NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.1% → 99.9%

Time: 10.9s

Alternatives: 15

Speedup: 0.4×

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

• could not determine a ground truth

  1. x = -2.0076647549317784e+132
  2. eps = 8.441077306517434e-199

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: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} - 1\\ t_1 := 1 + {\varepsilon}^{-1}\\ \mathbf{if}\;t\_1 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - t\_0 \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 1.9999999999999964:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot e^{\left(\varepsilon - 1\right) \cdot x} - t\_0 \cdot e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow eps -1.0) 1.0)) (t_1 (+ 1.0 (pow eps -1.0))))
   (if (<=
        (- (* t_1 (exp (* (+ -1.0 eps) x))) (* t_0 (exp (* (- -1.0 eps) x))))
        1.9999999999999964)
     (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
     (/
      (- (* t_1 (exp (* (- eps 1.0) x))) (* t_0 (exp (- (fma x eps x)))))
      2.0))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) - 1.0;
	double t_1 = 1.0 + pow(eps, -1.0);
	double tmp;
	if (((t_1 * exp(((-1.0 + eps) * x))) - (t_0 * exp(((-1.0 - eps) * x)))) <= 1.9999999999999964) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = ((t_1 * exp(((eps - 1.0) * x))) - (t_0 * exp(-fma(x, eps, x)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) - 1.0)
	t_1 = Float64(1.0 + (eps ^ -1.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(t_0 * exp(Float64(Float64(-1.0 - eps) * x)))) <= 1.9999999999999964)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = Float64(Float64(Float64(t_1 * exp(Float64(Float64(eps - 1.0) * x))) - Float64(t_0 * exp(Float64(-fma(x, eps, x))))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.9999999999999964], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$1 * N[Exp[N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $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))))) < 1.9999999999999964

   1. Initial program 27.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 100.0%

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

   if 1.9999999999999964 < (-.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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. /-rgt-identity N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 1.9999999999999964:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} - 1\\ t_1 := {\varepsilon}^{-1} + 1\\ \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - t\_0 \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot \mathsf{fma}\left(0.5 \cdot x, {\left(\varepsilon - 1\right)}^{2}, \varepsilon - 1\right), x, t\_1\right) - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow eps -1.0) 1.0)) (t_1 (+ (pow eps -1.0) 1.0)))
   (if (<=
        (-
         (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
         (* t_0 (exp (* (- -1.0 eps) x))))
        4.0)
     (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
     (/
      (-
       (fma (* t_1 (fma (* 0.5 x) (pow (- eps 1.0) 2.0) (- eps 1.0))) x t_1)
       t_0)
      2.0))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) - 1.0;
	double t_1 = pow(eps, -1.0) + 1.0;
	double tmp;
	if ((((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - (t_0 * exp(((-1.0 - eps) * x)))) <= 4.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = (fma((t_1 * fma((0.5 * x), pow((eps - 1.0), 2.0), (eps - 1.0))), x, t_1) - t_0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) - 1.0)
	t_1 = Float64((eps ^ -1.0) + 1.0)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(t_0 * exp(Float64(Float64(-1.0 - eps) * x)))) <= 4.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = Float64(Float64(fma(Float64(t_1 * fma(Float64(0.5 * x), (Float64(eps - 1.0) ^ 2.0), Float64(eps - 1.0))), x, t_1) - t_0) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(t$95$1 * N[(N[(0.5 * x), $MachinePrecision] * N[Power[N[(eps - 1.0), $MachinePrecision], 2.0], $MachinePrecision] + N[(eps - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + t$95$1), $MachinePrecision] - t$95$0), $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 48.1%

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

   3. Taylor expanded in 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 98.3%

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

   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

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   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 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

