NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.6% → 100.0%

Time: 13.6s

Alternatives: 11

Speedup: 14.2×

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

• could not determine a ground truth

  1. x = -1.99999999999999997e77
  2. eps = -2e-231

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: 72.6% 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: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = (x + 1.0) * exp((0.0 - x));
	} else {
		tmp = 0.5 * (exp((x * (-1.0 - eps_m))) + exp((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 <= 1.0d0) then
        tmp = (x + 1.0d0) * exp((0.0d0 - x))
    else
        tmp = 0.5d0 * (exp((x * ((-1.0d0) - eps_m))) + exp((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 <= 1.0) {
		tmp = (x + 1.0) * Math.exp((0.0 - x));
	} else {
		tmp = 0.5 * (Math.exp((x * (-1.0 - eps_m))) + Math.exp((x * eps_m)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.0:
		tmp = (x + 1.0) * math.exp((0.0 - x))
	else:
		tmp = 0.5 * (math.exp((x * (-1.0 - eps_m))) + math.exp((x * eps_m)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.0)
		tmp = Float64(Float64(x + 1.0) * exp(Float64(0.0 - x)));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(x * Float64(-1.0 - eps_m))) + exp(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 <= 1.0)
		tmp = (x + 1.0) * exp((0.0 - x));
	else
		tmp = 0.5 * (exp((x * (-1.0 - eps_m))) + exp((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, 1.0], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 62.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(x + 1\right), \color{blue}{\left(e^{-1 \cdot x}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(e^{\color{blue}{-1 \cdot x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(-1 \cdot x\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(0 - x\right)\right)\right) \]

      12. --lowering--.f64 72.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   7. Simplified 72.1%

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

   if 1 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

4. Final simplification 79.4%

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

Alternative 2: 98.8% accurate, 1.1× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (* 0.5 (+ (exp (* x (+ eps_m -1.0))) (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 + -1.0))) + 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 + (-1.0d0)))) + 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 + -1.0))) + 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 + -1.0))) + math.exp((x * (-1.0 - eps_m))))

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(0.5 * Float64(exp(Float64(x * Float64(eps_m + -1.0))) + 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 + -1.0))) + 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[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

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

3. Simplified 72.4%

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

5. Taylor expanded in eps around inf

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

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

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

   2. metadata-eval N/A

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

   3. distribute-lft-out N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

   14. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

   16. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

   18. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

   19. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

   20. --lowering--.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

7. Simplified 99.6%

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

Alternative 3: 92.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \frac{0.25 \cdot \left(eps\_m \cdot \left(eps\_m \cdot \left(x \cdot eps\_m\right)\right)\right)}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (* (+ x 1.0) (exp (- 0.0 x)))
   (+ 1.0 (* x (/ (* 0.25 (* eps_m (* eps_m (* x eps_m)))) eps_m)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = (x + 1.0) * exp((0.0 - x));
	} else {
		tmp = 1.0 + (x * ((0.25 * (eps_m * (eps_m * (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 (eps_m <= 1.0d0) then
        tmp = (x + 1.0d0) * exp((0.0d0 - x))
    else
        tmp = 1.0d0 + (x * ((0.25d0 * (eps_m * (eps_m * (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 (eps_m <= 1.0) {
		tmp = (x + 1.0) * Math.exp((0.0 - x));
	} else {
		tmp = 1.0 + (x * ((0.25 * (eps_m * (eps_m * (x * eps_m)))) / eps_m));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.0:
		tmp = (x + 1.0) * math.exp((0.0 - x))
	else:
		tmp = 1.0 + (x * ((0.25 * (eps_m * (eps_m * (x * eps_m)))) / eps_m))
	return tmp

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.0], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(N[(0.25 * N[(eps$95$m * N[(eps$95$m * N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 62.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(x + 1\right), \color{blue}{\left(e^{-1 \cdot x}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(e^{\color{blue}{-1 \cdot x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(-1 \cdot x\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(0 - x\right)\right)\right) \]

      12. --lowering--.f64 72.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   7. Simplified 72.1%

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

   if 1 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \frac{-1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      6. /-lowering-/.f64 69.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right) \]

   7. Simplified 69.7%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\varepsilon - 1\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon - 1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   10. Simplified 83.0%

