NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.0% → 99.9%

Time: 14.4s

Alternatives: 15

Speedup: 7.0×

• 38.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.629774212886285e+35
  2. eps = 2.6111365116815692e-107

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: 74.0% 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, 2.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.41:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh \left(x \cdot eps\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.41)
   (* 0.5 (* (exp (- x)) (+ x (+ x 2.0))))
   (* 0.5 (* 2.0 (cosh (* x eps_m))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.41) {
		tmp = 0.5 * (exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = 0.5 * (2.0 * cosh((x * 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 <= 0.41d0) then
        tmp = 0.5d0 * (exp(-x) * (x + (x + 2.0d0)))
    else
        tmp = 0.5d0 * (2.0d0 * cosh((x * 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 <= 0.41) {
		tmp = 0.5 * (Math.exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = 0.5 * (2.0 * Math.cosh((x * eps_m)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.41:
		tmp = 0.5 * (math.exp(-x) * (x + (x + 2.0)))
	else:
		tmp = 0.5 * (2.0 * math.cosh((x * eps_m)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.41)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) * Float64(x + Float64(x + 2.0))));
	else
		tmp = Float64(0.5 * Float64(2.0 * cosh(Float64(x * eps_m))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.41)
		tmp = 0.5 * (exp(-x) * (x + (x + 2.0)));
	else
		tmp = 0.5 * (2.0 * cosh((x * eps_m)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.41], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Cosh[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.409999999999999976

   1. Initial program 64.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Simplified 71.9%

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

   if 0.409999999999999976 < 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. Simplified 100.0%

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

   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. Simplified 100.0%

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

   9. Taylor expanded in eps around inf

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-neg-in N/A

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

       3. neg-mul-1 N/A

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

       4. lower-*.f64 N/A

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

       5. neg-mul-1 N/A

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

       6. lower-neg.f64 100.0

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

   11. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(e^{x \cdot \varepsilon} + e^{\color{blue}{\varepsilon \cdot \left(-x\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-exp.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-exp.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 100.0

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

       9. lift-+.f64 N/A

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

       10. lift-exp.f64 N/A

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

       11. lift-exp.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-neg.f64 N/A

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

       14. distribute-rgt-neg-out N/A

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

       15. *-commutative N/A

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

       16. lift-*.f64 N/A

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

       17. cosh-undef N/A

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

       18. lower-*.f64 N/A

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

       19. lower-cosh.f64 100.0

           \[\leadsto \left(2 \cdot \color{blue}{\cosh \left(x \cdot \varepsilon\right)}\right) \cdot 0.5 \]

   13. Applied egg-rr 100.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.41:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 0:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh \left(x \cdot eps\_m\right)\right)\\ \end{array} \end{array} \]

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))))
      0.0)
   (exp (- x))
   (* 0.5 (* 2.0 (cosh (* x eps_m))))))

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)))) <= 0.0) {
		tmp = exp(-x);
	} else {
		tmp = 0.5 * (2.0 * cosh((x * 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 ((((1.0d0 + (1.0d0 / eps_m)) * exp((x * (eps_m + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps_m))) * (1.0d0 + ((-1.0d0) / eps_m)))) <= 0.0d0) then
        tmp = exp(-x)
    else
        tmp = 0.5d0 * (2.0d0 * cosh((x * 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 ((((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)))) <= 0.0) {
		tmp = Math.exp(-x);
	} else {
		tmp = 0.5 * (2.0 * Math.cosh((x * eps_m)));
	}
	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)))) <= 0.0:
		tmp = math.exp(-x)
	else:
		tmp = 0.5 * (2.0 * math.cosh((x * eps_m)))
	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)))) <= 0.0)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(0.5 * Float64(2.0 * cosh(Float64(x * eps_m))));
	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)))) <= 0.0)
		tmp = exp(-x);
	else
		tmp = 0.5 * (2.0 * cosh((x * eps_m)));
	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], 0.0], N[Exp[(-x)], $MachinePrecision], N[(0.5 * N[(2.0 * N[Cosh[N[(x * eps$95$m), $MachinePrecision]], $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))))) < 0.0