   8. Applied rewrites 87.8%

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

4. Final simplification 94.1%

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

Alternative 3: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + {\varepsilon}^{-1}\\ \mathbf{if}\;t\_0 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 2.000000002:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot e^{x \cdot \varepsilon} - -1 \cdot e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (pow eps -1.0))))
   (if (<=
        (-
         (* t_0 (exp (* (+ -1.0 eps) x)))
         (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
        2.000000002)
     (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
     (/ (- (* t_0 (exp (* x eps))) (* -1.0 (exp (- (fma x eps x))))) 2.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + pow(eps, -1.0);
	double tmp;
	if (((t_0 * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 2.000000002) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = ((t_0 * exp((x * eps))) - (-1.0 * exp(-fma(x, eps, x)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + (eps ^ -1.0))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) <= 2.000000002)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(x * eps))) - Float64(-1.0 * exp(Float64(-fma(x, eps, x))))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.000000002], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$0 * N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $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))))) < 2.0000000020000002

   1. Initial program 46.4%

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

   3. Taylor expanded in eps around 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 100.0%

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

   if 2.0000000020000002 < (-.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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. /-rgt-identity N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in eps around inf

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 2.000000002:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{x \cdot \varepsilon} - -1 \cdot e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision] - N[(-1.0 * N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $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 48.1%

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

   3. Taylor expanded in 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 98.3%

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

   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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. /-rgt-identity N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in eps around inf

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 48.2

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

   12. Applied rewrites 48.2%

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

4. Final simplification 78.5%

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

Alternative 5: 79.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double t_0 = 1.0 + pow(eps, -1.0);
	double tmp;
	if (((t_0 * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 4.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = ((t_0 * exp((x * eps))) - -1.0) / 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 = 1.0d0 + (eps ** (-1.0d0))
    if (((t_0 * exp((((-1.0d0) + eps) * x))) - (((eps ** (-1.0d0)) - 1.0d0) * exp((((-1.0d0) - eps) * x)))) <= 4.0d0) then
        tmp = (2.0d0 * ((1.0d0 + x) / exp(x))) * 0.5d0
    else
        tmp = ((t_0 * exp((x * eps))) - (-1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$0 * N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $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 48.1%

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

   3. Taylor expanded in 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 98.3%

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

   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

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   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 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 54.9%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 54.9

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

   11. Applied rewrites 54.9%

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

4. Final simplification 81.2%

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

Alternative 6: 66.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 1.9999999999999964:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
       (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
      1.9999999999999964)
   (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
   (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 1.9999999999999964) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) <= 1.9999999999999964)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.9999999999999964], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 1.9999999999999964

   1. Initial program 27.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 100.0%

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

   if 1.9999999999999964 < (-.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 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 29.6%

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

   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 48.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 1.9999999999999964:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 1.9999999999999964:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
       (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
      1.9999999999999964)
   (*
    (*
     2.0
     (/ (+ 1.0 x) (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0)))
    0.5)
   (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 1.9999999999999964) {
		tmp = (2.0 * ((1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0))) * 0.5;
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) <= 1.9999999999999964)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0))) * 0.5);
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.9999999999999964], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 1.9999999999999964

   1. Initial program 27.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.1%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5 \]

   if 1.9999999999999964 < (-.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 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 29.6%

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

   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 48.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 1.9999999999999964:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 1.9999999999999964:\\ \;\;\;\;\frac{x + 1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
       (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
      1.9999999999999964)
   (/ (+ x 1.0) (fma (fma 0.5 x 1.0) x 1.0))
   (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 1.9999999999999964) {
		tmp = (x + 1.0) / fma(fma(0.5, x, 1.0), x, 1.0);
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) <= 1.9999999999999964)
		tmp = Float64(Float64(x + 1.0) / fma(fma(0.5, x, 1.0), x, 1.0));
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.9999999999999964], N[(N[(x + 1.0), $MachinePrecision] / N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 1.9999999999999964

   1. Initial program 27.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 87.9%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 87.9%

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

   if 1.9999999999999964 < (-.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 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 29.6%