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

   11. Taylor expanded in eps around 0

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(\frac{1}{4} \cdot x + \varepsilon \cdot \left(\frac{-1}{4} \cdot x + \varepsilon \cdot \left(\frac{1}{2} + \left(\frac{-1}{4} \cdot x + \frac{1}{4} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right) - \frac{1}{2}}{\varepsilon}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot x + \varepsilon \cdot \left(\frac{-1}{4} \cdot x + \varepsilon \cdot \left(\frac{1}{2} + \left(\frac{-1}{4} \cdot x + \frac{1}{4} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right) - \frac{1}{2}\right), \color{blue}{\varepsilon}\right)\right)\right) \]

   13. Simplified 66.1%

       \[\leadsto 1 + x \cdot \color{blue}{\frac{\left(x \cdot \left(0.25 + \varepsilon \cdot -0.25\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.5 + x \cdot \left(-0.25 + \varepsilon \cdot 0.25\right)\right)\right) + -0.5}{\varepsilon}} \]

   14. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot \left({\varepsilon}^{3} \cdot x\right)\right)}, \varepsilon\right)\right)\right) \]
   15. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\varepsilon}^{3} \cdot x\right)\right), \varepsilon\right)\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right)\right), \varepsilon\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(\varepsilon \cdot {\varepsilon}^{2}\right) \cdot x\right)\right), \varepsilon\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\varepsilon \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right), \varepsilon\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot x\right)\right)\right), \varepsilon\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right)\right)\right), \varepsilon\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)\right), \varepsilon\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot x\right)\right)\right)\right), \varepsilon\right)\right)\right) \]

       9. *-lowering-*.f64 93.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, x\right)\right)\right)\right), \varepsilon\right)\right)\right) \]

   16. Simplified 93.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \frac{0.25 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right)\right)\right)}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;1 + x \cdot \frac{0.25 \cdot \left(eps\_m \cdot \left(eps\_m \cdot \left(x \cdot eps\_m\right)\right)\right)}{eps\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(eps\_m \cdot eps\_m\right) \cdot \left(x \cdot \left(x \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.25e-5)
   (+ 1.0 (* x (/ (* 0.25 (* eps_m (* eps_m (* x eps_m)))) eps_m)))
   (* (* eps_m eps_m) (* x (* x 0.25)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.25e-5) {
		tmp = 1.0 + (x * ((0.25 * (eps_m * (eps_m * (x * eps_m)))) / eps_m));
	} else {
		tmp = (eps_m * eps_m) * (x * (x * 0.25));
	}
	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 <= 1.25d-5) then
        tmp = 1.0d0 + (x * ((0.25d0 * (eps_m * (eps_m * (x * eps_m)))) / eps_m))
    else
        tmp = (eps_m * eps_m) * (x * (x * 0.25d0))
    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 <= 1.25e-5) {
		tmp = 1.0 + (x * ((0.25 * (eps_m * (eps_m * (x * eps_m)))) / eps_m));
	} else {
		tmp = (eps_m * eps_m) * (x * (x * 0.25));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.25e-5:
		tmp = 1.0 + (x * ((0.25 * (eps_m * (eps_m * (x * eps_m)))) / eps_m))
	else:
		tmp = (eps_m * eps_m) * (x * (x * 0.25))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.25e-5)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(0.25 * Float64(eps_m * Float64(eps_m * Float64(x * eps_m)))) / eps_m)));
	else
		tmp = Float64(Float64(eps_m * eps_m) * Float64(x * Float64(x * 0.25)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.25e-5)
		tmp = 1.0 + (x * ((0.25 * (eps_m * (eps_m * (x * eps_m)))) / eps_m));
	else
		tmp = (eps_m * eps_m) * (x * (x * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.25e-5], N[(1.0 + N[(x * N[(N[(0.25 * N[(eps$95$m * N[(eps$95$m * N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25000000000000006e-5

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \frac{-1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      6. /-lowering-/.f64 40.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right) \]

   7. Simplified 40.2%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\varepsilon - 1\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon - 1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   10. Simplified 66.8%