   1. Initial program 44.3%

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

   3. Taylor expanded in 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. Simplified 97.9%

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

   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. neg-mul-1 N/A

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

      2. distribute-lft-out N/A

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

      3. distribute-rgt-out N/A

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

      4. neg-mul-1 N/A

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

      5. metadata-eval N/A

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

      6. 1-exp N/A

         \[\leadsto e^{-1 \cdot x} \cdot \color{blue}{e^{0}} \]

      7. exp-sum N/A

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

      8. *-commutative N/A

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

      9. mul0-rgt N/A

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

      10. distribute-lft-out N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-exp.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 97.9

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

   8. Simplified 97.9%

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

   if 0.0 < (-.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 99.3%

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

   3. Taylor expanded in 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. Simplified 99.2%

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

   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 99.3

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

   8. Simplified 99.3%

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

   9. Taylor expanded in eps around inf

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-neg-in N/A

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

       3. neg-mul-1 N/A

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

       4. lower-*.f64 N/A

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

       5. neg-mul-1 N/A

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

       6. lower-neg.f64 99.3

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

   11. Simplified 99.3%

       \[\leadsto 0.5 \cdot \left(e^{x \cdot \varepsilon} + e^{\color{blue}{\varepsilon \cdot \left(-x\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-exp.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-exp.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 99.3

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

       9. lift-+.f64 N/A

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

       10. lift-exp.f64 N/A

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

       11. lift-exp.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-neg.f64 N/A

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

       14. distribute-rgt-neg-out N/A

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

       15. *-commutative N/A

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

       16. lift-*.f64 N/A

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

       17. cosh-undef N/A

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

       18. lower-*.f64 N/A

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

       19. lower-cosh.f64 99.3

           \[\leadsto \left(2 \cdot \color{blue}{\cosh \left(x \cdot \varepsilon\right)}\right) \cdot 0.5 \]

   13. Applied egg-rr 99.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 500:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(0.5 \cdot \left(eps\_m \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\right)}{eps\_m}\\ \end{array} \end{array} \]

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))))
      500.0)
   (exp (- x))
   (/ (* x (* 0.5 (* eps_m (* x (* eps_m eps_m))))) eps_m)))

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)))) <= 500.0) {
		tmp = exp(-x);
	} else {
		tmp = (x * (0.5 * (eps_m * (x * (eps_m * eps_m))))) / 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 ((((1.0d0 + (1.0d0 / eps_m)) * exp((x * (eps_m + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps_m))) * (1.0d0 + ((-1.0d0) / eps_m)))) <= 500.0d0) then
        tmp = exp(-x)
    else
        tmp = (x * (0.5d0 * (eps_m * (x * (eps_m * eps_m))))) / 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 ((((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)))) <= 500.0) {
		tmp = Math.exp(-x);
	} else {
		tmp = (x * (0.5 * (eps_m * (x * (eps_m * eps_m))))) / eps_m;
	}
	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)))) <= 500.0:
		tmp = math.exp(-x)
	else:
		tmp = (x * (0.5 * (eps_m * (x * (eps_m * eps_m))))) / eps_m
	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)))) <= 500.0)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(Float64(x * Float64(0.5 * Float64(eps_m * Float64(x * Float64(eps_m * eps_m))))) / eps_m);
	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)))) <= 500.0)
		tmp = exp(-x);
	else
		tmp = (x * (0.5 * (eps_m * (x * (eps_m * eps_m))))) / eps_m;
	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], 500.0], N[Exp[(-x)], $MachinePrecision], N[(N[(x * N[(0.5 * N[(eps$95$m * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $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))))) < 500

