NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.9% → 99.2%

Time: 11.8s

Alternatives: 15

Speedup: 1.1×

• 38.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -9.253723058519385e+78
  2. eps = 2.757532079034447e-246

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. neg-mul-1 N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 73.5

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

4. Applied rewrites 73.5%

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

5. Taylor expanded in eps around inf

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

   1. exp-neg N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. *-lft-identity N/A

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

   9. lower-fma.f64 71.7

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

7. Applied rewrites 71.7%

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

8. Taylor expanded in eps around inf

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

    \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(x \cdot \left(1 - \varepsilon\right)\right)} \cdot 1 - \frac{-1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2} \]
12. Add Preprocessing

Alternative 2: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-1 + \varepsilon\right) \cdot x}\\ \mathbf{if}\;\left(\frac{1}{\varepsilon} + 1\right) \cdot t\_0 - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 4:\\ \;\;\;\;\left(\frac{x + 1}{e^{x}} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot 1 - x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* (+ -1.0 eps) x))))
   (if (<=
        (-
         (* (+ (/ 1.0 eps) 1.0) t_0)
         (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
        4.0)
     (* (* (/ (+ x 1.0) (exp x)) 2.0) 0.5)
     (/ (- (* t_0 1.0) (* x eps)) 2.0))))

C

double code(double x, double eps) {
	double t_0 = exp(((-1.0 + eps) * x));
	double tmp;
	if (((((1.0 / eps) + 1.0) * t_0) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 4.0) {
		tmp = (((x + 1.0) / exp(x)) * 2.0) * 0.5;
	} else {
		tmp = ((t_0 * 1.0) - (x * eps)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((((-1.0d0) + eps) * x))
    if (((((1.0d0 / eps) + 1.0d0) * t_0) - (exp((((-1.0d0) - eps) * x)) * ((1.0d0 / eps) - 1.0d0))) <= 4.0d0) then
        tmp = (((x + 1.0d0) / exp(x)) * 2.0d0) * 0.5d0
    else
        tmp = ((t_0 * 1.0d0) - (x * eps)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp(((-1.0 + eps) * x));
	double tmp;
	if (((((1.0 / eps) + 1.0) * t_0) - (Math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 4.0) {
		tmp = (((x + 1.0) / Math.exp(x)) * 2.0) * 0.5;
	} else {
		tmp = ((t_0 * 1.0) - (x * eps)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp(((-1.0 + eps) * x))
	tmp = 0
	if ((((1.0 / eps) + 1.0) * t_0) - (math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 4.0:
		tmp = (((x + 1.0) / math.exp(x)) * 2.0) * 0.5
	else:
		tmp = ((t_0 * 1.0) - (x * eps)) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(Float64(-1.0 + eps) * x))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(1.0 / eps) + 1.0) * t_0) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) <= 4.0)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) * 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(t_0 * 1.0) - Float64(x * eps)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp(((-1.0 + eps) * x));
	tmp = 0.0;
	if (((((1.0 / eps) + 1.0) * t_0) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 4.0)
		tmp = (((x + 1.0) / exp(x)) * 2.0) * 0.5;
	else
		tmp = ((t_0 * 1.0) - (x * eps)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$0 * 1.0), $MachinePrecision] - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 49.8%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.2%