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

   11. Taylor expanded in eps around 0

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(\frac{1}{4} \cdot x + \varepsilon \cdot \left(\frac{-1}{4} \cdot x + \varepsilon \cdot \left(\frac{1}{2} + \left(\frac{-1}{4} \cdot x + \frac{1}{4} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right) - \frac{1}{2}}{\varepsilon}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot x + \varepsilon \cdot \left(\frac{-1}{4} \cdot x + \varepsilon \cdot \left(\frac{1}{2} + \left(\frac{-1}{4} \cdot x + \frac{1}{4} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right) - \frac{1}{2}\right), \color{blue}{\varepsilon}\right)\right)\right) \]

   13. Simplified 63.0%

       \[\leadsto 1 + x \cdot \color{blue}{\frac{\left(x \cdot \left(0.25 + \varepsilon \cdot -0.25\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.5 + x \cdot \left(-0.25 + \varepsilon \cdot 0.25\right)\right)\right) + -0.5}{\varepsilon}} \]

   14. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot \left({\varepsilon}^{3} \cdot x\right)\right)}, \varepsilon\right)\right)\right) \]
   15. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\varepsilon}^{3} \cdot x\right)\right), \varepsilon\right)\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right)\right), \varepsilon\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(\varepsilon \cdot {\varepsilon}^{2}\right) \cdot x\right)\right), \varepsilon\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\varepsilon \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right), \varepsilon\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot x\right)\right)\right), \varepsilon\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right)\right)\right), \varepsilon\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)\right), \varepsilon\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot x\right)\right)\right)\right), \varepsilon\right)\right)\right) \]

       9. *-lowering-*.f64 96.5%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, x\right)\right)\right)\right), \varepsilon\right)\right)\right) \]

   16. Simplified 96.5%

       \[\leadsto 1 + x \cdot \frac{\color{blue}{0.25 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)}}{\varepsilon} \]

   if 1.25000000000000006e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \frac{-1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      6. /-lowering-/.f64 29.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right) \]

   7. Simplified 29.3%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\varepsilon - 1\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon - 1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   10. Simplified 45.4%

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

   11. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left(\frac{1}{4} \cdot {x}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{\frac{1}{4}} \cdot {x}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{\frac{1}{4}} \cdot {x}^{2}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({x}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\left(x \cdot x\right) \cdot \frac{1}{4}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{4}\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(x \cdot \left(\frac{1}{4} \cdot \color{blue}{x}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot x\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{4}}\right)\right)\right) \]

       13. *-lowering-*.f64 58.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{4}}\right)\right)\right) \]

   13. Simplified 58.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;1 + x \cdot \frac{0.25 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right)\right)\right)}{\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(x \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;0.25 \cdot \left(eps\_m \cdot \left(eps\_m \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-13}:\\ \;\;\;\;1 + x \cdot \left(x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(eps\_m \cdot eps\_m\right) \cdot \left(x \cdot \left(x \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5.2e-5)
   (* 0.25 (* eps_m (* eps_m (* x x))))
   (if (<= x 3.2e-13)
     (+ 1.0 (* x (* x -0.5)))
     (* (* eps_m eps_m) (* x (* x 0.25))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5.2e-5) {
		tmp = 0.25 * (eps_m * (eps_m * (x * x)));
	} else if (x <= 3.2e-13) {
		tmp = 1.0 + (x * (x * -0.5));
	} else {
		tmp = (eps_m * eps_m) * (x * (x * 0.25));
	}
	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 <= (-5.2d-5)) then
        tmp = 0.25d0 * (eps_m * (eps_m * (x * x)))
    else if (x <= 3.2d-13) then
        tmp = 1.0d0 + (x * (x * (-0.5d0)))
    else
        tmp = (eps_m * eps_m) * (x * (x * 0.25d0))
    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 <= -5.2e-5) {
		tmp = 0.25 * (eps_m * (eps_m * (x * x)));
	} else if (x <= 3.2e-13) {
		tmp = 1.0 + (x * (x * -0.5));
	} else {
		tmp = (eps_m * eps_m) * (x * (x * 0.25));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5.2e-5:
		tmp = 0.25 * (eps_m * (eps_m * (x * x)))
	elif x <= 3.2e-13:
		tmp = 1.0 + (x * (x * -0.5))
	else:
		tmp = (eps_m * eps_m) * (x * (x * 0.25))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5.2e-5)
		tmp = Float64(0.25 * Float64(eps_m * Float64(eps_m * Float64(x * x))));
	elseif (x <= 3.2e-13)
		tmp = Float64(1.0 + Float64(x * Float64(x * -0.5)));
	else
		tmp = Float64(Float64(eps_m * eps_m) * Float64(x * Float64(x * 0.25)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -5.2e-5)
		tmp = 0.25 * (eps_m * (eps_m * (x * x)));
	elseif (x <= 3.2e-13)
		tmp = 1.0 + (x * (x * -0.5));
	else
		tmp = (eps_m * eps_m) * (x * (x * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5.2e-5], N[(0.25 * N[(eps$95$m * N[(eps$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e-13], N[(1.0 + N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.19999999999999968e-5

