NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 100.0%

Time: 12.9s

Alternatives: 11

Speedup: 14.2×

• 38.1% 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: 73.4% 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

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (/ (+ x 1.0) (exp 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(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(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(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(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(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(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[x], $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 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in eps around 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 70.2%

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

   7. Simplified 70.2%

      \[\leadsto \color{blue}{\left(x + 1\right) \cdot e^{0 - x}} \]
   8. Step-by-step derivation

      1. exp-diff N/A

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

      2. 1-exp N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. exp-lowering-exp.f64 70.3%

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

   9. Applied egg-rr 70.3%

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

   if 1 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in 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. *-commutative N/A

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

      12. associate-*r* N/A

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

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

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

      14. *-commutative 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 99.8%

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

   7. Simplified 99.8%

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

   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 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 78.4%

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

Alternative 2: 98.7% 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 75.0%

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

3. Simplified 75.0%

   \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
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. *-commutative N/A

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

   12. associate-*r* N/A

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

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

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

   14. *-commutative 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 98.8%

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

7. Simplified 98.8%

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

Alternative 3: 91.5% accurate, 2.1× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (/ (+ x 1.0) (exp x))
   (/
    (+ (* x -0.5) (* eps_m (+ 1.0 (* 0.5 (* x (* x (* eps_m 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(x);
	} else {
		tmp = ((x * -0.5) + (eps_m * (1.0 + (0.5 * (x * (x * (eps_m * eps_m))))))) / eps_m;
	}
	return tmp;
}

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = ((x * -0.5) + (eps_m * (1.0 + (0.5 * (x * (x * (eps_m * 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(x)
	else:
		tmp = ((x * -0.5) + (eps_m * (1.0 + (0.5 * (x * (x * (eps_m * 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(x));
	else
		tmp = Float64(Float64(Float64(x * -0.5) + Float64(eps_m * Float64(1.0 + Float64(0.5 * Float64(x * Float64(x * Float64(eps_m * eps_m))))))) / eps_m);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in eps around 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 70.2%

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

   7. Simplified 70.2%

      \[\leadsto \color{blue}{\left(x + 1\right) \cdot e^{0 - x}} \]
   8. Step-by-step derivation

      1. exp-diff N/A

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

      2. 1-exp N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. exp-lowering-exp.f64 70.3%

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

   9. Applied egg-rr 70.3%

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

   if 1 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 80.3%

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

   8. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. distribute-lft-out-- N/A

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

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

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

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

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

      8. /-lowering-/.f64 76.3%

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

   10. Simplified 76.3%

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

   11. Taylor expanded in eps around 0

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

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

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

   13. Simplified 81.6%

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

Alternative 4: 90.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 45:\\ \;\;\;\;e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -0.5 + eps\_m \cdot \left(1 + 0.5 \cdot \left(x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\right)}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 45.0)
   (exp (- 0.0 x))
   (/
    (+ (* x -0.5) (* eps_m (+ 1.0 (* 0.5 (* x (* x (* eps_m eps_m)))))))
    eps_m)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 45.0) {
		tmp = exp((0.0 - x));
	} else {
		tmp = ((x * -0.5) + (eps_m * (1.0 + (0.5 * (x * (x * (eps_m * eps_m))))))) / eps_m;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 45.0d0) then
        tmp = exp((0.0d0 - x))
    else
        tmp = ((x * (-0.5d0)) + (eps_m * (1.0d0 + (0.5d0 * (x * (x * (eps_m * eps_m))))))) / eps_m
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 45.0) {
		tmp = Math.exp((0.0 - x));
	} else {
		tmp = ((x * -0.5) + (eps_m * (1.0 + (0.5 * (x * (x * (eps_m * eps_m))))))) / eps_m;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 45.0:
		tmp = math.exp((0.0 - x))
	else:
		tmp = ((x * -0.5) + (eps_m * (1.0 + (0.5 * (x * (x * (eps_m * eps_m))))))) / eps_m
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 45.0)
		tmp = exp(Float64(0.0 - x));
	else
		tmp = Float64(Float64(Float64(x * -0.5) + Float64(eps_m * Float64(1.0 + Float64(0.5 * Float64(x * Float64(x * Float64(eps_m * eps_m))))))) / eps_m);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 45.0)
		tmp = exp((0.0 - x));
	else
		tmp = ((x * -0.5) + (eps_m * (1.0 + (0.5 * (x * (x * (eps_m * eps_m))))))) / eps_m;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 45.0], N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x * -0.5), $MachinePrecision] + N[(eps$95$m * N[(1.0 + N[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 45

