- x))
   (fma
    0.5
    (*
     x
     (/
      (+
       1.0
       (+
        (* x -0.5)
        (fma
         eps_m
         (fma
          x
          -0.5
          (fma eps_m (fma x -0.5 (fma x eps_m (* 0.5 x))) (* x -0.5)))
         (fma 0.5 x -1.0))))
      eps_m))
    1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 2.0000004) {
		tmp = exp(-x);
	} else {
		tmp = fma(0.5, (x * ((1.0 + ((x * -0.5) + fma(eps_m, fma(x, -0.5, fma(eps_m, fma(x, -0.5, fma(x, eps_m, (0.5 * x))), (x * -0.5))), fma(0.5, x, -1.0)))) / eps_m)), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) <= 2.0000004)
		tmp = exp(Float64(-x));
	else
		tmp = fma(0.5, Float64(x * Float64(Float64(1.0 + Float64(Float64(x * -0.5) + fma(eps_m, fma(x, -0.5, fma(eps_m, fma(x, -0.5, fma(x, eps_m, Float64(0.5 * x))), Float64(x * -0.5))), fma(0.5, x, -1.0)))) / eps_m)), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0000004], N[Exp[(-x)], $MachinePrecision], N[(0.5 * N[(x * N[(N[(1.0 + N[(N[(x * -0.5), $MachinePrecision] + N[(eps$95$m * N[(x * -0.5 + N[(eps$95$m * N[(x * -0.5 + N[(x * eps$95$m + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 2.00000039999999979

   1. Initial program 63.6%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-exp.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 98.6

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

   8. Applied rewrites 98.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. *-rgt-identity 98.6

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

   10. Applied rewrites 98.6%

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

   if 2.00000039999999979 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 89.9%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.0000004:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{1 + \left(x \cdot -0.5 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, -0.5, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, -0.5, \mathsf{fma}\left(x, \varepsilon, 0.5 \cdot x\right)\right), x \cdot -0.5\right)\right), \mathsf{fma}\left(0.5, x, -1\right)\right)\right)}{\varepsilon}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 10:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(eps\_m \cdot eps\_m\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      10.0)
   1.0
   (* 0.25 (* (* eps_m eps_m) (* x x)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 10.0) {
		tmp = 1.0;
	} else {
		tmp = 0.25 * ((eps_m * eps_m) * (x * x));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if ((((1.0d0 + (1.0d0 / eps_m)) * exp((x * (eps_m + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps_m))) * (1.0d0 + ((-1.0d0) / eps_m)))) <= 10.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.25d0 * ((eps_m * eps_m) * (x * x))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * Math.exp((x * (eps_m + -1.0)))) + (Math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 10.0) {
		tmp = 1.0;
	} else {
		tmp = 0.25 * ((eps_m * eps_m) * (x * x));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if (((1.0 + (1.0 / eps_m)) * math.exp((x * (eps_m + -1.0)))) + (math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 10.0:
		tmp = 1.0
	else:
		tmp = 0.25 * ((eps_m * eps_m) * (x * x))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) <= 10.0)
		tmp = 1.0;
	else
		tmp = Float64(0.25 * Float64(Float64(eps_m * eps_m) * Float64(x * x)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 10.0)
		tmp = 1.0;
	else
		tmp = 0.25 * ((eps_m * eps_m) * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 10.0], 1.0, N[(0.25 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 10

   1. Initial program 64.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.3%

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

   if 10 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft1-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 89.0

          \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.5 \cdot x, -1 + \varepsilon, 1\right) \cdot \color{blue}{\left(-1 + \varepsilon\right)}, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   5. Applied rewrites 89.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5 \cdot x, -1 + \varepsilon, 1\right) \cdot \left(-1 + \varepsilon\right), 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   6. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-out N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-out N/A

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

      14. associate-*r* N/A

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

      15. distribute-rgt-out N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 N/A