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \frac{-1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      6. /-lowering-/.f64 56.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right) \]

   7. Simplified 56.6%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\varepsilon - 1\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon - 1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   10. Simplified 93.9%

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

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{4} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot {\varepsilon}^{2}\right) \cdot x\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot x\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\varepsilon}^{2}}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot {\color{blue}{\varepsilon}}^{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 93.9%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

   13. Simplified 93.9%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.25\right)\right)} \]

   14. Taylor expanded in x around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left({\varepsilon}^{2} \cdot {x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 93.9%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   16. Simplified 93.9%

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

   if -5.19999999999999968e-5 < x < 3.2e-13

   1. Initial program 47.9%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 67.9%

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

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \left(1 + {x}^{2} \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\frac{1}{12} \cdot x + \frac{1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right) + {x}^{2} \cdot \left(\left(\frac{1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) - \frac{1}{4}\right)}{\varepsilon}} \]

   10. Simplified 70.1%

       \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot 0.08333333333333333 + \left(\left(0.25 + 0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + -0.25\right)\right) + \varepsilon \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(-0.25 + -0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + \left(x \cdot 0.25 + -0.25\right)\right)\right)}{\varepsilon}} \]

   11. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2}\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{2}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \frac{-1}{2}\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{-1}{2} \cdot \color{blue}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       8. *-lowering-*.f64 78.5%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   13. Simplified 78.5%

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

   if 3.2e-13 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \frac{-1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      6. /-lowering-/.f64 28.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right) \]

   7. Simplified 28.7%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\varepsilon - 1\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon - 1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   10. Simplified 45.5%

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

   11. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left(\frac{1}{4} \cdot {x}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{\frac{1}{4}} \cdot {x}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{\frac{1}{4}} \cdot {x}^{2}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({x}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\left(x \cdot x\right) \cdot \frac{1}{4}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{4}\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(x \cdot \left(\frac{1}{4} \cdot \color{blue}{x}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot x\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{4}}\right)\right)\right) \]

       13. *-lowering-*.f64 58.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{4}}\right)\right)\right) \]

   13. Simplified 58.3%

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

Alternative 6: 69.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 0.25 \cdot \left(eps\_m \cdot \left(eps\_m \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-12}:\\ \;\;\;\;1 + x \cdot \left(x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* 0.25 (* eps_m (* eps_m (* x x))))))
   (if (<= x -0.0023) t_0 (if (<= x 2.2e-12) (+ 1.0 (* x (* x -0.5))) t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 0.25 * (eps_m * (eps_m * (x * x)));
	double tmp;
	if (x <= -0.0023) {
		tmp = t_0;
	} else if (x <= 2.2e-12) {
		tmp = 1.0 + (x * (x * -0.5));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 0.25d0 * (eps_m * (eps_m * (x * x)))
    if (x <= (-0.0023d0)) then
        tmp = t_0
    else if (x <= 2.2d-12) then
        tmp = 1.0d0 + (x * (x * (-0.5d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 0.25 * (eps_m * (eps_m * (x * x)));
	double tmp;
	if (x <= -0.0023) {
		tmp = t_0;
	} else if (x <= 2.2e-12) {
		tmp = 1.0 + (x * (x * -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 0.25 * (eps_m * (eps_m * (x * x)))
	tmp = 0
	if x <= -0.0023:
		tmp = t_0
	elif x <= 2.2e-12:
		tmp = 1.0 + (x * (x * -0.5))
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(0.25 * Float64(eps_m * Float64(eps_m * Float64(x * x))))
	tmp = 0.0
	if (x <= -0.0023)
		tmp = t_0;
	elseif (x <= 2.2e-12)
		tmp = Float64(1.0 + Float64(x * Float64(x * -0.5)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 0.25 * (eps_m * (eps_m * (x * x)));
	tmp = 0.0;
	if (x <= -0.0023)
		tmp = t_0;
	elseif (x <= 2.2e-12)
		tmp = 1.0 + (x * (x * -0.5));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(0.25 * N[(eps$95$m * N[(eps$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0023], t$95$0, If[LessEqual[x, 2.2e-12], N[(1.0 + N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0023 or 2.19999999999999992e-12 < x