   1. Initial program 65.6%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in eps around 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. *-commutative N/A

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

      12. associate-*r* N/A

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

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

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

      14. *-commutative 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 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in eps around 0

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

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

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

      2. neg-mul-1 N/A

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 78.6%

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

   10. Simplified 78.6%

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

   if 45 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 81.4%

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

   8. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. distribute-lft-out-- N/A

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

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

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

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

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

      8. /-lowering-/.f64 77.3%

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

   10. Simplified 77.3%

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

   11. Taylor expanded in eps around 0

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

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

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

   13. Simplified 82.7%

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

Alternative 5: 84.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(eps\_m + 1\right) \cdot \left(-0.5 + \frac{0.5}{eps\_m}\right)\\ t_1 := 0.5 + \frac{0.5}{eps\_m}\\ \mathbf{if}\;x \leq 22000:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m + -1\right) \cdot t\_1 + \frac{0.5 + eps\_m \cdot \left(x \cdot -0.5 + eps\_m \cdot \left(-0.5 + 0.5 \cdot \left(x \cdot eps\_m\right)\right)\right)}{eps\_m}\right)\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+159}:\\ \;\;\;\;\left(1 + \left(x \cdot \left(eps\_m + -1\right)\right) \cdot t\_1\right) + x \cdot \left(t\_0 + 0.5 \cdot \left(x \cdot \left(t\_1 \cdot \left(\left(eps\_m + -1\right) \cdot \left(eps\_m + -1\right)\right) + \left(-1 - eps\_m\right) \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (+ eps_m 1.0) (+ -0.5 (/ 0.5 eps_m))))
        (t_1 (+ 0.5 (/ 0.5 eps_m))))
   (if (<= x 22000.0)
     (+
      1.0
      (*
       x
       (+
        (* (+ eps_m -1.0) t_1)
        (/
         (+
          0.5
          (* eps_m (+ (* x -0.5) (* eps_m (+ -0.5 (* 0.5 (* x eps_m)))))))
         eps_m))))
     (if (<= x 2.15e+159)
       (+
        (+ 1.0 (* (* x (+ eps_m -1.0)) t_1))
        (*
         x
         (+
          t_0
          (*
           0.5
           (*
            x
            (+
             (* t_1 (* (+ eps_m -1.0) (+ eps_m -1.0)))
             (* (- -1.0 eps_m) t_0)))))))
       (* 0.5 (* x (* x (* eps_m eps_m))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (eps_m + 1.0) * (-0.5 + (0.5 / eps_m));
	double t_1 = 0.5 + (0.5 / eps_m);
	double tmp;
	if (x <= 22000.0) {
		tmp = 1.0 + (x * (((eps_m + -1.0) * t_1) + ((0.5 + (eps_m * ((x * -0.5) + (eps_m * (-0.5 + (0.5 * (x * eps_m))))))) / eps_m)));