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

   8. Applied rewrites 87.3%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 84.7

         \[\leadsto 0.25 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

   10. Applied rewrites 84.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 10:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0.5 \cdot \left(e^{\mathsf{fma}\left(x, eps\_m, -x\right)} + e^{x \cdot \left(-1 - eps\_m\right)}\right) \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (* 0.5 (+ (exp (fma x eps_m (- x))) (exp (* x (- -1.0 eps_m))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.5 * (exp(fma(x, eps_m, -x)) + exp((x * (-1.0 - eps_m))));
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(0.5 * Float64(exp(fma(x, eps_m, Float64(-x))) + exp(Float64(x * Float64(-1.0 - eps_m)))))
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(0.5 * N[(N[Exp[N[(x * eps$95$m + (-x)), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower-+.f64 N/A

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

5. Applied rewrites 99.0%

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

Alternative 6: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \left(e^{-x} \cdot \left(x + 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.5 \cdot x, eps\_m + -1, 1\right) \cdot \left(eps\_m + -1\right), 1\right) + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (* 0.5 (* 2.0 (* (exp (- x)) (+ x 1.0))))
   (/
    (+
     (*
      (+ 1.0 (/ 1.0 eps_m))
      (fma x (* (fma (* 0.5 x) (+ eps_m -1.0) 1.0) (+ eps_m -1.0)) 1.0))
     (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = 0.5 * (2.0 * (exp(-x) * (x + 1.0)));
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * fma(x, (fma((0.5 * x), (eps_m + -1.0), 1.0) * (eps_m + -1.0)), 1.0)) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.0)
		tmp = Float64(0.5 * Float64(2.0 * Float64(exp(Float64(-x)) * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * fma(x, Float64(fma(Float64(0.5 * x), Float64(eps_m + -1.0), 1.0) * Float64(eps_m + -1.0)), 1.0)) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.0], N[(0.5 * N[(2.0 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(N[(0.5 * x), $MachinePrecision] * N[(eps$95$m + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 70.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-out N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-+.f64 69.1

          \[\leadsto 0.5 \cdot \left(2 \cdot \left(e^{-x} \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 \cdot \left(e^{-x} \cdot \left(x + 1\right)\right)\right)} \]

   if 1 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft1-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 90.9

          \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.5 \cdot x, -1 + \varepsilon, 1\right) \cdot \color{blue}{\left(-1 + \varepsilon\right)}, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   5. Applied rewrites 90.9%

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

4. Final simplification 75.6%

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

Alternative 7: 84.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 720:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{1 + \left(x \cdot -0.5 + \mathsf{fma}\left(eps\_m, \mathsf{fma}\left(x, -0.5, \mathsf{fma}\left(eps\_m, \mathsf{fma}\left(x, -0.5, \mathsf{fma}\left(x, eps\_m, 0.5 \cdot x\right)\right), x \cdot -0.5\right)\right), \mathsf{fma}\left(0.5, x, -1\right)\right)\right)}{eps\_m}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 720.0)
   (fma
    0.5
    (*
     x
     (/
      (+
       1.0
       (+
        (* x -0.5)
        (fma
         eps_m
         (fma
          x
          -0.5
          (fma eps_m (fma x -0.5 (fma x eps_m (* 0.5 x))) (* x -0.5)))
         (fma 0.5 x -1.0))))
      eps_m))
    1.0)
   (* 0.5 (* x (* x (* eps_m eps_m))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 720.0) {
		tmp = fma(0.5, (x * ((1.0 + ((x * -0.5) + fma(eps_m, fma(x, -0.5, fma(eps_m, fma(x, -0.5, fma(x, eps_m, (0.5 * x))), (x * -0.5))), fma(0.5, x, -1.0)))) / eps_m)), 1.0);
	} else {
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 720.0)
		tmp = fma(0.5, Float64(x * Float64(Float64(1.0 + Float64(Float64(x * -0.5) + fma(eps_m, fma(x, -0.5, fma(eps_m, fma(x, -0.5, fma(x, eps_m, Float64(0.5 * x))), Float64(x * -0.5))), fma(0.5, x, -1.0)))) / eps_m)), 1.0);
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps_m * eps_m))));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 720.0], N[(0.5 * N[(x * N[(N[(1.0 + N[(N[(x * -0.5), $MachinePrecision] + N[(eps$95$m * N[(x * -0.5 + N[(eps$95$m * N[(x * -0.5 + N[(x * eps$95$m + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 720