   1. Initial program 55.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. Simplified 98.2%

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

   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. neg-mul-1 N/A

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

      2. distribute-lft-out N/A

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

      3. distribute-rgt-out N/A

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

      4. neg-mul-1 N/A

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

      5. metadata-eval N/A

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

      6. 1-exp N/A

         \[\leadsto e^{-1 \cdot x} \cdot \color{blue}{e^{0}} \]

      7. exp-sum N/A

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

      8. *-commutative N/A

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

      9. mul0-rgt N/A

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

      10. distribute-lft-out N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-exp.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 97.0

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

   8. Simplified 97.0%

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

   if 500 < (-.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. Simplified 87.5%

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

   6. Taylor expanded in eps around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{2} \cdot \left(\left(\left(-1 - \varepsilon\right) + \frac{\varepsilon + 1}{\varepsilon}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + -1 \cdot \frac{\color{blue}{0} \cdot x - 1}{\varepsilon}\right)\right), 1\right) \]

      6. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{2} \cdot \left(\left(\left(-1 - \varepsilon\right) + \frac{\varepsilon + 1}{\varepsilon}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + -1 \cdot \frac{\color{blue}{0} - 1}{\varepsilon}\right)\right), 1\right) \]

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 87.6

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

   8. Simplified 87.6%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

   11. Simplified 87.4%

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

   12. Taylor expanded in eps around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 90.8

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

   14. Simplified 90.8%

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

4. Final simplification 94.3%

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

Alternative 4: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      500.0)
   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 ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * (x * (eps_m * 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 ((((1.0d0 + (1.0d0 / eps_m)) * exp((x * (eps_m + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps_m))) * (1.0d0 + ((-1.0d0) / eps_m)))) <= 500.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (x * (x * (eps_m * 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 ((((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)))) <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	}
	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)))) <= 500.0:
		tmp = 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 (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)))) <= 500.0)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps_m * eps_m))));
	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)))) <= 500.0)
		tmp = 1.0;
	else
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	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], 500.0], 1.0, 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 (-.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))))) < 500

   1. Initial program 55.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 64.8%

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

   if 500 < (-.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. Simplified 87.5%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 87.6

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

   8. Simplified 87.6%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 87.6

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

   11. Simplified 87.6%

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

4. Final simplification 74.9%

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

Alternative 5: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.5d0 * (exp(((x * eps_m) - x)) + exp((x * ((-1.0d0) - eps_m))))
end function

Java

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.5 * (math.exp(((x * eps_m) - x)) + math.exp((x * (-1.0 - eps_m))))

Julia

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

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.5 * (exp(((x * eps_m) - x)) + exp((x * (-1.0 - eps_m))));
end

Wolfram

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

Derivation

1. Initial program 75.0%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
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. Simplified 98.6%

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

Alternative 6: 84.4% accurate, 5.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(eps\_m, \mathsf{fma}\left(eps\_m, 0.5 \cdot \left(x \cdot eps\_m\right), -0.5 \cdot x\right), 0\right)}{eps\_m}, 1\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+252}:\\ \;\;\;\;\frac{x \cdot \left(0.5 \cdot \left(eps\_m \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\right)}{eps\_m}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.4)
   (fma
    x
    (/ (fma eps_m (fma eps_m (* 0.5 (* x eps_m)) (- (* 0.5 x))) 0.0) eps_m)
    1.0)
   (if (<= x 3.9e+252)
     (/ (* x (* 0.5 (* eps_m (* x (* eps_m eps_m))))) eps_m)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.4) {
		tmp = fma(x, (fma(eps_m, fma(eps_m, (0.5 * (x * eps_m)), -(0.5 * x)), 0.0) / eps_m), 1.0);
	} else if (x <= 3.9e+252) {
		tmp = (x * (0.5 * (eps_m * (x * (eps_m * eps_m))))) / eps_m;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.4)
		tmp = fma(x, Float64(fma(eps_m, fma(eps_m, Float64(0.5 * Float64(x * eps_m)), Float64(-Float64(0.5 * x))), 0.0) / eps_m), 1.0);
	elseif (x <= 3.9e+252)
		tmp = Float64(Float64(x * Float64(0.5 * Float64(eps_m * Float64(x * Float64(eps_m * eps_m))))) / eps_m);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.4], N[(x * N[(N[(eps$95$m * N[(eps$95$m * N[(0.5 * N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] + (-N[(0.5 * x), $MachinePrecision])), $MachinePrecision] + 0.0), $MachinePrecision] / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 3.9e+252], N[(N[(x * N[(0.5 * N[(eps$95$m * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 1.3999999999999999

