NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 99.8%

Time: 16.3s

Alternatives: 19

Speedup: 5.5×

• 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 = -9.689634551509207e+24
  2. eps = -5.990615598879327e-55

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: 73.8% 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: 99.8% accurate, 0.6× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      0.0)
   (/ (* (exp (- x)) (fma x 2.0 2.0)) 2.0)
   (/ (+ (exp (- (* x eps_m) x)) (exp (* eps_m (- x)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 0.0) {
		tmp = (exp(-x) * fma(x, 2.0, 2.0)) / 2.0;
	} else {
		tmp = (exp(((x * eps_m) - x)) + exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(x * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(x * eps$95$m), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $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))))) < 0.0

   1. Initial program 35.7%

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

   3. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 99.0

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.1

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

   8. Simplified 99.1%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 99.8

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

   8. Simplified 99.8%

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

4. Final simplification 99.5%

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

Alternative 2: 99.8% accurate, 0.6× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      0.0)
   (/ (* (exp (- x)) (fma x 2.0 2.0)) 2.0)
   (/ (+ (exp (- (fma x eps_m x))) (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 0.0) {
		tmp = (exp(-x) * fma(x, 2.0, 2.0)) / 2.0;
	} else {
		tmp = (exp(-fma(x, eps_m, x)) + exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(x * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision] + N[Exp[N[(x * eps$95$m), $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))))) < 0.0

   1. Initial program 35.7%

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

   3. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 99.0

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.1

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

   8. Simplified 99.1%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.8

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

   8. Simplified 99.8%

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

4. Final simplification 99.5%

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

Alternative 3: 99.9% accurate, 0.6× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      0.0)
   (/ (* (exp (- x)) (fma x 2.0 2.0)) 2.0)
   (/ (+ (exp (* eps_m (- x))) (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 0.0) {
		tmp = (exp(-x) * fma(x, 2.0, 2.0)) / 2.0;
	} else {
		tmp = (exp((eps_m * -x)) + exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(x * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * eps$95$m), $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))))) < 0.0

   1. Initial program 35.7%

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

   3. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 99.0

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.1

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

   8. Simplified 99.1%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in eps around inf

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

       1. lower-*.f64 99.8

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

   11. Simplified 99.8%

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

4. Final simplification 99.5%

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

Alternative 4: 92.3% accurate, 0.7× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ -1.0 eps_m))) (t_1 (+ 1.0 (/ 1.0 eps_m))))
   (if (<=
        (+
         (* t_1 (exp (* x (+ eps_m -1.0))))
         (* (exp (* x (- -1.0 eps_m))) t_0))
        2.0)
     (/ (* (exp (- x)) (fma x 2.0 2.0)) 2.0)
     (/
      (fma
       x
       (fma
        x
        (*
         0.5
         (fma
          (+ eps_m 1.0)
          (* (+ eps_m 1.0) t_0)
          (* t_1 (* (- 1.0 eps_m) (- 1.0 eps_m)))))
        (fma (+ eps_m 1.0) (+ (/ 1.0 eps_m) -1.0) (* t_1 (+ eps_m -1.0))))
       2.0)
      2.0))))

C

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

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(x * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(x * N[(0.5 * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(N[(eps$95$m + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(t$95$1 * N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] + -1.0), $MachinePrecision] + N[(t$95$1 * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $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))))) < 2

   1. Initial program 50.3%

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

   3. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 99.3

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.3

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

   8. Simplified 99.3%

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

   if 2 < (-.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

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

   5. Simplified 85.8%

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

4. Final simplification 93.7%

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

Alternative 5: 92.3% accurate, 0.7× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0)
   (/ (* (exp (- x)) (fma x 2.0 2.0)) 2.0)
   (/
    (fma
     x
     (fma (* x 0.5) (fma eps_m eps_m (* (- 1.0 eps_m) (- 1.0 eps_m))) -1.0)
     2.0)
    2.0)))

C

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

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(x * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(x * 0.5), $MachinePrecision] * N[(eps$95$m * eps$95$m + N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $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))))) < 2