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 89.8%

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

   if 720 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 40.7%

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

   9. Taylor expanded in eps around inf

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

       1. lower-*.f64 N/A

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

       2. distribute-rgt-out-- N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. distribute-rgt-out N/A

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

       6. associate-*r* N/A

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

       7. distribute-rgt-out N/A

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

       8. metadata-eval N/A

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

       9. *-rgt-identity N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 71.3

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

   11. Applied rewrites 71.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 720:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{1 + \left(x \cdot -0.5 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, -0.5, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, -0.5, \mathsf{fma}\left(x, \varepsilon, 0.5 \cdot x\right)\right), x \cdot -0.5\right)\right), \mathsf{fma}\left(0.5, x, -1\right)\right)\right)}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-85}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(eps\_m, eps\_m, eps\_m\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(eps\_m, 0.5, 0.25\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.75e-85)
   (* 0.5 (* x (* x (fma eps_m eps_m eps_m))))
   (if (<= x 1.45e-45)
     (fma (* x x) (fma eps_m 0.5 0.25) 1.0)
     (* 0.5 (* x (* x (* eps_m eps_m)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.75e-85) {
		tmp = 0.5 * (x * (x * fma(eps_m, eps_m, eps_m)));
	} else if (x <= 1.45e-45) {
		tmp = fma((x * x), fma(eps_m, 0.5, 0.25), 1.0);
	} else {
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.75e-85)
		tmp = Float64(0.5 * Float64(x * Float64(x * fma(eps_m, eps_m, eps_m))));
	elseif (x <= 1.45e-45)
		tmp = fma(Float64(x * x), fma(eps_m, 0.5, 0.25), 1.0);
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps_m * eps_m))));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.75e-85], N[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-45], N[(N[(x * x), $MachinePrecision] * N[(eps$95$m * 0.5 + 0.25), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.74999999999999989e-85

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 78.6

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

   8. Applied rewrites 78.6%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 84.3%

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

   12. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. distribute-rgt-in N/A

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

       16. *-lft-identity N/A

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

       17. unpow2 N/A

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

       18. unpow2 N/A

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

   14. Applied rewrites 75.9%

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

   if -1.74999999999999989e-85 < x < 1.45e-45

   1. Initial program 54.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 93.2%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-+.f64 92.4

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

   11. Applied rewrites 92.4%

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

   12. Taylor expanded in eps around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 85.5

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.5, 0.25\right)}, 1\right) \]

   14. Applied rewrites 85.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\varepsilon, 0.5, 0.25\right), 1\right)} \]

   if 1.45e-45 < x

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 42.8%

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

   9. Taylor expanded in eps around inf

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

       1. lower-*.f64 N/A

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

       2. distribute-rgt-out-- N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. distribute-rgt-out N/A

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

       6. associate-*r* N/A

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

       7. distribute-rgt-out N/A

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

       8. metadata-eval N/A

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

       9. *-rgt-identity N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 69.2

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

   11. Applied rewrites 69.2%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-85}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon, \varepsilon\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\varepsilon, 0.5, 0.25\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(eps\_m, 0.5, 0.25\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* x (* x (* eps_m eps_m))))))
   (if (<= x -1.75e-85)
     t_0
     (if (<= x 1.45e-45) (fma (* x x) (fma eps_m 0.5 0.25) 1.0) t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 0.5 * (x * (x * (eps_m * eps_m)));
	double tmp;
	if (x <= -1.75e-85) {
		tmp = t_0;
	} else if (x <= 1.45e-45) {
		tmp = fma((x * x), fma(eps_m, 0.5, 0.25), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(0.5 * Float64(x * Float64(x * Float64(eps_m * eps_m))))
	tmp = 0.0
	if (x <= -1.75e-85)
		tmp = t_0;
	elseif (x <= 1.45e-45)
		tmp = fma(Float64(x * x), fma(eps_m, 0.5, 0.25), 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.75e-85], t$95$0, If[LessEqual[x, 1.45e-45], N[(N[(x * x), $MachinePrecision] * N[(eps$95$m * 0.5 + 0.25), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.74999999999999989e-85 or 1.45e-45 < x