   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 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. Simplified 89.7%

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

   6. Taylor expanded in eps around 0

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

   8. Simplified 92.5%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.5 \cdot \left(x \cdot \varepsilon\right), 0.5 \cdot \left(-x\right)\right), 0\right)}{\varepsilon}}, 1\right) \]

   if 1.3999999999999999 < x < 3.89999999999999965e252

   1. Initial program 98.6%

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

   3. Taylor expanded in 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. Simplified 38.8%

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

   6. Taylor expanded in eps around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{2} \cdot \left(\left(\left(-1 - \varepsilon\right) + \frac{\varepsilon + 1}{\varepsilon}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + -1 \cdot \frac{\color{blue}{0} \cdot x - 1}{\varepsilon}\right)\right), 1\right) \]

      6. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{2} \cdot \left(\left(\left(-1 - \varepsilon\right) + \frac{\varepsilon + 1}{\varepsilon}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + -1 \cdot \frac{\color{blue}{0} - 1}{\varepsilon}\right)\right), 1\right) \]

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 38.8

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

   8. Simplified 38.8%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

   11. Simplified 40.1%

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

   12. Taylor expanded in eps around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 74.0

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

   14. Simplified 74.0%

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

   if 3.89999999999999965e252 < 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. Simplified 26.2%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      6. mul0-rgt N/A

         \[\leadsto \left(x \cdot \color{blue}{0}\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{0} \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \frac{1}{2}}{\varepsilon}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \]

      10. mul0-lft N/A

          \[\leadsto \frac{\color{blue}{0 \cdot x}}{\varepsilon} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{0 \cdot \frac{x}{\varepsilon}} \]

      12. mul0-lft 75.4

          \[\leadsto \color{blue}{0} \]

   8. Simplified 75.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.5 \cdot \left(x \cdot \varepsilon\right), -0.5 \cdot x\right), 0\right)}{\varepsilon}, 1\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+252}:\\ \;\;\;\;\frac{x \cdot \left(0.5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)}{\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 51:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot \left(\left(eps\_m \cdot eps\_m\right) \cdot \left(x \cdot \left(0.5 + \mathsf{fma}\left(x, -0.6666666666666666, 0.5\right)\right)\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+252}:\\ \;\;\;\;\frac{x \cdot \left(0.5 \cdot \left(eps\_m \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\right)}{eps\_m}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 51.0)
   (fma
    x
    (* 0.5 (* (* eps_m eps_m) (* x (+ 0.5 (fma x -0.6666666666666666 0.5)))))
    1.0)
   (if (<= x 3.9e+252)
     (/ (* x (* 0.5 (* eps_m (* x (* eps_m eps_m))))) eps_m)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 51.0) {
		tmp = fma(x, (0.5 * ((eps_m * eps_m) * (x * (0.5 + fma(x, -0.6666666666666666, 0.5))))), 1.0);
	} else if (x <= 3.9e+252) {
		tmp = (x * (0.5 * (eps_m * (x * (eps_m * eps_m))))) / eps_m;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 51.0)
		tmp = fma(x, Float64(0.5 * Float64(Float64(eps_m * eps_m) * Float64(x * Float64(0.5 + fma(x, -0.6666666666666666, 0.5))))), 1.0);
	elseif (x <= 3.9e+252)
		tmp = Float64(Float64(x * Float64(0.5 * Float64(eps_m * Float64(x * Float64(eps_m * eps_m))))) / eps_m);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 51.0], N[(x * N[(0.5 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * N[(x * N[(0.5 + N[(x * -0.6666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 3.9e+252], N[(N[(x * N[(0.5 * N[(eps$95$m * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 51