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

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 59.0

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 59.0%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 59.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 4:\\ \;\;\;\;\left(\frac{x + 1}{e^{x}} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} \cdot 1 - x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 4:\\ \;\;\;\;\left(\frac{x + 1}{e^{x}} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} \cdot 1 - x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ (/ 1.0 eps) 1.0) (exp (* (+ -1.0 eps) x)))
       (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
      4.0)
   (* (* (/ (+ x 1.0) (exp x)) 2.0) 0.5)
   (/ (- (* (exp (* x eps)) 1.0) (* x eps)) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (((((1.0 / eps) + 1.0) * exp(((-1.0 + eps) * x))) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 4.0) {
		tmp = (((x + 1.0) / exp(x)) * 2.0) * 0.5;
	} else {
		tmp = ((exp((x * eps)) * 1.0) - (x * eps)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (((((1.0d0 / eps) + 1.0d0) * exp((((-1.0d0) + eps) * x))) - (exp((((-1.0d0) - eps) * x)) * ((1.0d0 / eps) - 1.0d0))) <= 4.0d0) then
        tmp = (((x + 1.0d0) / exp(x)) * 2.0d0) * 0.5d0
    else
        tmp = ((exp((x * eps)) * 1.0d0) - (x * eps)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (((((1.0 / eps) + 1.0) * Math.exp(((-1.0 + eps) * x))) - (Math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 4.0) {
		tmp = (((x + 1.0) / Math.exp(x)) * 2.0) * 0.5;
	} else {
		tmp = ((Math.exp((x * eps)) * 1.0) - (x * eps)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if ((((1.0 / eps) + 1.0) * math.exp(((-1.0 + eps) * x))) - (math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 4.0:
		tmp = (((x + 1.0) / math.exp(x)) * 2.0) * 0.5
	else:
		tmp = ((math.exp((x * eps)) * 1.0) - (x * eps)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(1.0 / eps) + 1.0) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) <= 4.0)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) * 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(exp(Float64(x * eps)) * 1.0) - Float64(x * eps)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (((((1.0 / eps) + 1.0) * exp(((-1.0 + eps) * x))) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 4.0)
		tmp = (((x + 1.0) / exp(x)) * 2.0) * 0.5;
	else
		tmp = ((exp((x * eps)) * 1.0) - (x * eps)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision] - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 49.8%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.2%

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

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 59.0

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 59.0%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 59.0%

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

   12. Taylor expanded in eps around inf

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

       1. lower-*.f64 59.0

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

   14. Applied rewrites 59.0%

       \[\leadsto \frac{1 \cdot e^{\color{blue}{\varepsilon \cdot x}} - \varepsilon \cdot x}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 4:\\ \;\;\;\;\left(\frac{x + 1}{e^{x}} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} \cdot 1 - x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. neg-mul-1 N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 73.5

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

4. Applied rewrites 73.5%

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

5. Taylor expanded in eps around inf

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

   1. exp-neg N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. *-lft-identity N/A

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

   9. lower-fma.f64 71.7

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

7. Applied rewrites 71.7%

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

8. Taylor expanded in eps around inf

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

10. Applied rewrites 99.7%

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

    1. lift-pow.f64 N/A

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

    2. lift-exp.f64 N/A

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

    3. pow-exp N/A

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

    4. neg-mul-1 N/A

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

    5. lift-neg.f64 N/A

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

    6. lift-exp.f64 99.7

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

    7. lift-neg.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. distribute-lft-neg-in N/A