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

   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 48.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 1.9999999999999964:\\ \;\;\;\;\frac{x + 1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} - \mathsf{fma}\left(-1 - \varepsilon, x, 1\right) \cdot \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\varepsilon - 1, x, 1\right) \cdot \left({\varepsilon}^{-1} + 1\right) - -1 \cdot e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.0)
   (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
   (if (<= eps 6e+46)
     (/
      (-
       (* (+ 1.0 (pow eps -1.0)) (exp (* (- eps 1.0) x)))
       (* (fma (- -1.0 eps) x 1.0) (- (pow eps -1.0) 1.0)))
      2.0)
     (/
      (-
       (* (fma (- eps 1.0) x 1.0) (+ (pow eps -1.0) 1.0))
       (* -1.0 (exp (- (fma x eps x)))))
      2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else if (eps <= 6e+46) {
		tmp = (((1.0 + pow(eps, -1.0)) * exp(((eps - 1.0) * x))) - (fma((-1.0 - eps), x, 1.0) * (pow(eps, -1.0) - 1.0))) / 2.0;
	} else {
		tmp = ((fma((eps - 1.0), x, 1.0) * (pow(eps, -1.0) + 1.0)) - (-1.0 * exp(-fma(x, eps, x)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	elseif (eps <= 6e+46)
		tmp = Float64(Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(eps - 1.0) * x))) - Float64(fma(Float64(-1.0 - eps), x, 1.0) * Float64((eps ^ -1.0) - 1.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(fma(Float64(eps - 1.0), x, 1.0) * Float64((eps ^ -1.0) + 1.0)) - Float64(-1.0 * exp(Float64(-fma(x, eps, x))))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[eps, 6e+46], N[(N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-1.0 - eps), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1

   1. Initial program 55.6%

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

   3. Taylor expanded in eps around 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 72.2%

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

   if 1 < eps < 6.00000000000000047e46

   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-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. /-rgt-identity N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

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

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 76.0

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

   7. Applied rewrites 76.0%

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

   if 6.00000000000000047e46 < 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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. /-rgt-identity N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in eps around inf

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. sub-neg N/A

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

       6. remove-double-neg N/A

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

       7. mul-1-neg N/A

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

       8. distribute-neg-in N/A

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

       9. +-commutative N/A

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

       10. mul-1-neg N/A

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

       11. sub-neg N/A

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

       12. sub-neg N/A

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

       13. mul-1-neg N/A

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

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

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

       15. distribute-rgt1-in N/A

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

   12. Applied rewrites 71.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} - \mathsf{fma}\left(-1 - \varepsilon, x, 1\right) \cdot \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\varepsilon - 1, x, 1\right) \cdot \left({\varepsilon}^{-1} + 1\right) - -1 \cdot e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\varepsilon - 1, x, 1\right) \cdot \left({\varepsilon}^{-1} + 1\right) - -1 \cdot e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.0)
   (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
   (if (<= eps 1.7e+46)
     (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0)
     (/
      (-
       (* (fma (- eps 1.0) x 1.0) (+ (pow eps -1.0) 1.0))
       (* -1.0 (exp (- (fma x eps x)))))
      2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else if (eps <= 1.7e+46) {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	} else {
		tmp = ((fma((eps - 1.0), x, 1.0) * (pow(eps, -1.0) + 1.0)) - (-1.0 * exp(-fma(x, eps, x)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	elseif (eps <= 1.7e+46)
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(Float64(fma(Float64(eps - 1.0), x, 1.0) * Float64((eps ^ -1.0) + 1.0)) - Float64(-1.0 * exp(Float64(-fma(x, eps, x))))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[eps, 1.7e+46], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1

   1. Initial program 55.6%

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

   3. Taylor expanded in eps around 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 72.2%

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

   if 1 < eps < 1.6999999999999999e46

   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 50.0%

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

   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 74.6%

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

   if 1.6999999999999999e46 < 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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. /-rgt-identity N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in eps around inf