   1. Initial program 63.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 51.6%

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

   6. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 90.8

          \[\leadsto \mathsf{fma}\left(x, 0.5 \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(0.5 + \color{blue}{\mathsf{fma}\left(x, -0.6666666666666666, 0.5\right)}\right)\right)\right), 1\right) \]

   8. Simplified 90.8%

      \[\leadsto \mathsf{fma}\left(x, 0.5 \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(0.5 + \mathsf{fma}\left(x, -0.6666666666666666, 0.5\right)\right)\right)\right)}, 1\right) \]

   if 51 < x < 3.89999999999999965e252

   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. Simplified 39.4%

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

   6. Taylor expanded in eps around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{2} \cdot \left(\left(\left(-1 - \varepsilon\right) + \frac{\varepsilon + 1}{\varepsilon}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + -1 \cdot \frac{\color{blue}{0} \cdot x - 1}{\varepsilon}\right)\right), 1\right) \]

      6. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{2} \cdot \left(\left(\left(-1 - \varepsilon\right) + \frac{\varepsilon + 1}{\varepsilon}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + -1 \cdot \frac{\color{blue}{0} - 1}{\varepsilon}\right)\right), 1\right) \]

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 39.4

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

   8. Simplified 39.4%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

   11. Simplified 40.7%

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

   12. Taylor expanded in eps around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 75.1

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

   14. Simplified 75.1%

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

   if 3.89999999999999965e252 < 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. Simplified 26.2%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      6. mul0-rgt N/A

         \[\leadsto \left(x \cdot \color{blue}{0}\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{0} \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \frac{1}{2}}{\varepsilon}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \]

      10. mul0-lft N/A

          \[\leadsto \frac{\color{blue}{0 \cdot x}}{\varepsilon} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{0 \cdot \frac{x}{\varepsilon}} \]

      12. mul0-lft 75.4

          \[\leadsto \color{blue}{0} \]

   8. Simplified 75.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 51:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot \left(\left(eps\_m \cdot eps\_m\right) \cdot \left(x \cdot \left(0.5 + \mathsf{fma}\left(x, -0.6666666666666666, 0.5\right)\right)\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+252}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 51.0)
   (fma
    x
    (* 0.5 (* (* eps_m eps_m) (* x (+ 0.5 (fma x -0.6666666666666666 0.5)))))
    1.0)
   (if (<= x 3.9e+252) (* 0.5 (* x (* x (* eps_m eps_m)))) 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 51.0) {
		tmp = fma(x, (0.5 * ((eps_m * eps_m) * (x * (0.5 + fma(x, -0.6666666666666666, 0.5))))), 1.0);
	} else if (x <= 3.9e+252) {
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 51.0)
		tmp = fma(x, Float64(0.5 * Float64(Float64(eps_m * eps_m) * Float64(x * Float64(0.5 + fma(x, -0.6666666666666666, 0.5))))), 1.0);
	elseif (x <= 3.9e+252)
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps_m * eps_m))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 51.0], N[(x * N[(0.5 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * N[(x * N[(0.5 + N[(x * -0.6666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 3.9e+252], N[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 51

   1. Initial program 63.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 51.6%

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

   6. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 90.8

          \[\leadsto \mathsf{fma}\left(x, 0.5 \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(0.5 + \color{blue}{\mathsf{fma}\left(x, -0.6666666666666666, 0.5\right)}\right)\right)\right), 1\right) \]