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

    10. lower-*.f64 N/A

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

    11. lift--.f64 N/A

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

    12. flip-- N/A

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

    13. distribute-neg-frac2 N/A

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

    14. metadata-eval N/A

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

    15. metadata-eval N/A

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

    16. distribute-neg-in N/A

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

    17. metadata-eval N/A

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

    18. sub-neg N/A

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

    19. flip-+ N/A

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

    20. lower-+.f64 99.7

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

12. Applied rewrites 99.7%

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

13. Final simplification 99.7%

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

Alternative 5: 69.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-289}:\\ \;\;\;\;\frac{1 \cdot 1 - \frac{-1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} \cdot 1 - \left(x - \frac{x - \left(x - 1\right)}{\varepsilon}\right) \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5e-289)
   (/ (- (* 1.0 1.0) (/ -1.0 (exp (fma eps x x)))) 2.0)
   (if (<= x 0.0032)
     (/
      (-
       (* (exp (* (+ -1.0 eps) x)) 1.0)
       (* (- x (/ (- x (- x 1.0)) eps)) eps))
      2.0)
     (/ (- (* (exp (* x eps)) (+ (/ 1.0 eps) 1.0)) (- (/ 1.0 eps) 1.0)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5e-289) {
		tmp = ((1.0 * 1.0) - (-1.0 / exp(fma(eps, x, x)))) / 2.0;
	} else if (x <= 0.0032) {
		tmp = ((exp(((-1.0 + eps) * x)) * 1.0) - ((x - ((x - (x - 1.0)) / eps)) * eps)) / 2.0;
	} else {
		tmp = ((exp((x * eps)) * ((1.0 / eps) + 1.0)) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5e-289)
		tmp = Float64(Float64(Float64(1.0 * 1.0) - Float64(-1.0 / exp(fma(eps, x, x)))) / 2.0);
	elseif (x <= 0.0032)
		tmp = Float64(Float64(Float64(exp(Float64(Float64(-1.0 + eps) * x)) * 1.0) - Float64(Float64(x - Float64(Float64(x - Float64(x - 1.0)) / eps)) * eps)) / 2.0);
	else
		tmp = Float64(Float64(Float64(exp(Float64(x * eps)) * Float64(Float64(1.0 / eps) + 1.0)) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5e-289], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[(-1.0 / N[Exp[N[(eps * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.0032], N[(N[(N[(N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(x - N[(N[(x - N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000029e-289

   1. Initial program 70.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in eps around inf

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

   10. Applied rewrites 99.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 72.7%

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

   if -5.00000000000000029e-289 < x < 0.00320000000000000015

   1. Initial program 51.5%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 42.4

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

   5. Applied rewrites 42.4%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 20.6%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 28.8%

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

   12. Taylor expanded in eps around inf

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

   14. Applied rewrites 90.3%

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

   if 0.00320000000000000015 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 32.9

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in eps around inf

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

      1. lower-*.f64 48.7

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

   8. Applied rewrites 48.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-289}:\\ \;\;\;\;\frac{1 \cdot 1 - \frac{-1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} \cdot 1 - \left(x - \frac{x - \left(x - 1\right)}{\varepsilon}\right) \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-9}:\\ \;\;\;\;\frac{1 \cdot 1 - \frac{-1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.15e-9)
   (/ (- (* 1.0 1.0) (/ -1.0 (exp (fma eps x x)))) 2.0)
   (/ (- (* (exp (* x eps)) (+ (/ 1.0 eps) 1.0)) (- (/ 1.0 eps) 1.0)) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.15e-9) {
		tmp = ((1.0 * 1.0) - (-1.0 / exp(fma(eps, x, x)))) / 2.0;
	} else {
		tmp = ((exp((x * eps)) * ((1.0 / eps) + 1.0)) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.15e-9)
		tmp = Float64(Float64(Float64(1.0 * 1.0) - Float64(-1.0 / exp(fma(eps, x, x)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(exp(Float64(x * eps)) * Float64(Float64(1.0 / eps) + 1.0)) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.15e-9], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[(-1.0 / N[Exp[N[(eps * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.14999999999999981e-9

   1. Initial program 62.7%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 62.7

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 60.1

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

   7. Applied rewrites 60.1%

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

   8. Taylor expanded in eps around inf

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

   10. Applied rewrites 99.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 77.4%

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

   if 2.14999999999999981e-9 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 32.9

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in eps around inf

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

      1. lower-*.f64 48.7

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

   8. Applied rewrites 48.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-9}:\\ \;\;\;\;\frac{1 \cdot 1 - \frac{-1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-9}:\\ \;\;\;\;\frac{1 \cdot 1 - \frac{-1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} \cdot 1 - \frac{1 - x}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.15e-9)
   (/ (- (* 1.0 1.0) (/ -1.0 (exp (fma eps x x)))) 2.0)
   (/ (- (* (exp (* (+ -1.0 eps) x)) 1.0) (/ (- 1.0 x) eps)) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.15e-9) {
		tmp = ((1.0 * 1.0) - (-1.0 / exp(fma(eps, x, x)))) / 2.0;
	} else {
		tmp = ((exp(((-1.0 + eps) * x)) * 1.0) - ((1.0 - x) / eps)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.15e-9)
		tmp = Float64(Float64(Float64(1.0 * 1.0) - Float64(-1.0 / exp(fma(eps, x, x)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(exp(Float64(Float64(-1.0 + eps) * x)) * 1.0) - Float64(Float64(1.0 - x) / eps)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.15e-9], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[(-1.0 / N[Exp[N[(eps * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(1.0 - x), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.14999999999999981e-9