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. sub-neg N/A

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

       6. remove-double-neg N/A

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

       7. mul-1-neg N/A

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

       8. distribute-neg-in N/A

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

       9. +-commutative N/A

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

       10. mul-1-neg N/A

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

       11. sub-neg N/A

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

       12. sub-neg N/A

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

       13. mul-1-neg N/A

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

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

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

       15. distribute-rgt1-in N/A

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

   12. Applied rewrites 71.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\varepsilon - 1, x, 1\right) \cdot \left({\varepsilon}^{-1} + 1\right) - -1 \cdot e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{-x} - -1}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.0)
   (/ (- (* (+ 1.0 (pow eps -1.0)) (exp (- x))) -1.0) 2.0)
   (if (<= x 2e+105)
     (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
     (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.0) {
		tmp = (((1.0 + pow(eps, -1.0)) * exp(-x)) - -1.0) / 2.0;
	} else if (x <= 2e+105) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(-x))) - -1.0) / 2.0);
	elseif (x <= 2e+105)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.0], N[(N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+105], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2

   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 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 56.0%

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

   9. Taylor expanded in eps around 0

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

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 94.1

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

   11. Applied rewrites 94.1%

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

   if -2 < x < 1.9999999999999999e105

   1. Initial program 54.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 76.6%

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

   if 1.9999999999999999e105 < 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 40.1%

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

   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 61.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{-x} - -1}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(-1, 1 + \varepsilon, \frac{-1}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;\frac{x + 1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+105}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.0)
   (fma (* 0.5 x) (fma -1.0 (+ 1.0 eps) (/ -1.0 eps)) 1.0)
   (if (<= x 480.0)
     (/ (+ x 1.0) (fma (fma 0.5 x 1.0) x 1.0))
     (if (<= x 2e+105)
       0.0
       (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.0) {
		tmp = fma((0.5 * x), fma(-1.0, (1.0 + eps), (-1.0 / eps)), 1.0);
	} else if (x <= 480.0) {
		tmp = (x + 1.0) / fma(fma(0.5, x, 1.0), x, 1.0);
	} else if (x <= 2e+105) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.0)
		tmp = fma(Float64(0.5 * x), fma(-1.0, Float64(1.0 + eps), Float64(-1.0 / eps)), 1.0);
	elseif (x <= 480.0)
		tmp = Float64(Float64(x + 1.0) / fma(fma(0.5, x, 1.0), x, 1.0));
	elseif (x <= 2e+105)
		tmp = 0.0;
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.0], N[(N[(0.5 * x), $MachinePrecision] * N[(-1.0 * N[(1.0 + eps), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 480.0], N[(N[(x + 1.0), $MachinePrecision] / N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+105], 0.0, N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 16.9%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 16.9%

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

   if -2 < x < 480

   1. Initial program 50.7%

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

   3. Taylor expanded in 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 77.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.6%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 77.6%

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

   if 480 < x < 1.9999999999999999e105

   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-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. /-rgt-identity N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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 64.8

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

   7. Applied rewrites 64.8%

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

   if 1.9999999999999999e105 < 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 40.1%

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

   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 61.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(-1, 1 + \varepsilon, \frac{-1}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;\frac{x + 1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+105}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.5% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+105}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 480.0)
   1.0
   (if (<= x 2e+105) 0.0 (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 2e+105) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 2e+105)
		tmp = 0.0;
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 480.0], 1.0, If[LessEqual[x, 2e+105], 0.0, N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 480

   1. Initial program 59.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.0%

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

   if 480 < x < 1.9999999999999999e105

   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-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. /-rgt-identity N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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 64.8

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

   7. Applied rewrites 64.8%

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

   if 1.9999999999999999e105 < 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 40.1%

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

   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 61.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+105}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.7% accurate, 38.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 480.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 480.0) {
		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 <= 480.0d0) 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 <= 480.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 480.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 480.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 480.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 480.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 480

   1. Initial program 59.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.0%

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

   if 480 < 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-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. /-rgt-identity N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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 45.9

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

   7. Applied rewrites 45.9%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.6% 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 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 x around 0

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

5. Applied rewrites 50.5%

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

6. Final simplification 50.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(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))