   8. Simplified 90.8%

      \[\leadsto \mathsf{fma}\left(x, 0.5 \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(0.5 + \mathsf{fma}\left(x, -0.6666666666666666, 0.5\right)\right)\right)\right)}, 1\right) \]

   if 51 < x < 3.89999999999999965e252

   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. Simplified 39.4%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 40.0

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

   8. Simplified 40.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 66.6

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

   11. Simplified 66.6%

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

   if 3.89999999999999965e252 < 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. Simplified 26.2%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      6. mul0-rgt N/A

         \[\leadsto \left(x \cdot \color{blue}{0}\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{0} \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \frac{1}{2}}{\varepsilon}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \]

      10. mul0-lft N/A

          \[\leadsto \frac{\color{blue}{0 \cdot x}}{\varepsilon} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{0 \cdot \frac{x}{\varepsilon}} \]

      12. mul0-lft 75.4

          \[\leadsto \color{blue}{0} \]

   8. Simplified 75.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(eps\_m \cdot eps\_m\right)\\ \mathbf{if}\;x \leq 47000:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot t\_0, 1\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+109}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+252}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* eps_m eps_m))))
   (if (<= x 47000.0)
     (fma x (* 0.5 t_0) 1.0)
     (if (<= x 8e+109) 0.0 (if (<= x 3.9e+252) (* 0.5 (* x t_0)) 0.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (eps_m * eps_m);
	double tmp;
	if (x <= 47000.0) {
		tmp = fma(x, (0.5 * t_0), 1.0);
	} else if (x <= 8e+109) {
		tmp = 0.0;
	} else if (x <= 3.9e+252) {
		tmp = 0.5 * (x * t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(eps_m * eps_m))
	tmp = 0.0
	if (x <= 47000.0)
		tmp = fma(x, Float64(0.5 * t_0), 1.0);
	elseif (x <= 8e+109)
		tmp = 0.0;
	elseif (x <= 3.9e+252)
		tmp = Float64(0.5 * Float64(x * t_0));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 47000.0], N[(x * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 8e+109], 0.0, If[LessEqual[x, 3.9e+252], N[(0.5 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < 47000

   1. Initial program 63.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. Simplified 89.2%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 89.7

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

   8. Simplified 89.7%

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

   if 47000 < x < 7.99999999999999985e109 or 3.89999999999999965e252 < 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. Simplified 22.8%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      6. mul0-rgt N/A

         \[\leadsto \left(x \cdot \color{blue}{0}\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{0} \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \frac{1}{2}}{\varepsilon}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \]

      10. mul0-lft N/A

          \[\leadsto \frac{\color{blue}{0 \cdot x}}{\varepsilon} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{0 \cdot \frac{x}{\varepsilon}} \]

      12. mul0-lft 69.2

          \[\leadsto \color{blue}{0} \]

   8. Simplified 69.2%

      \[\leadsto \color{blue}{0} \]

   if 7.99999999999999985e109 < x < 3.89999999999999965e252

   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. Simplified 55.3%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 55.8

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

   8. Simplified 55.8%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 79.3

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

   11. Simplified 79.3%

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

Alternative 10: 69.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot \left(eps\_m \cdot \left(x \cdot \left(x \cdot eps\_m\right)\right)\right)\\ \mathbf{elif}\;x \leq 840:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2.25e-13)
   (* 0.5 (* eps_m (* x (* x eps_m))))
   (if (<= x 840.0) 1.0 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.25e-13) {
		tmp = 0.5 * (eps_m * (x * (x * eps_m)));
	} else if (x <= 840.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	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 <= (-2.25d-13)) then
        tmp = 0.5d0 * (eps_m * (x * (x * eps_m)))
    else if (x <= 840.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    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 <= -2.25e-13) {
		tmp = 0.5 * (eps_m * (x * (x * eps_m)));
	} else if (x <= 840.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2.25e-13:
		tmp = 0.5 * (eps_m * (x * (x * eps_m)))
	elif x <= 840.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2.25e-13)
		tmp = Float64(0.5 * Float64(eps_m * Float64(x * Float64(x * eps_m))));
	elseif (x <= 840.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2.25e-13)
		tmp = 0.5 * (eps_m * (x * (x * eps_m)));
	elseif (x <= 840.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2.25e-13], N[(0.5 * N[(eps$95$m * N[(x * N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 840.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.25e-13