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 57.2%

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

   9. Taylor expanded in eps around inf

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

       1. lower-*.f64 N/A

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

       2. distribute-rgt-out-- N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. distribute-rgt-out N/A

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

       6. associate-*r* N/A

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

       7. distribute-rgt-out N/A

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

       8. metadata-eval N/A

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

       9. *-rgt-identity N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 71.9

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

   11. Applied rewrites 71.9%

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

   if -1.74999999999999989e-85 < x < 1.45e-45

   1. Initial program 54.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 93.2%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-+.f64 92.4

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

   11. Applied rewrites 92.4%

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

   12. Taylor expanded in eps around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 85.5

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.5, 0.25\right)}, 1\right) \]

   14. Applied rewrites 85.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\varepsilon, 0.5, 0.25\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-85}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\varepsilon, 0.5, 0.25\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-85}:\\ \;\;\;\;0.25 \cdot \left(\left(eps\_m \cdot eps\_m\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(eps\_m, 0.5, 0.25\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.75e-85)
   (* 0.25 (* (* eps_m eps_m) (* x x)))
   (if (<= x 1.45e-45)
     (fma (* x x) (fma eps_m 0.5 0.25) 1.0)
     (* (* x (* x (* eps_m eps_m))) 0.25))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.75e-85) {
		tmp = 0.25 * ((eps_m * eps_m) * (x * x));
	} else if (x <= 1.45e-45) {
		tmp = fma((x * x), fma(eps_m, 0.5, 0.25), 1.0);
	} else {
		tmp = (x * (x * (eps_m * eps_m))) * 0.25;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.75e-85)
		tmp = Float64(0.25 * Float64(Float64(eps_m * eps_m) * Float64(x * x)));
	elseif (x <= 1.45e-45)
		tmp = fma(Float64(x * x), fma(eps_m, 0.5, 0.25), 1.0);
	else
		tmp = Float64(Float64(x * Float64(x * Float64(eps_m * eps_m))) * 0.25);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.75e-85], N[(0.25 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-45], N[(N[(x * x), $MachinePrecision] * N[(eps$95$m * 0.5 + 0.25), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.74999999999999989e-85

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft1-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 85.7

          \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.5 \cdot x, -1 + \varepsilon, 1\right) \cdot \color{blue}{\left(-1 + \varepsilon\right)}, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   5. Applied rewrites 85.7%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5 \cdot x, -1 + \varepsilon, 1\right) \cdot \left(-1 + \varepsilon\right), 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   6. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-out N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-out N/A

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

      14. associate-*r* N/A

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

      15. distribute-rgt-out N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 N/A

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

   8. Applied rewrites 76.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 76.0

         \[\leadsto 0.25 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

   10. Applied rewrites 76.0%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   if -1.74999999999999989e-85 < x < 1.45e-45

   1. Initial program 54.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 93.2%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-+.f64 92.4

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

   11. Applied rewrites 92.4%

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

   12. Taylor expanded in eps around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 85.5

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.5, 0.25\right)}, 1\right) \]

   14. Applied rewrites 85.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\varepsilon, 0.5, 0.25\right), 1\right)} \]

   if 1.45e-45 < x

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft1-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 42.8

          \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.5 \cdot x, -1 + \varepsilon, 1\right) \cdot \color{blue}{\left(-1 + \varepsilon\right)}, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   5. Applied rewrites 42.8%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5 \cdot x, -1 + \varepsilon, 1\right) \cdot \left(-1 + \varepsilon\right), 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   6. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-out N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-out N/A

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

      14. associate-*r* N/A

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

      15. distribute-rgt-out N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 N/A