	} else if (x <= 2.15e+159) {
		tmp = (1.0 + ((x * (eps_m + -1.0)) * t_1)) + (x * (t_0 + (0.5 * (x * ((t_1 * ((eps_m + -1.0) * (eps_m + -1.0))) + ((-1.0 - eps_m) * t_0))))));
	} else {
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (eps_m + 1.0d0) * ((-0.5d0) + (0.5d0 / eps_m))
    t_1 = 0.5d0 + (0.5d0 / eps_m)
    if (x <= 22000.0d0) then
        tmp = 1.0d0 + (x * (((eps_m + (-1.0d0)) * t_1) + ((0.5d0 + (eps_m * ((x * (-0.5d0)) + (eps_m * ((-0.5d0) + (0.5d0 * (x * eps_m))))))) / eps_m)))
    else if (x <= 2.15d+159) then
        tmp = (1.0d0 + ((x * (eps_m + (-1.0d0))) * t_1)) + (x * (t_0 + (0.5d0 * (x * ((t_1 * ((eps_m + (-1.0d0)) * (eps_m + (-1.0d0)))) + (((-1.0d0) - eps_m) * t_0))))))
    else
        tmp = 0.5d0 * (x * (x * (eps_m * eps_m)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (eps_m + 1.0) * (-0.5 + (0.5 / eps_m));
	double t_1 = 0.5 + (0.5 / eps_m);
	double tmp;
	if (x <= 22000.0) {
		tmp = 1.0 + (x * (((eps_m + -1.0) * t_1) + ((0.5 + (eps_m * ((x * -0.5) + (eps_m * (-0.5 + (0.5 * (x * eps_m))))))) / eps_m)));
	} else if (x <= 2.15e+159) {
		tmp = (1.0 + ((x * (eps_m + -1.0)) * t_1)) + (x * (t_0 + (0.5 * (x * ((t_1 * ((eps_m + -1.0) * (eps_m + -1.0))) + ((-1.0 - eps_m) * t_0))))));
	} else {
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (eps_m + 1.0) * (-0.5 + (0.5 / eps_m))
	t_1 = 0.5 + (0.5 / eps_m)
	tmp = 0
	if x <= 22000.0:
		tmp = 1.0 + (x * (((eps_m + -1.0) * t_1) + ((0.5 + (eps_m * ((x * -0.5) + (eps_m * (-0.5 + (0.5 * (x * eps_m))))))) / eps_m)))
	elif x <= 2.15e+159:
		tmp = (1.0 + ((x * (eps_m + -1.0)) * t_1)) + (x * (t_0 + (0.5 * (x * ((t_1 * ((eps_m + -1.0) * (eps_m + -1.0))) + ((-1.0 - eps_m) * t_0))))))
	else:
		tmp = 0.5 * (x * (x * (eps_m * eps_m)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(eps_m + 1.0) * Float64(-0.5 + Float64(0.5 / eps_m)))
	t_1 = Float64(0.5 + Float64(0.5 / eps_m))
	tmp = 0.0
	if (x <= 22000.0)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(Float64(eps_m + -1.0) * t_1) + Float64(Float64(0.5 + Float64(eps_m * Float64(Float64(x * -0.5) + Float64(eps_m * Float64(-0.5 + Float64(0.5 * Float64(x * eps_m))))))) / eps_m))));
	elseif (x <= 2.15e+159)
		tmp = Float64(Float64(1.0 + Float64(Float64(x * Float64(eps_m + -1.0)) * t_1)) + Float64(x * Float64(t_0 + Float64(0.5 * Float64(x * Float64(Float64(t_1 * Float64(Float64(eps_m + -1.0) * Float64(eps_m + -1.0))) + Float64(Float64(-1.0 - eps_m) * t_0)))))));
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps_m * eps_m))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (eps_m + 1.0) * (-0.5 + (0.5 / eps_m));
	t_1 = 0.5 + (0.5 / eps_m);
	tmp = 0.0;
	if (x <= 22000.0)
		tmp = 1.0 + (x * (((eps_m + -1.0) * t_1) + ((0.5 + (eps_m * ((x * -0.5) + (eps_m * (-0.5 + (0.5 * (x * eps_m))))))) / eps_m)));
	elseif (x <= 2.15e+159)
		tmp = (1.0 + ((x * (eps_m + -1.0)) * t_1)) + (x * (t_0 + (0.5 * (x * ((t_1 * ((eps_m + -1.0) * (eps_m + -1.0))) + ((-1.0 - eps_m) * t_0))))));
	else
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(-0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 22000.0], N[(1.0 + N[(x * N[(N[(N[(eps$95$m + -1.0), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(0.5 + N[(eps$95$m * N[(N[(x * -0.5), $MachinePrecision] + N[(eps$95$m * N[(-0.5 + N[(0.5 * N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.15e+159], N[(N[(1.0 + N[(N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(x * N[(t$95$0 + N[(0.5 * N[(x * N[(N[(t$95$1 * N[(N[(eps$95$m + -1.0), $MachinePrecision] * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - eps$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 22000