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 24.6%

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

   6. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 90.8

          \[\leadsto 0.5 \cdot \left(\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot \left(0.5 + \color{blue}{\mathsf{fma}\left(x, -0.6666666666666666, 0.5\right)}\right)\right)\right) \]

   8. Simplified 90.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot \left(0.5 + \mathsf{fma}\left(x, -0.6666666666666666, 0.5\right)\right)\right)\right)} \]

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 86.1

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

   11. Simplified 86.1%

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

   if -2.25e-13 < x < 840

   1. Initial program 53.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.9%

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

   if 840 < 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. Simplified 36.0%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      6. mul0-rgt N/A

         \[\leadsto \left(x \cdot \color{blue}{0}\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{0} \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \frac{1}{2}}{\varepsilon}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \]

      10. mul0-lft N/A

          \[\leadsto \frac{\color{blue}{0 \cdot x}}{\varepsilon} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{0 \cdot \frac{x}{\varepsilon}} \]

      12. mul0-lft 59.9

          \[\leadsto \color{blue}{0} \]

   8. Simplified 59.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 66.4% accurate, 10.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.6) (fma x (fma x (fma x -0.16666666666666666 0.5) -1.0) 1.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.6) {
		tmp = fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.6)
		tmp = fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.6], N[(x * N[(x * N[(x * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   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. Simplified 98.5%

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

   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. neg-mul-1 N/A

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

      2. distribute-lft-out N/A

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

      3. distribute-rgt-out N/A

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

      4. neg-mul-1 N/A

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

      5. metadata-eval N/A

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

      6. 1-exp N/A

         \[\leadsto e^{-1 \cdot x} \cdot \color{blue}{e^{0}} \]

      7. exp-sum N/A

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

      8. *-commutative N/A

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

      9. mul0-rgt N/A

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

      10. distribute-lft-out N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-exp.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 74.5

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

   8. Simplified 74.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 70.2

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

   11. Simplified 70.2%

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

   if 1.6000000000000001 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 36.4%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      6. mul0-rgt N/A

         \[\leadsto \left(x \cdot \color{blue}{0}\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{0} \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \frac{1}{2}}{\varepsilon}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \]

      10. mul0-lft N/A

          \[\leadsto \frac{\color{blue}{0 \cdot x}}{\varepsilon} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{0 \cdot \frac{x}{\varepsilon}} \]

      12. mul0-lft 58.5

          \[\leadsto \color{blue}{0} \]

   8. Simplified 58.5%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 64.2% accurate, 14.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 840:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 840.0) (fma x (fma 0.5 x -1.0) 1.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 840.0) {
		tmp = fma(x, fma(0.5, x, -1.0), 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 840.0)
		tmp = fma(x, fma(0.5, x, -1.0), 1.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 840.0], N[(x * N[(0.5 * x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 840

   1. Initial program 63.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. Simplified 98.0%

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

   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. neg-mul-1 N/A

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

      2. distribute-lft-out N/A

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

      3. distribute-rgt-out N/A

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

      4. neg-mul-1 N/A

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

      5. metadata-eval N/A

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

      6. 1-exp N/A

         \[\leadsto e^{-1 \cdot x} \cdot \color{blue}{e^{0}} \]

      7. exp-sum N/A

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

      8. *-commutative N/A

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

      9. mul0-rgt N/A

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

      10. distribute-lft-out N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-exp.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 73.8