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \frac{-1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      6. /-lowering-/.f64 35.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right) \]

   7. Simplified 35.4%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\varepsilon - 1\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon - 1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   10. Simplified 57.0%

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

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{4} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot {\varepsilon}^{2}\right) \cdot x\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot x\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\varepsilon}^{2}}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot {\color{blue}{\varepsilon}}^{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 58.2%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

   13. Simplified 58.2%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.25\right)\right)} \]

   14. Taylor expanded in x around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left({\varepsilon}^{2} \cdot {x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 62.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   16. Simplified 62.3%

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

   if -0.0023 < x < 2.19999999999999992e-12

   1. Initial program 47.9%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 67.9%

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

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \left(1 + {x}^{2} \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\frac{1}{12} \cdot x + \frac{1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right) + {x}^{2} \cdot \left(\left(\frac{1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) - \frac{1}{4}\right)}{\varepsilon}} \]

   10. Simplified 70.1%

       \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot 0.08333333333333333 + \left(\left(0.25 + 0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + -0.25\right)\right) + \varepsilon \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(-0.25 + -0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + \left(x \cdot 0.25 + -0.25\right)\right)\right)}{\varepsilon}} \]

   11. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2}\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{2}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \frac{-1}{2}\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{-1}{2} \cdot \color{blue}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       8. *-lowering-*.f64 78.5%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   13. Simplified 78.5%

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

Alternative 7: 57.1% accurate, 13.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4500:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+145}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 4500.0)
   1.0
   (if (<= x 4e+145) 0.0 (* x (* (* x x) 0.3333333333333333)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 4500.0) {
		tmp = 1.0;
	} else if (x <= 4e+145) {
		tmp = 0.0;
	} else {
		tmp = x * ((x * x) * 0.3333333333333333);
	}
	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 <= 4500.0d0) then
        tmp = 1.0d0
    else if (x <= 4d+145) then
        tmp = 0.0d0
    else
        tmp = x * ((x * x) * 0.3333333333333333d0)
    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 <= 4500.0) {
		tmp = 1.0;
	} else if (x <= 4e+145) {
		tmp = 0.0;
	} else {
		tmp = x * ((x * x) * 0.3333333333333333);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 4500.0:
		tmp = 1.0
	elif x <= 4e+145:
		tmp = 0.0
	else:
		tmp = x * ((x * x) * 0.3333333333333333)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 4500.0)
		tmp = 1.0;
	elseif (x <= 4e+145)
		tmp = 0.0;
	else
		tmp = Float64(x * Float64(Float64(x * x) * 0.3333333333333333));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 4500.0)
		tmp = 1.0;
	elseif (x <= 4e+145)
		tmp = 0.0;
	else
		tmp = x * ((x * x) * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 4500.0], 1.0, If[LessEqual[x, 4e+145], 0.0, N[(x * N[(N[(x * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4500

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 64.1%

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

   if 4500 < x < 4e145

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 3.8%

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

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \left(1 + {x}^{2} \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\frac{1}{12} \cdot x + \frac{1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right) + {x}^{2} \cdot \left(\left(\frac{1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) - \frac{1}{4}\right)}{\varepsilon}} \]