   1. Initial program 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 86.0%

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

   8. Taylor expanded in eps around 0

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

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

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

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

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 88.6%

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

   10. Simplified 88.6%

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

   if 22000 < x < 2.1500000000000001e159

   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

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

   7. Simplified 37.3%

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

   9. Applied egg-rr 73.2%

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

   if 2.1500000000000001e159 < 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

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

   7. Simplified 40.3%

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

   8. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. distribute-lft-out-- N/A

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

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

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

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

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

      8. /-lowering-/.f64 39.4%

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

   10. Simplified 39.4%

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

   11. Taylor expanded in x around inf

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 67.3%

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

   13. Simplified 67.3%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 22000:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon + -1\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right) + \frac{0.5 + \varepsilon \cdot \left(x \cdot -0.5 + \varepsilon \cdot \left(-0.5 + 0.5 \cdot \left(x \cdot \varepsilon\right)\right)\right)}{\varepsilon}\right)\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+159}:\\ \;\;\;\;\left(1 + \left(x \cdot \left(\varepsilon + -1\right)\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right)\right) + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + \frac{0.5}{\varepsilon}\right) + 0.5 \cdot \left(x \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) + \left(-1 - \varepsilon\right) \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + \frac{0.5}{\varepsilon}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m + -1\right) \cdot \left(0.5 + \frac{0.5}{eps\_m}\right) + \frac{0.5 + eps\_m \cdot \left(x \cdot -0.5 + eps\_m \cdot \left(-0.5 + 0.5 \cdot \left(x \cdot eps\_m\right)\right)\right)}{eps\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 350.0)
   (+
    1.0
    (*
     x
     (+
      (* (+ eps_m -1.0) (+ 0.5 (/ 0.5 eps_m)))
      (/
       (+ 0.5 (* eps_m (+ (* x -0.5) (* eps_m (+ -0.5 (* 0.5 (* x eps_m)))))))
       eps_m))))
   (* 0.5 (* x (* x (* eps_m eps_m))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 350.0) {
		tmp = 1.0 + (x * (((eps_m + -1.0) * (0.5 + (0.5 / eps_m))) + ((0.5 + (eps_m * ((x * -0.5) + (eps_m * (-0.5 + (0.5 * (x * eps_m))))))) / eps_m)));
	} else {
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 350.0d0) then
        tmp = 1.0d0 + (x * (((eps_m + (-1.0d0)) * (0.5d0 + (0.5d0 / eps_m))) + ((0.5d0 + (eps_m * ((x * (-0.5d0)) + (eps_m * ((-0.5d0) + (0.5d0 * (x * eps_m))))))) / eps_m)))
    else
        tmp = 0.5d0 * (x * (x * (eps_m * eps_m)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 350.0) {
		tmp = 1.0 + (x * (((eps_m + -1.0) * (0.5 + (0.5 / eps_m))) + ((0.5 + (eps_m * ((x * -0.5) + (eps_m * (-0.5 + (0.5 * (x * eps_m))))))) / eps_m)));
	} else {
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 350.0:
		tmp = 1.0 + (x * (((eps_m + -1.0) * (0.5 + (0.5 / eps_m))) + ((0.5 + (eps_m * ((x * -0.5) + (eps_m * (-0.5 + (0.5 * (x * eps_m))))))) / eps_m)))
	else:
		tmp = 0.5 * (x * (x * (eps_m * eps_m)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 350.0)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(Float64(eps_m + -1.0) * Float64(0.5 + Float64(0.5 / eps_m))) + Float64(Float64(0.5 + Float64(eps_m * Float64(Float64(x * -0.5) + Float64(eps_m * Float64(-0.5 + Float64(0.5 * Float64(x * eps_m))))))) / eps_m))));
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps_m * eps_m))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 350.0)
		tmp = 1.0 + (x * (((eps_m + -1.0) * (0.5 + (0.5 / eps_m))) + ((0.5 + (eps_m * ((x * -0.5) + (eps_m * (-0.5 + (0.5 * (x * eps_m))))))) / eps_m)));
	else
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 350.0], N[(1.0 + N[(x * N[(N[(N[(eps$95$m + -1.0), $MachinePrecision] * N[(0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 + N[(eps$95$m * N[(N[(x * -0.5), $MachinePrecision] + N[(eps$95$m * N[(-0.5 + N[(0.5 * N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 350

   1. Initial program 63.6%

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

   3. Simplified 63.6%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 86.5%

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

   8. Taylor expanded in eps around 0

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

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

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

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

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 89.1%

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

   10. Simplified 89.1%

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

   if 350 < 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

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

   7. Simplified 38.3%

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

   8. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. distribute-lft-out-- N/A