   1. Initial program 50.3%

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

   3. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 99.3

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.3

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

   8. Simplified 99.3%

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

   if 2 < (-.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 inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. distribute-rgt1-in N/A

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

       4. metadata-eval N/A

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

       5. mul0-lft N/A

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

       6. +-lft-identity N/A

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

       7. lower-fma.f64 N/A

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

   11. Simplified 85.8%

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

4. Final simplification 93.7%

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

Alternative 6: 92.3% accurate, 0.7× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0)
   (/ (* (exp (- x)) (+ x (+ x 2.0))) 2.0)
   (/
    (fma
     x
     (fma (* x 0.5) (fma eps_m eps_m (* (- 1.0 eps_m) (- 1.0 eps_m))) -1.0)
     2.0)
    2.0)))

C

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

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(x * 0.5), $MachinePrecision] * N[(eps$95$m * eps$95$m + N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $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))))) < 2

   1. Initial program 50.3%

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

   3. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 99.3

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

   5. Simplified 99.3%

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

   if 2 < (-.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 inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. distribute-rgt1-in N/A

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

       4. metadata-eval N/A

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

       5. mul0-lft N/A

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

       6. +-lft-identity N/A

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

       7. lower-fma.f64 N/A

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

   11. Simplified 85.8%

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

4. Final simplification 93.7%

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

Alternative 7: 91.4% accurate, 0.7× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0)
   (exp (- x))
   (/
    (fma
     x
     (fma (* x 0.5) (fma eps_m eps_m (* (- 1.0 eps_m) (- 1.0 eps_m))) -1.0)
     2.0)
    2.0)))

C

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

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[Exp[(-x)], $MachinePrecision], N[(N[(x * N[(N[(x * 0.5), $MachinePrecision] * N[(eps$95$m * eps$95$m + N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $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))))) < 2

   1. Initial program 50.3%

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

   3. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 99.3

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 97.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(x\right)}} \]
   10. Step-by-step derivation

       1. neg-mul-1 N/A

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

       2. lower-exp.f64 N/A

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

       3. neg-mul-1 N/A

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

       4. lower-neg.f64 97.1

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

   11. Simplified 97.1%

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

   if 2 < (-.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 inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. distribute-rgt1-in N/A

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

       4. metadata-eval N/A

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

       5. mul0-lft N/A

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

       6. +-lft-identity N/A

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

       7. lower-fma.f64 N/A

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

   11. Simplified 85.8%

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

4. Final simplification 92.5%

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

Alternative 8: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 2:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.6666666666666666\right), -1\right), 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot 0.5, \mathsf{fma}\left(eps\_m, eps\_m, \left(1 - eps\_m\right) \cdot \left(1 - eps\_m\right)\right), -1\right), 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0)
   (/ (fma x (* x (fma x (fma x -0.25 0.6666666666666666) -1.0)) 2.0) 2.0)
   (/
    (fma
     x
     (fma (* x 0.5) (fma eps_m eps_m (* (- 1.0 eps_m) (- 1.0 eps_m))) -1.0)
     2.0)
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 2.0) {
		tmp = fma(x, (x * fma(x, fma(x, -0.25, 0.6666666666666666), -1.0)), 2.0) / 2.0;
	} else {
		tmp = fma(x, fma((x * 0.5), fma(eps_m, eps_m, ((1.0 - eps_m) * (1.0 - eps_m))), -1.0), 2.0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) <= 2.0)
		tmp = Float64(fma(x, Float64(x * fma(x, fma(x, -0.25, 0.6666666666666666), -1.0)), 2.0) / 2.0);
	else
		tmp = Float64(fma(x, fma(Float64(x * 0.5), fma(eps_m, eps_m, Float64(Float64(1.0 - eps_m) * Float64(1.0 - eps_m))), -1.0), 2.0) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(x * N[(x * N[(x * N[(x * -0.25 + 0.6666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(x * 0.5), $MachinePrecision] * N[(eps$95$m * eps$95$m + N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $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))))) < 2