   10. Simplified 31.3%

       \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot 0.08333333333333333 + \left(\left(0.25 + 0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + -0.25\right)\right) + \varepsilon \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(-0.25 + -0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + \left(x \cdot 0.25 + -0.25\right)\right)\right)}{\varepsilon}} \]

   11. Taylor expanded in eps around 0

       \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)}{\varepsilon}} \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)}{\varepsilon} \]

       2. distribute-rgt-out N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(\frac{-1}{12} + \frac{1}{12}\right)\right)}{\varepsilon} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot 0\right)}{\varepsilon} \]

       4. mul0-rgt N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot 0}{\varepsilon} \]

       5. associate-*l* N/A

          \[\leadsto \frac{x \cdot \left(x \cdot 0\right)}{\varepsilon} \]

       6. mul0-rgt N/A

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

       7. mul0-rgt N/A

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

       8. /-lowering-/.f64 55.3%

          \[\leadsto \mathsf{/.f64}\left(0, \color{blue}{\varepsilon}\right) \]

   13. Simplified 55.3%

       \[\leadsto \color{blue}{\frac{0}{\varepsilon}} \]
   14. Step-by-step derivation

       1. div0 55.3%

          \[\leadsto 0 \]

   15. Applied egg-rr 55.3%

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

   if 4e145 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 2.5%

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

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \left(1 + {x}^{2} \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\frac{1}{12} \cdot x + \frac{1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right) + {x}^{2} \cdot \left(\left(\frac{1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) - \frac{1}{4}\right)}{\varepsilon}} \]

   10. Simplified 4.4%

       \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot 0.08333333333333333 + \left(\left(0.25 + 0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + -0.25\right)\right) + \varepsilon \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(-0.25 + -0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + \left(x \cdot 0.25 + -0.25\right)\right)\right)}{\varepsilon}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{3} \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)}, \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot {x}^{3}\right) \cdot \frac{1}{3}\right), \varepsilon\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\varepsilon \cdot \left({x}^{3} \cdot \frac{1}{3}\right)\right), \varepsilon\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\varepsilon \cdot \left(\frac{1}{3} \cdot {x}^{3}\right)\right), \varepsilon\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{3} \cdot {x}^{3}\right)\right), \varepsilon\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({x}^{3} \cdot \frac{1}{3}\right)\right), \varepsilon\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left({x}^{3}\right), \frac{1}{3}\right)\right), \varepsilon\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \frac{1}{3}\right)\right), \varepsilon\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), \frac{1}{3}\right)\right), \varepsilon\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), \frac{1}{3}\right)\right), \varepsilon\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \frac{1}{3}\right)\right), \varepsilon\right) \]

       11. *-lowering-*.f64 58.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{3}\right)\right), \varepsilon\right) \]

   13. Simplified 58.4%

       \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.3333333333333333\right)}}{\varepsilon} \]

   14. Taylor expanded in eps around 0

       \[\leadsto \color{blue}{\frac{1}{3} \cdot {x}^{3}} \]
   15. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {x}^{3} \cdot \color{blue}{\frac{1}{3}} \]

       2. cube-mult N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{3} \]

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \frac{1}{3}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{3}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{3}\right)\right) \]

       8. *-lowering-*.f64 58.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{3}\right)\right) \]

   16. Simplified 58.4%

       \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.1% accurate, 14.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;1 + x \cdot \left(\left(x \cdot eps\_m\right) \cdot \left(eps\_m \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(eps\_m \cdot eps\_m\right) \cdot \left(x \cdot \left(x \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.25e-5)
   (+ 1.0 (* x (* (* x eps_m) (* eps_m 0.25))))
   (* (* eps_m eps_m) (* x (* x 0.25)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.25e-5) {
		tmp = 1.0 + (x * ((x * eps_m) * (eps_m * 0.25)));
	} else {
		tmp = (eps_m * eps_m) * (x * (x * 0.25));
	}
	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 <= 1.25d-5) then
        tmp = 1.0d0 + (x * ((x * eps_m) * (eps_m * 0.25d0)))
    else
        tmp = (eps_m * eps_m) * (x * (x * 0.25d0))
    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 <= 1.25e-5) {
		tmp = 1.0 + (x * ((x * eps_m) * (eps_m * 0.25)));
	} else {
		tmp = (eps_m * eps_m) * (x * (x * 0.25));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.25e-5:
		tmp = 1.0 + (x * ((x * eps_m) * (eps_m * 0.25)))
	else:
		tmp = (eps_m * eps_m) * (x * (x * 0.25))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.25e-5)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * eps_m) * Float64(eps_m * 0.25))));
	else
		tmp = Float64(Float64(eps_m * eps_m) * Float64(x * Float64(x * 0.25)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.25e-5)
		tmp = 1.0 + (x * ((x * eps_m) * (eps_m * 0.25)));
	else
		tmp = (eps_m * eps_m) * (x * (x * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.25e-5], N[(1.0 + N[(x * N[(N[(x * eps$95$m), $MachinePrecision] * N[(eps$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25000000000000006e-5