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

   8. Applied rewrites 69.2%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-85}:\\ \;\;\;\;0.25 \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\varepsilon, 0.5, 0.25\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\\ \mathbf{if}\;x \leq 29:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_0, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* x (* eps_m eps_m)))))
   (if (<= x 29.0) (fma 0.5 t_0 1.0) (* 0.5 t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (x * (eps_m * eps_m));
	double tmp;
	if (x <= 29.0) {
		tmp = fma(0.5, t_0, 1.0);
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(x * Float64(eps_m * eps_m)))
	tmp = 0.0
	if (x <= 29.0)
		tmp = fma(0.5, t_0, 1.0);
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 29.0], N[(0.5 * t$95$0 + 1.0), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 29

   1. Initial program 69.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in eps around inf

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

      1. distribute-rgt-out-- N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 87.0

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

   8. Applied rewrites 87.0%

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

   if 29 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 40.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 40.2%

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

   9. Taylor expanded in eps around inf

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

       1. lower-*.f64 N/A

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

       2. distribute-rgt-out-- N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. distribute-rgt-out N/A

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

       6. associate-*r* N/A

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

       7. distribute-rgt-out N/A

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

       8. metadata-eval N/A

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

       9. *-rgt-identity N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 70.5

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

   11. Applied rewrites 70.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 29:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(eps\_m, 0.5, 0.25\right), 1\right) \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (fma (* x x) (fma eps_m 0.5 0.25) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return fma((x * x), fma(eps_m, 0.5, 0.25), 1.0);
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return fma(Float64(x * x), fma(eps_m, 0.5, 0.25), 1.0)
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(x * x), $MachinePrecision] * N[(eps$95$m * 0.5 + 0.25), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 71.8%

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

6. Taylor expanded in eps around inf

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

8. Applied rewrites 71.2%

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

9. Taylor expanded in x around 0

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

    1. sub-neg N/A

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

    2. metadata-eval N/A

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

    3. lower-+.f64 71.5

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

11. Applied rewrites 71.5%

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

12. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. +-commutative N/A

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

    3. associate-*r* N/A

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

    4. distribute-rgt-out N/A

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

    5. lower-fma.f64 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-fma.f64 49.3

       \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.5, 0.25\right)}, 1\right) \]

14. Applied rewrites 49.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\varepsilon, 0.5, 0.25\right), 1\right)} \]
15. Add Preprocessing

Alternative 13: 51.2% accurate, 16.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \left(eps\_m \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0) (* x (* eps_m -0.5)) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = x * (eps_m * -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x * (eps_m * (-0.5d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = x * (eps_m * -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = x * (eps_m * -0.5)
	else:
		tmp = 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x * Float64(eps_m * -0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x * (eps_m * -0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(x * N[(eps$95$m * -0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 97.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 28.8%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 28.8

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

   11. Applied rewrites 28.8%

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

   if -1 < x

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 45.3%

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

4. Final simplification 42.5%

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

Alternative 14: 57.4% accurate, 22.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \mathsf{fma}\left(x, x \cdot 0.25, 1\right) \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (fma x (* x 0.25) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return fma(x, (x * 0.25), 1.0);
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return fma(x, Float64(x * 0.25), 1.0)
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(x * N[(x * 0.25), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 71.8%

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

6. Taylor expanded in eps around inf

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

8. Applied rewrites 71.2%

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

9. Taylor expanded in x around 0

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

    1. sub-neg N/A

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

    2. metadata-eval N/A

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

    3. lower-+.f64 71.5

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

11. Applied rewrites 71.5%

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

12. Taylor expanded in eps around 0

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

    1. +-commutative N/A

       \[\leadsto \color{blue}{\frac{1}{4} \cdot {x}^{2} + 1} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{4}} + 1 \]

    3. unpow2 N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 55.2

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot 0.25}, 1\right) \]

14. Applied rewrites 55.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.25, 1\right)} \]
15. Add Preprocessing

Alternative 15: 43.6% accurate, 273.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 1.0;
}

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 1.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 38.1%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))