   1. Initial program 50.3%

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

   3. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 99.3

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + {x}^{2} \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) - 1\right)}}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) - 1\right) + 2}}{2} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) - 1\right) + 2}{2} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) - 1\right)\right)} + 2}{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) - 1\right), 2\right)}}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) - 1\right)}, 2\right)}{2} \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{2} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) + \color{blue}{-1}\right), 2\right)}{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{2}{3} + \frac{-1}{4} \cdot x, -1\right)}, 2\right)}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{\frac{-1}{4} \cdot x + \frac{2}{3}}, -1\right), 2\right)}{2} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{4}} + \frac{2}{3}, -1\right), 2\right)}{2} \]

      11. lower-fma.f64 73.2

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.25, 0.6666666666666666\right)}, -1\right), 2\right)}{2} \]

   8. Simplified 73.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.6666666666666666\right), -1\right), 2\right)}}{2} \]

   if 2 < (-.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 inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. distribute-rgt1-in N/A

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

       4. metadata-eval N/A

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

       5. mul0-lft N/A

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

       6. +-lft-identity N/A

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

       7. lower-fma.f64 N/A

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

   11. Simplified 85.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.6666666666666666\right), -1\right), 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot 0.5, \mathsf{fma}\left(\varepsilon, \varepsilon, \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right), -1\right), 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 10000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.6666666666666666\right), -1\right), 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      10000.0)
   (/ (fma x (* x (fma x (fma x -0.25 0.6666666666666666) -1.0)) 2.0) 2.0)
   (/ (* (* x x) (* 0.5 (* eps_m eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 10000.0) {
		tmp = fma(x, (x * fma(x, fma(x, -0.25, 0.6666666666666666), -1.0)), 2.0) / 2.0;
	} else {
		tmp = ((x * x) * (0.5 * (eps_m * eps_m))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) <= 10000.0)
		tmp = Float64(fma(x, Float64(x * fma(x, fma(x, -0.25, 0.6666666666666666), -1.0)), 2.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(0.5 * Float64(eps_m * eps_m))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 10000.0], N[(N[(x * N[(x * N[(x * N[(x * -0.25 + 0.6666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(eps$95$m * eps$95$m), $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))))) < 1e4

   1. Initial program 51.8%

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

   3. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + {x}^{2} \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) - 1\right)}}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) - 1\right) + 2}}{2} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) - 1\right) + 2}{2} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) - 1\right)\right)} + 2}{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) - 1\right), 2\right)}}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) - 1\right)}, 2\right)}{2} \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{2} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(\frac{2}{3} + \frac{-1}{4} \cdot x\right) + \color{blue}{-1}\right), 2\right)}{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{2}{3} + \frac{-1}{4} \cdot x, -1\right)}, 2\right)}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{\frac{-1}{4} \cdot x + \frac{2}{3}}, -1\right), 2\right)}{2} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{4}} + \frac{2}{3}, -1\right), 2\right)}{2} \]

      11. lower-fma.f64 71.6

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.25, 0.6666666666666666\right)}, -1\right), 2\right)}{2} \]

   8. Simplified 71.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.6666666666666666\right), -1\right), 2\right)}}{2} \]

   if 1e4 < (-.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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 89.6

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

   5. Simplified 89.6%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 81.8

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

   8. Simplified 81.8%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 10000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.6666666666666666\right), -1\right), 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 10000:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      10000.0)
   (fma (* x x) (fma x 0.3333333333333333 -0.5) 1.0)
   (/ (* (* x x) (* 0.5 (* eps_m eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 10000.0) {
		tmp = fma((x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	} else {
		tmp = ((x * x) * (0.5 * (eps_m * eps_m))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) <= 10000.0)
		tmp = fma(Float64(x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(0.5 * Float64(eps_m * eps_m))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 10000.0], N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(eps$95$m * eps$95$m), $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))))) < 1e4

   1. Initial program 51.8%

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

   3. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 71.3

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.6666666666666666, -1\right)}, 2\right)}{2} \]

   8. Simplified 71.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.6666666666666666, -1\right), 2\right)}}{2} \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. sub-neg N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