   1. Initial program 62.7%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 62.7

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 60.1

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

   7. Applied rewrites 60.1%

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

   8. Taylor expanded in eps around inf

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

   10. Applied rewrites 99.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 77.4%

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

   if 2.14999999999999981e-9 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 35.7

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

   5. Applied rewrites 35.7%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 31.9%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 31.9%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-9}:\\ \;\;\;\;\frac{1 \cdot 1 - \frac{-1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} \cdot 1 - \frac{1 - x}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-9}:\\ \;\;\;\;\frac{1 \cdot 1 - \frac{-1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} \cdot 1 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.15e-9)
   (/ (- (* 1.0 1.0) (/ -1.0 (exp (fma eps x x)))) 2.0)
   (if (<= x 8.4e+102)
     (/ (- (* (exp (* (+ -1.0 eps) x)) 1.0) (* x eps)) 2.0)
     (fma (* 0.3333333333333333 x) (* x x) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.15e-9) {
		tmp = ((1.0 * 1.0) - (-1.0 / exp(fma(eps, x, x)))) / 2.0;
	} else if (x <= 8.4e+102) {
		tmp = ((exp(((-1.0 + eps) * x)) * 1.0) - (x * eps)) / 2.0;
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.15e-9)
		tmp = Float64(Float64(Float64(1.0 * 1.0) - Float64(-1.0 / exp(fma(eps, x, x)))) / 2.0);
	elseif (x <= 8.4e+102)
		tmp = Float64(Float64(Float64(exp(Float64(Float64(-1.0 + eps) * x)) * 1.0) - Float64(x * eps)) / 2.0);
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.15e-9], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[(-1.0 / N[Exp[N[(eps * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.4e+102], N[(N[(N[(N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision] - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.14999999999999981e-9

   1. Initial program 62.7%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 62.7

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 60.1

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

   7. Applied rewrites 60.1%

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

   8. Taylor expanded in eps around inf

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

   10. Applied rewrites 99.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 77.4%

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

   if 2.14999999999999981e-9 < x < 8.40000000000000006e102

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 37.6

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

   5. Applied rewrites 37.6%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 38.5%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 38.5%

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

   if 8.40000000000000006e102 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 62.0%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-9}:\\ \;\;\;\;\frac{1 \cdot 1 - \frac{-1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} \cdot 1 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -900:\\ \;\;\;\;\frac{e^{-x} \cdot 1 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{x + 1}{e^{x}} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -900.0)
   (/ (- (* (exp (- x)) 1.0) (* x eps)) 2.0)
   (if (<= x 5e+102)
     (* (* (/ (+ x 1.0) (exp x)) 2.0) 0.5)
     (fma (* 0.3333333333333333 x) (* x x) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -900.0) {
		tmp = ((exp(-x) * 1.0) - (x * eps)) / 2.0;
	} else if (x <= 5e+102) {
		tmp = (((x + 1.0) / exp(x)) * 2.0) * 0.5;
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -900.0)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) * 1.0) - Float64(x * eps)) / 2.0);
	elseif (x <= 5e+102)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) * 2.0) * 0.5);
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -900.0], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] * 1.0), $MachinePrecision] - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+102], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -900

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 71.1%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 71.1%