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \frac{-1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      6. /-lowering-/.f64 40.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right) \]

   7. Simplified 40.2%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\varepsilon - 1\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon - 1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   10. Simplified 66.8%

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

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{4} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot {\varepsilon}^{2}\right) \cdot x\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot x\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\varepsilon}^{2}}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot {\color{blue}{\varepsilon}}^{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 91.2%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

   13. Simplified 91.2%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.25\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)}\right)\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot \varepsilon\right), \color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right), \left(\color{blue}{\varepsilon} \cdot \frac{1}{4}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 92.1%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

   15. Applied egg-rr 92.1%

       \[\leadsto 1 + x \cdot \color{blue}{\left(\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 0.25\right)\right)} \]

   if 1.25000000000000006e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \frac{-1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      6. /-lowering-/.f64 29.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right) \]

   7. Simplified 29.3%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\varepsilon - 1\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon - 1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   10. Simplified 45.4%

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

   11. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left(\frac{1}{4} \cdot {x}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{\frac{1}{4}} \cdot {x}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{\frac{1}{4}} \cdot {x}^{2}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({x}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\left(x \cdot x\right) \cdot \frac{1}{4}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{4}\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(x \cdot \left(\frac{1}{4} \cdot \color{blue}{x}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot x\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{4}}\right)\right)\right) \]

       13. *-lowering-*.f64 58.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{4}}\right)\right)\right) \]

   13. Simplified 58.4%

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

Alternative 9: 78.2% accurate, 14.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(eps\_m \cdot eps\_m\right) \cdot \left(x \cdot \left(x \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.25e-5)
   (+ 1.0 (* x (* x (* 0.25 (* eps_m eps_m)))))
   (* (* eps_m eps_m) (* x (* x 0.25)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.25e-5) {
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	} else {
		tmp = (eps_m * eps_m) * (x * (x * 0.25));
	}
	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 <= 1.25d-5) then
        tmp = 1.0d0 + (x * (x * (0.25d0 * (eps_m * eps_m))))
    else
        tmp = (eps_m * eps_m) * (x * (x * 0.25d0))
    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 <= 1.25e-5) {
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	} else {
		tmp = (eps_m * eps_m) * (x * (x * 0.25));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.25e-5:
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))))
	else:
		tmp = (eps_m * eps_m) * (x * (x * 0.25))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.25e-5)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.25 * Float64(eps_m * eps_m)))));
	else
		tmp = Float64(Float64(eps_m * eps_m) * Float64(x * Float64(x * 0.25)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.25e-5)
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	else
		tmp = (eps_m * eps_m) * (x * (x * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.25e-5], N[(1.0 + N[(x * N[(x * N[(0.25 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25000000000000006e-5

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \frac{-1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      6. /-lowering-/.f64 40.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right) \]

   7. Simplified 40.2%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\varepsilon - 1\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon - 1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   10. Simplified 66.8%

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

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{4} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot {\varepsilon}^{2}\right) \cdot x\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot x\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\varepsilon}^{2}}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot {\color{blue}{\varepsilon}}^{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 91.2%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

   13. Simplified 91.2%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.25\right)\right)} \]

   if 1.25000000000000006e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \frac{-1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{\varepsilon}\right), \frac{-1}{2}\right)\right) \]

      6. /-lowering-/.f64 29.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right) \]