       5. *-inverses N/A

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

       6. associate-/l* N/A

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. metadata-eval N/A

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

       10. associate-*r/ N/A

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

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

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

       12. distribute-lft-in N/A

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

       13. sub-neg N/A

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

       14. lower-fma.f64 N/A

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

   11. Simplified 71.3%

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

   if 1e4 < (-.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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 89.6

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

   5. Simplified 89.6%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 81.8

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

   8. Simplified 81.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 10000:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.0% accurate, 1.2× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (+ (exp (- (* x eps_m) x)) (exp (- (fma x eps_m x)))) 2.0))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

3. Taylor expanded in eps around inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower-+.f64 N/A

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

5. Simplified 98.2%

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

Alternative 12: 93.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{e^{-x} \cdot \mathsf{fma}\left(x, 2, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m - x} + \mathsf{fma}\left(x, \mathsf{fma}\left(0.5, \mathsf{fma}\left(x, eps\_m, x\right) \cdot \left(eps\_m + 1\right), -1 - eps\_m\right), 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2.6e-13)
   (/ (* (exp (- x)) (fma x 2.0 2.0)) 2.0)
   (/
    (+
     (exp (- (* x eps_m) x))
     (fma x (fma 0.5 (* (fma x eps_m x) (+ eps_m 1.0)) (- -1.0 eps_m)) 1.0))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.6e-13) {
		tmp = (exp(-x) * fma(x, 2.0, 2.0)) / 2.0;
	} else {
		tmp = (exp(((x * eps_m) - x)) + fma(x, fma(0.5, (fma(x, eps_m, x) * (eps_m + 1.0)), (-1.0 - eps_m)), 1.0)) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2.6e-13)
		tmp = Float64(Float64(exp(Float64(-x)) * fma(x, 2.0, 2.0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(Float64(x * eps_m) - x)) + fma(x, fma(0.5, Float64(fma(x, eps_m, x) * Float64(eps_m + 1.0)), Float64(-1.0 - eps_m)), 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 2.6e-13], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(x * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(x * eps$95$m), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[(x * N[(0.5 * N[(N[(x * eps$95$m + x), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.6e-13

   1. Initial program 60.9%

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

   3. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 69.5

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

   5. Simplified 69.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 69.5

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

   8. Simplified 69.5%

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

   if 2.6e-13 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. sub-neg N/A

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

      19. lower--.f64 88.4

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

   8. Simplified 88.4%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{e^{-x} \cdot \mathsf{fma}\left(x, 2, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + \mathsf{fma}\left(x, \mathsf{fma}\left(0.5, \mathsf{fma}\left(x, \varepsilon, x\right) \cdot \left(\varepsilon + 1\right), -1 - \varepsilon\right), 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{if}\;x \leq 760:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, \mathsf{fma}\left(x, eps\_m, x\right) \cdot \left(eps\_m + 1\right), -1 - eps\_m\right), 1\right) + \mathsf{fma}\left(x, \left(eps\_m + -1\right) \cdot \mathsf{fma}\left(x \cdot 0.5, eps\_m + -1, 1\right), 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+268}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)))
   (if (<= x 760.0)
     (/
      (+
       (fma x (fma 0.5 (* (fma x eps_m x) (+ eps_m 1.0)) (- -1.0 eps_m)) 1.0)
       (fma x (* (+ eps_m -1.0) (fma (* x 0.5) (+ eps_m -1.0) 1.0)) 1.0))
      2.0)
     (if (<= x 1.6e+24)
       t_0
       (if (<= x 9.5e+268) (/ (* (* x x) (* 0.5 (* eps_m eps_m))) 2.0) t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	double tmp;
	if (x <= 760.0) {
		tmp = (fma(x, fma(0.5, (fma(x, eps_m, x) * (eps_m + 1.0)), (-1.0 - eps_m)), 1.0) + fma(x, ((eps_m + -1.0) * fma((x * 0.5), (eps_m + -1.0), 1.0)), 1.0)) / 2.0;
	} else if (x <= 1.6e+24) {
		tmp = t_0;
	} else if (x <= 9.5e+268) {
		tmp = ((x * x) * (0.5 * (eps_m * eps_m))) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0)
	tmp = 0.0
	if (x <= 760.0)
		tmp = Float64(Float64(fma(x, fma(0.5, Float64(fma(x, eps_m, x) * Float64(eps_m + 1.0)), Float64(-1.0 - eps_m)), 1.0) + fma(x, Float64(Float64(eps_m + -1.0) * fma(Float64(x * 0.5), Float64(eps_m + -1.0), 1.0)), 1.0)) / 2.0);
	elseif (x <= 1.6e+24)
		tmp = t_0;
	elseif (x <= 9.5e+268)
		tmp = Float64(Float64(Float64(x * x) * Float64(0.5 * Float64(eps_m * eps_m))) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, 760.0], N[(N[(N[(x * N[(0.5 * N[(N[(x * eps$95$m + x), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(x * N[(N[(eps$95$m + -1.0), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] * N[(eps$95$m + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.6e+24], t$95$0, If[LessEqual[x, 9.5e+268], N[(N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < 760