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

   8. Simplified 73.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 67.3

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

   11. Simplified 67.3%

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

   if 840 < 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. Simplified 36.0%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      6. mul0-rgt N/A

         \[\leadsto \left(x \cdot \color{blue}{0}\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{0} \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \frac{1}{2}}{\varepsilon}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \]

      10. mul0-lft N/A

          \[\leadsto \frac{\color{blue}{0 \cdot x}}{\varepsilon} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{0 \cdot \frac{x}{\varepsilon}} \]

      12. mul0-lft 59.9

          \[\leadsto \color{blue}{0} \]

   8. Simplified 59.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.9% accurate, 27.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 1.0) (- 1.0 x) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 - x;
	} else {
		tmp = 0.0;
	}
	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 = 1.0d0 - x
    else
        tmp = 0.0d0
    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 = 1.0 - x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 - x
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.0], N[(1.0 - x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   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. Simplified 98.5%

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

   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. neg-mul-1 N/A

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

      2. distribute-lft-out N/A

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

      3. distribute-rgt-out N/A

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

      4. neg-mul-1 N/A

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

      5. metadata-eval N/A

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

      6. 1-exp N/A

         \[\leadsto e^{-1 \cdot x} \cdot \color{blue}{e^{0}} \]

      7. exp-sum N/A

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

      8. *-commutative N/A

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

      9. mul0-rgt N/A

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

      10. distribute-lft-out N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-exp.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 74.5

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

   8. Simplified 74.5%

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

   9. Taylor expanded in x around 0

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

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 53.8

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

   11. Simplified 53.8%

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

   if 1 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 36.4%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      6. mul0-rgt N/A

         \[\leadsto \left(x \cdot \color{blue}{0}\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{0} \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \frac{1}{2}}{\varepsilon}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \]

      10. mul0-lft N/A

          \[\leadsto \frac{\color{blue}{0 \cdot x}}{\varepsilon} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{0 \cdot \frac{x}{\varepsilon}} \]

      12. mul0-lft 58.5

          \[\leadsto \color{blue}{0} \]

   8. Simplified 58.5%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 56.9% accurate, 38.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 840:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 840.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 840.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	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 <= 840.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    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 <= 840.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 840.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 840.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 840.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 840.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 840

   1. Initial program 63.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 53.1%

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

   if 840 < 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. Simplified 36.0%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      6. mul0-rgt N/A

         \[\leadsto \left(x \cdot \color{blue}{0}\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{0} \cdot \frac{\frac{1}{2}}{\varepsilon} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \frac{1}{2}}{\varepsilon}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \]

      10. mul0-lft N/A

          \[\leadsto \frac{\color{blue}{0 \cdot x}}{\varepsilon} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{0 \cdot \frac{x}{\varepsilon}} \]

      12. mul0-lft 59.9

          \[\leadsto \color{blue}{0} \]

   8. Simplified 59.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 16.7% accurate, 273.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 0.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 75.0%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
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. Simplified 72.4%

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

6. Taylor expanded in eps around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. distribute-rgt-out N/A

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

   5. metadata-eval N/A

      \[\leadsto \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

   6. mul0-rgt N/A

      \[\leadsto \left(x \cdot \color{blue}{0}\right) \cdot \frac{\frac{1}{2}}{\varepsilon} \]

   7. mul0-rgt N/A

      \[\leadsto \color{blue}{0} \cdot \frac{\frac{1}{2}}{\varepsilon} \]

   8. associate-/l* N/A

      \[\leadsto \color{blue}{\frac{0 \cdot \frac{1}{2}}{\varepsilon}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \]

   10. mul0-lft N/A

       \[\leadsto \frac{\color{blue}{0 \cdot x}}{\varepsilon} \]

   11. associate-*r/ N/A

       \[\leadsto \color{blue}{0 \cdot \frac{x}{\varepsilon}} \]

   12. mul0-lft 20.6

       \[\leadsto \color{blue}{0} \]

8. Simplified 20.6%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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))