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

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

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

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

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

      8. /-lowering-/.f64 37.6%

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

   10. Simplified 37.6%

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

   11. Taylor expanded in x around inf

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 63.5%

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

   13. Simplified 63.5%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon + -1\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right) + \frac{0.5 + \varepsilon \cdot \left(x \cdot -0.5 + \varepsilon \cdot \left(-0.5 + 0.5 \cdot \left(x \cdot \varepsilon\right)\right)\right)}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0018:\\ \;\;\;\;1 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot \left(x \cdot -0.3333333333333333 + \left(1 + eps\_m \cdot \left(eps\_m \cdot \left(1 - x\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.0018)
   (+
    1.0
    (*
     x
     (+
      -1.0
      (*
       0.5
       (*
        x
        (+
         (* x -0.3333333333333333)
         (+ 1.0 (* eps_m (* eps_m (- 1.0 x))))))))))
   (* 0.5 (* x (* x (* eps_m eps_m))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.0018) {
		tmp = 1.0 + (x * (-1.0 + (0.5 * (x * ((x * -0.3333333333333333) + (1.0 + (eps_m * (eps_m * (1.0 - x)))))))));
	} else {
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 0.0018d0) then
        tmp = 1.0d0 + (x * ((-1.0d0) + (0.5d0 * (x * ((x * (-0.3333333333333333d0)) + (1.0d0 + (eps_m * (eps_m * (1.0d0 - x)))))))))
    else
        tmp = 0.5d0 * (x * (x * (eps_m * eps_m)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.0018) {
		tmp = 1.0 + (x * (-1.0 + (0.5 * (x * ((x * -0.3333333333333333) + (1.0 + (eps_m * (eps_m * (1.0 - x)))))))));
	} else {
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 0.0018:
		tmp = 1.0 + (x * (-1.0 + (0.5 * (x * ((x * -0.3333333333333333) + (1.0 + (eps_m * (eps_m * (1.0 - x)))))))))
	else:
		tmp = 0.5 * (x * (x * (eps_m * eps_m)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 0.0018)
		tmp = Float64(1.0 + Float64(x * Float64(-1.0 + Float64(0.5 * Float64(x * Float64(Float64(x * -0.3333333333333333) + Float64(1.0 + Float64(eps_m * Float64(eps_m * Float64(1.0 - x))))))))));
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps_m * eps_m))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 0.0018)
		tmp = 1.0 + (x * (-1.0 + (0.5 * (x * ((x * -0.3333333333333333) + (1.0 + (eps_m * (eps_m * (1.0 - x)))))))));
	else
		tmp = 0.5 * (x * (x * (eps_m * eps_m)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 0.0018], N[(1.0 + N[(x * N[(-1.0 + N[(0.5 * N[(x * N[(N[(x * -0.3333333333333333), $MachinePrecision] + N[(1.0 + N[(eps$95$m * N[(eps$95$m * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0018

   1. Initial program 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in 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. *-commutative N/A

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

      12. associate-*r* N/A

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

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

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

      14. *-commutative 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 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 51.6%

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

   11. Taylor expanded in eps around 0

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot \left(x \cdot \left(1 + -1 \cdot x\right)\right)\right) + \frac{1}{2} \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. distribute-lft-out N/A

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

       2. sub-neg N/A

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

       3. associate-+r+ N/A

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

       4. +-commutative N/A

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

       5. +-commutative N/A

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

       6. distribute-lft-in N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

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

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

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

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

   13. Simplified 87.6%

       \[\leadsto 1 + x \cdot \color{blue}{\left(-1 + 0.5 \cdot \left(x \cdot \left(x \cdot -0.3333333333333333 + \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(1 - x\right)\right)\right)\right)\right)\right)} \]

   if 0.0018 < 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

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

   7. Simplified 38.7%

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

   8. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. distribute-lft-out-- N/A

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

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

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

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

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

      8. /-lowering-/.f64 38.0%

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

   10. Simplified 38.0%

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

   11. Taylor expanded in x around inf

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 63.3%

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

   13. Simplified 63.3%

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

Alternative 8: 74.2% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* x (* x (* eps_m eps_m))))))
   (if (<= x -6.6e-15) t_0 (if (<= x 2.8e-15) 1.0 t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 0.5 * (x * (x * (eps_m * eps_m)));
	double tmp;
	if (x <= -6.6e-15) {
		tmp = t_0;
	} else if (x <= 2.8e-15) {
		tmp = 1.0;
	} 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 = 0.5d0 * (x * (x * (eps_m * eps_m)))
    if (x <= (-6.6d-15)) then
        tmp = t_0
    else if (x <= 2.8d-15) then
        tmp = 1.0d0
    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 = 0.5 * (x * (x * (eps_m * eps_m)));
	double tmp;
	if (x <= -6.6e-15) {
		tmp = t_0;
	} else if (x <= 2.8e-15) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 0.5 * (x * (x * (eps_m * eps_m)))
	tmp = 0
	if x <= -6.6e-15:
		tmp = t_0
	elif x <= 2.8e-15:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(0.5 * Float64(x * Float64(x * Float64(eps_m * eps_m))))
	tmp = 0.0
	if (x <= -6.6e-15)
		tmp = t_0;
	elseif (x <= 2.8e-15)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 0.5 * (x * (x * (eps_m * eps_m)));
	tmp = 0.0;
	if (x <= -6.6e-15)
		tmp = t_0;
	elseif (x <= 2.8e-15)
		tmp = 1.0;
	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[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.6e-15], t$95$0, If[LessEqual[x, 2.8e-15], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.6e-15 or 2.80000000000000014e-15 < x