   1. Initial program 58.5%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 97.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. sub-neg N/A

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

      19. lower--.f64 89.9

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

   8. Simplified 89.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. associate--l+ N/A

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

       5. associate-*r* N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. distribute-lft1-in N/A

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

       9. lower-*.f64 N/A

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

       10. lower-fma.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. sub-neg N/A

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

       14. metadata-eval N/A

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

       15. +-commutative N/A

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

       16. lower-+.f64 N/A

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

       17. sub-neg N/A

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

       18. metadata-eval N/A

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

       19. +-commutative N/A

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

       20. lower-+.f64 86.3

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

   11. Simplified 86.3%

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

   if 760 < x < 1.5999999999999999e24 or 9.49999999999999956e268 < 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

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

   5. Simplified 7.2%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 80.6

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

   8. Simplified 80.6%

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

   if 1.5999999999999999e24 < x < 9.49999999999999956e268

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 61.4

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

   5. Simplified 61.4%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 72.0

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

   8. Simplified 72.0%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 760:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, \mathsf{fma}\left(x, \varepsilon, x\right) \cdot \left(\varepsilon + 1\right), -1 - \varepsilon\right), 1\right) + \mathsf{fma}\left(x, \left(\varepsilon + -1\right) \cdot \mathsf{fma}\left(x \cdot 0.5, \varepsilon + -1, 1\right), 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+268}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{-159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot 0.5, \mathsf{fma}\left(eps\_m, eps\_m, \left(1 - eps\_m\right) \cdot \left(1 - eps\_m\right)\right), -1\right), 2\right)}{2}\\ \mathbf{elif}\;x \leq 760:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(x, \mathsf{fma}\left(0.5, \mathsf{fma}\left(x, eps\_m, x\right) \cdot \left(eps\_m + 1\right), -1 - eps\_m\right), 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+268}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)))
   (if (<= x -3.3e-159)
     (/
      (fma
       x
       (fma (* x 0.5) (fma eps_m eps_m (* (- 1.0 eps_m) (- 1.0 eps_m))) -1.0)
       2.0)
      2.0)
     (if (<= x 760.0)
       (/
        (+
         1.0
         (fma
          x
          (fma 0.5 (* (fma x eps_m x) (+ eps_m 1.0)) (- -1.0 eps_m))
          1.0))
        2.0)
       (if (<= x 1.6e+24)
         t_0
         (if (<= x 9.5e+268)
           (/ (* (* x x) (* 0.5 (* eps_m eps_m))) 2.0)
           t_0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	double tmp;
	if (x <= -3.3e-159) {
		tmp = fma(x, fma((x * 0.5), fma(eps_m, eps_m, ((1.0 - eps_m) * (1.0 - eps_m))), -1.0), 2.0) / 2.0;
	} else if (x <= 760.0) {
		tmp = (1.0 + fma(x, fma(0.5, (fma(x, eps_m, x) * (eps_m + 1.0)), (-1.0 - eps_m)), 1.0)) / 2.0;
	} else if (x <= 1.6e+24) {
		tmp = t_0;
	} else if (x <= 9.5e+268) {
		tmp = ((x * x) * (0.5 * (eps_m * eps_m))) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0)
	tmp = 0.0
	if (x <= -3.3e-159)
		tmp = Float64(fma(x, fma(Float64(x * 0.5), fma(eps_m, eps_m, Float64(Float64(1.0 - eps_m) * Float64(1.0 - eps_m))), -1.0), 2.0) / 2.0);
	elseif (x <= 760.0)
		tmp = Float64(Float64(1.0 + fma(x, fma(0.5, Float64(fma(x, eps_m, x) * Float64(eps_m + 1.0)), Float64(-1.0 - eps_m)), 1.0)) / 2.0);
	elseif (x <= 1.6e+24)
		tmp = t_0;
	elseif (x <= 9.5e+268)
		tmp = Float64(Float64(Float64(x * x) * Float64(0.5 * Float64(eps_m * eps_m))) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -3.3e-159], N[(N[(x * N[(N[(x * 0.5), $MachinePrecision] * N[(eps$95$m * eps$95$m + N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 760.0], N[(N[(1.0 + N[(x * N[(0.5 * N[(N[(x * eps$95$m + x), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.6e+24], t$95$0, If[LessEqual[x, 9.5e+268], N[(N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.3000000000000002e-159