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

   12. Taylor expanded in eps around 0

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

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 85.0

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

   14. Applied rewrites 85.0%

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

   if -900 < x < 5e102

   1. Initial program 57.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.2%

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

   if 5e102 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 62.0%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -900:\\ \;\;\;\;\frac{e^{-x} \cdot 1 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{x + 1}{e^{x}} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 44:\\ \;\;\;\;\left(\frac{x + 1}{e^{x}} \cdot 2\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 2.5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, x, \left(x - 1\right) - x\right), \varepsilon, 1 - x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 44.0)
   (* (* (/ (+ x 1.0) (exp x)) 2.0) 0.5)
   (if (<= eps 2.5e+68)
     (fma (* 0.3333333333333333 x) (* x x) 1.0)
     (/
      (-
       (+ (/ 1.0 eps) 1.0)
       (/ (fma (fma eps x (- (- x 1.0) x)) eps (- 1.0 x)) eps))
      2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 44.0) {
		tmp = (((x + 1.0) / exp(x)) * 2.0) * 0.5;
	} else if (eps <= 2.5e+68) {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	} else {
		tmp = (((1.0 / eps) + 1.0) - (fma(fma(eps, x, ((x - 1.0) - x)), eps, (1.0 - x)) / eps)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 44.0)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) * 2.0) * 0.5);
	elseif (eps <= 2.5e+68)
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) + 1.0) - Float64(fma(fma(eps, x, Float64(Float64(x - 1.0) - x)), eps, Float64(1.0 - x)) / eps)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 44.0], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[eps, 2.5e+68], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(N[(eps * x + N[(N[(x - 1.0), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] * eps + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 44

   1. Initial program 60.8%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.2%

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

   if 44 < eps < 2.5000000000000002e68

   1. Initial program 99.8%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 86.5%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]

   if 2.5000000000000002e68 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 32.1

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

   8. Applied rewrites 32.1%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 47.3%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 44:\\ \;\;\;\;\left(\frac{x + 1}{e^{x}} \cdot 2\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 2.5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, x, \left(x - 1\right) - x\right), \varepsilon, 1 - x\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, x, \left(x - 1\right) - x\right), \varepsilon, 1 - x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.9e-5)
   (/
    (-
     (+ (/ 1.0 eps) 1.0)
     (/ (fma (fma eps x (- (- x 1.0) x)) eps (- 1.0 x)) eps))
    2.0)
   (fma (* 0.3333333333333333 x) (* x x) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.9e-5) {
		tmp = (((1.0 / eps) + 1.0) - (fma(fma(eps, x, ((x - 1.0) - x)), eps, (1.0 - x)) / eps)) / 2.0;
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.9e-5)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) + 1.0) - Float64(fma(fma(eps, x, Float64(Float64(x - 1.0) - x)), eps, Float64(1.0 - x)) / eps)) / 2.0);
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.9e-5], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(N[(eps * x + N[(N[(x - 1.0), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] * eps + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9000000000000001e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 28.7

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

   8. Applied rewrites 28.7%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 35.5%

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

   if -1.9000000000000001e-5 < x

   1. Initial program 68.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 65.8%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 56.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -118:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -118.0)
   (/ (- (+ (/ 1.0 eps) 1.0) (* x eps)) 2.0)
   (fma (* 0.3333333333333333 x) (* x x) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -118.0) {
		tmp = (((1.0 / eps) + 1.0) - (x * eps)) / 2.0;
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -118.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) + 1.0) - Float64(x * eps)) / 2.0);
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -118.0], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -118

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 28.7

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

   8. Applied rewrites 28.7%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 28.7%

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

   if -118 < x

   1. Initial program 68.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 65.8%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -118:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.4% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma (* 0.3333333333333333 x) (* x x) 1.0))

C

double code(double x, double eps) {
	return fma((0.3333333333333333 * x), (x * x), 1.0);
}

Julia

function code(x, eps)
	return fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0)
end

Wolfram

code[x_, eps_] := N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 53.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 55.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

9. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
10. Step-by-step derivation

11. Applied rewrites 55.5%

    \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
12. Add Preprocessing

Alternative 14: 57.3% accurate, 38.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 490:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 490.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 490.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 490.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 490.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 490.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 490.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 490.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 490.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 490

   1. Initial program 63.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.4%

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

   if 490 < x

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in eps around 0

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

      1. div-sub N/A

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

      2. neg-mul-1 N/A

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

      3. +-inverses N/A

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

      4. metadata-eval 41.2

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

   7. Applied rewrites 41.2%

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

Alternative 15: 43.4% accurate, 273.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 42.8%

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

Reproduce

herbie shell --seed 2024273 
(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))