   1. Initial program 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 52.7%

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

   8. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. distribute-lft-out-- N/A

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

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

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

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

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

      8. /-lowering-/.f64 51.0%

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

   10. Simplified 51.0%

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

   11. Taylor expanded in x around inf

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 67.3%

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

   13. Simplified 67.3%

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

   if -6.6e-15 < x < 2.80000000000000014e-15

   1. Initial program 53.3%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 70.9%

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

Alternative 9: 81.5% accurate, 14.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(eps\_m \cdot eps\_m\right)\\ \mathbf{if}\;x \leq 220:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* eps_m eps_m))))
   (if (<= x 220.0) (+ 1.0 (* x (* 0.5 t_0))) (* 0.5 (* x t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (eps_m * eps_m);
	double tmp;
	if (x <= 220.0) {
		tmp = 1.0 + (x * (0.5 * t_0));
	} else {
		tmp = 0.5 * (x * 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 = x * (eps_m * eps_m)
    if (x <= 220.0d0) then
        tmp = 1.0d0 + (x * (0.5d0 * t_0))
    else
        tmp = 0.5d0 * (x * 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 = x * (eps_m * eps_m);
	double tmp;
	if (x <= 220.0) {
		tmp = 1.0 + (x * (0.5 * t_0));
	} else {
		tmp = 0.5 * (x * t_0);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (eps_m * eps_m)
	tmp = 0
	if x <= 220.0:
		tmp = 1.0 + (x * (0.5 * t_0))
	else:
		tmp = 0.5 * (x * t_0)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(eps_m * eps_m))
	tmp = 0.0
	if (x <= 220.0)
		tmp = Float64(1.0 + Float64(x * Float64(0.5 * t_0)));
	else
		tmp = Float64(0.5 * Float64(x * t_0));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (eps_m * eps_m);
	tmp = 0.0;
	if (x <= 220.0)
		tmp = 1.0 + (x * (0.5 * t_0));
	else
		tmp = 0.5 * (x * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 220.0], N[(1.0 + N[(x * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 220

   1. Initial program 63.6%

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

   3. Simplified 63.6%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 86.5%

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

   8. Taylor expanded in eps around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 86.5%

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

   10. Simplified 86.5%

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

   if 220 < 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

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

   7. Simplified 38.3%

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

   8. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. distribute-lft-out-- N/A

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

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

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

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

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

      8. /-lowering-/.f64 37.6%

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

   10. Simplified 37.6%

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

   11. Taylor expanded in x around inf

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 63.5%

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

   13. Simplified 63.5%

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

Alternative 10: 53.4% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 57000:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 57000.0) (- 1.0 x) (* 0.5 (* x (* x x)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 57000.0) {
		tmp = 1.0 - x;
	} else {
		tmp = 0.5 * (x * (x * x));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 57000.0d0) then
        tmp = 1.0d0 - x
    else
        tmp = 0.5d0 * (x * (x * x))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 57000.0) {
		tmp = 1.0 - x;
	} else {
		tmp = 0.5 * (x * (x * x));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 57000.0:
		tmp = 1.0 - x
	else:
		tmp = 0.5 * (x * (x * x))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 57000.0)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * x)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 57000.0)
		tmp = 1.0 - x;
	else
		tmp = 0.5 * (x * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 57000.0], N[(1.0 - x), $MachinePrecision], N[(0.5 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 57000

   1. Initial program 64.0%

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

   3. Simplified 64.0%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   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. *-commutative N/A

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

      12. associate-*r* N/A

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

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

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

      14. *-commutative 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 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 50.5%

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

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. --lowering--.f64 53.3%

          \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   13. Simplified 53.3%

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

   if 57000 < 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 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 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-lowering-*.f64 26.5%

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

   10. Simplified 26.5%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{3}} \]
   12. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 26.5%

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

   13. Simplified 26.5%

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

Alternative 11: 43.8% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Simplified 75.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 37.9%

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

Reproduce

herbie shell --seed 2024156 
(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))