   1. Initial program 71.9%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 96.4%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 96.4

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

   8. Simplified 96.4%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. distribute-rgt1-in N/A

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

       4. metadata-eval N/A

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

       5. mul0-lft N/A

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

       6. +-lft-identity N/A

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

       7. lower-fma.f64 N/A

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

   11. Simplified 85.5%

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

   if -3.3000000000000002e-159 < x < 760

   1. Initial program 50.2%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. sub-neg N/A

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

      19. lower--.f64 92.9

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

   8. Simplified 92.9%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 89.8%

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

   if 760 < x < 1.5999999999999999e24 or 9.49999999999999956e268 < 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

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

   5. Simplified 7.2%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 80.6

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

   8. Simplified 80.6%

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

   if 1.5999999999999999e24 < x < 9.49999999999999956e268

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 61.4

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

   5. Simplified 61.4%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 72.0

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

   8. Simplified 72.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot 0.5, \mathsf{fma}\left(\varepsilon, \varepsilon, \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right), -1\right), 2\right)}{2}\\ \mathbf{elif}\;x \leq 760:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(x, \mathsf{fma}\left(0.5, \mathsf{fma}\left(x, \varepsilon, x\right) \cdot \left(\varepsilon + 1\right), -1 - \varepsilon\right), 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+268}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.038:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(x, \mathsf{fma}\left(0.5, \mathsf{fma}\left(x, eps\_m, x\right) \cdot \left(eps\_m + 1\right), -1 - eps\_m\right), 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.038)
   (/
    (+
     1.0
     (fma x (fma 0.5 (* (fma x eps_m x) (+ eps_m 1.0)) (- -1.0 eps_m)) 1.0))
    2.0)
   (/ (* (* x x) (* 0.5 (* eps_m eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.038) {
		tmp = (1.0 + fma(x, fma(0.5, (fma(x, eps_m, x) * (eps_m + 1.0)), (-1.0 - eps_m)), 1.0)) / 2.0;
	} else {
		tmp = ((x * x) * (0.5 * (eps_m * eps_m))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 0.038)
		tmp = Float64(Float64(1.0 + fma(x, fma(0.5, Float64(fma(x, eps_m, x) * Float64(eps_m + 1.0)), Float64(-1.0 - eps_m)), 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(0.5 * Float64(eps_m * eps_m))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 0.038], N[(N[(1.0 + N[(x * N[(0.5 * N[(N[(x * eps$95$m + x), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0379999999999999991

   1. Initial program 58.3%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 97.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. sub-neg N/A

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

      19. lower--.f64 89.8

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

   8. Simplified 89.8%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 86.1%

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

   if 0.0379999999999999991 < 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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 50.2