   7. Simplified 29.3%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\varepsilon - 1\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon - 1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \left(\frac{1}{2} \cdot \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   10. Simplified 45.4%

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

   11. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left(\frac{1}{4} \cdot {x}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{\frac{1}{4}} \cdot {x}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{\frac{1}{4}} \cdot {x}^{2}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({x}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\left(x \cdot x\right) \cdot \frac{1}{4}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{4}\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(x \cdot \left(\frac{1}{4} \cdot \color{blue}{x}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot x\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{4}}\right)\right)\right) \]

       13. *-lowering-*.f64 58.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{4}}\right)\right)\right) \]

   13. Simplified 58.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(x \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.6% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 4500.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 4500.0) {
		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 <= 4500.0d0) 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 <= 4500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 4500.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 4500.0)
		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 <= 4500.0)
		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, 4500.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4500

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 64.1%

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

   if 4500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 3.1%

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

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \left(1 + {x}^{2} \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\frac{1}{12} \cdot x + \frac{1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right) + {x}^{2} \cdot \left(\left(\frac{1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) - \frac{1}{4}\right)}{\varepsilon}} \]

   10. Simplified 17.7%

       \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot 0.08333333333333333 + \left(\left(0.25 + 0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + -0.25\right)\right) + \varepsilon \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(-0.25 + -0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + \left(x \cdot 0.25 + -0.25\right)\right)\right)}{\varepsilon}} \]

   11. Taylor expanded in eps around 0

       \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)}{\varepsilon}} \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)}{\varepsilon} \]

       2. distribute-rgt-out N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(\frac{-1}{12} + \frac{1}{12}\right)\right)}{\varepsilon} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot 0\right)}{\varepsilon} \]

       4. mul0-rgt N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot 0}{\varepsilon} \]

       5. associate-*l* N/A

          \[\leadsto \frac{x \cdot \left(x \cdot 0\right)}{\varepsilon} \]

       6. mul0-rgt N/A

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

       7. mul0-rgt N/A

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

       8. /-lowering-/.f64 49.1%

          \[\leadsto \mathsf{/.f64}\left(0, \color{blue}{\varepsilon}\right) \]

   13. Simplified 49.1%

       \[\leadsto \color{blue}{\frac{0}{\varepsilon}} \]
   14. Step-by-step derivation

       1. div0 49.1%

          \[\leadsto 0 \]

   15. Applied egg-rr 49.1%

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

Alternative 11: 15.8% accurate, 227.0× 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 72.4%

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

3. Simplified 72.4%

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

5. Taylor expanded in x around 0

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

7. Simplified 39.0%

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

8. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{\varepsilon \cdot \left(1 + {x}^{2} \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\frac{1}{12} \cdot x + \frac{1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right) + {x}^{2} \cdot \left(\left(\frac{1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) - \frac{1}{4}\right)}{\varepsilon}} \]

10. Simplified 42.9%

    \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot 0.08333333333333333 + \left(\left(0.25 + 0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + -0.25\right)\right) + \varepsilon \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(-0.25 + -0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + \left(x \cdot 0.25 + -0.25\right)\right)\right)}{\varepsilon}} \]

11. Taylor expanded in eps around 0

    \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)}{\varepsilon}} \]
12. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)}{\varepsilon} \]

    2. distribute-rgt-out N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(\frac{-1}{12} + \frac{1}{12}\right)\right)}{\varepsilon} \]

    3. metadata-eval N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot 0\right)}{\varepsilon} \]

    4. mul0-rgt N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot 0}{\varepsilon} \]

    5. associate-*l* N/A

       \[\leadsto \frac{x \cdot \left(x \cdot 0\right)}{\varepsilon} \]

    6. mul0-rgt N/A

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

    7. mul0-rgt N/A

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

    8. /-lowering-/.f64 18.7%

       \[\leadsto \mathsf{/.f64}\left(0, \color{blue}{\varepsilon}\right) \]

13. Simplified 18.7%

    \[\leadsto \color{blue}{\frac{0}{\varepsilon}} \]
14. Step-by-step derivation

    1. div0 18.7%

       \[\leadsto 0 \]

15. Applied egg-rr 18.7%

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

Reproduce

herbie shell --seed 2024155 
(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))