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

   5. Simplified 50.2%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 58.1

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

   8. Simplified 58.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.038:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(x, \mathsf{fma}\left(0.5, \mathsf{fma}\left(x, \varepsilon, x\right) \cdot \left(\varepsilon + 1\right), -1 - \varepsilon\right), 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.8% accurate, 10.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -460:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -460.0)
   (fma x (fma x (fma x -0.16666666666666666 0.5) -1.0) 1.0)
   (fma (* x x) (fma x 0.3333333333333333 -0.5) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -460.0) {
		tmp = fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = fma((x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -460.0)
		tmp = fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	else
		tmp = fma(Float64(x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -460.0], N[(x * N[(x * N[(x * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -460

   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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 97.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

   11. Simplified 63.5%

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

   if -460 < x

   1. Initial program 66.8%

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

   3. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-+l- N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 67.4

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

   5. Simplified 67.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} \cdot x - 1, 2\right)}{2} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{3} \cdot x + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{2} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \frac{2}{3}} + \left(\mathsf{neg}\left(1\right)\right), 2\right)}{2} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, x \cdot \frac{2}{3} + \color{blue}{-1}, 2\right)}{2} \]

      8. lower-fma.f64 61.4

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.6666666666666666, -1\right)}, 2\right)}{2} \]

   8. Simplified 61.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.6666666666666666, -1\right), 2\right)}}{2} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1} \]

       2. sub-neg N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} + 1 \]

       3. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \left(\color{blue}{x \cdot \frac{1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) + 1 \]

       4. metadata-eval N/A

          \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot 1}\right)\right)\right) + 1 \]

       5. *-inverses N/A

          \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{\frac{x}{x}}\right)\right)\right) + 1 \]

       6. associate-/l* N/A

          \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot x}{x}}\right)\right)\right) + 1 \]

       7. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\frac{\color{blue}{x \cdot \frac{1}{2}}}{x}\right)\right)\right) + 1 \]

       8. associate-/l* N/A

          \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\frac{1}{2}}{x}}\right)\right)\right) + 1 \]

       9. metadata-eval N/A

          \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(x \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{x}\right)\right)\right) + 1 \]

       10. associate-*r/ N/A

           \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x}\right)}\right)\right)\right) + 1 \]

       11. distribute-rgt-neg-out N/A

           \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right) + 1 \]

       12. distribute-lft-in N/A

           \[\leadsto {x}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{3} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)} + 1 \]

       13. sub-neg N/A

           \[\leadsto {x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{1}{3} - \frac{1}{2} \cdot \frac{1}{x}\right)}\right) + 1 \]

       14. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, x \cdot \left(\frac{1}{3} - \frac{1}{2} \cdot \frac{1}{x}\right), 1\right)} \]

   11. Simplified 61.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 59.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -460:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -460.0)
   (fma x (fma x 0.5 -1.0) 1.0)
   (fma (* x x) (fma x 0.3333333333333333 -0.5) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -460.0) {
		tmp = fma(x, fma(x, 0.5, -1.0), 1.0);
	} else {
		tmp = fma((x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -460.0)
		tmp = fma(x, fma(x, 0.5, -1.0), 1.0);
	else
		tmp = fma(Float64(x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -460.0], N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -460

   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

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}\right)}{2} \]

      2. unsub-neg N/A

         \[\leadsto \frac{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \color{blue}{\left(-1 \cdot e^{\mathsf{neg}\left(x\right)} - x \cdot e^{\mathsf{neg}\left(x\right)}\right)}}{2} \]

      3. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) + x \cdot e^{\mathsf{neg}\left(x\right)}}}{2} \]

      4. distribute-rgt1-in N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x + 1\right) \cdot e^{\mathsf{neg}\left(x\right)}} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) + x \cdot e^{\mathsf{neg}\left(x\right)}}{2} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(x + 1\right) - -1\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}}{2} \]

      6. *-commutative N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(x + 1\right) - -1\right) + \color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot x}}{2} \]

      7. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}}{2} \]

      9. lower-exp.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}{2} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(x\right)}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}{2} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \color{blue}{\left(\left(\left(x + 1\right) - -1\right) + x\right)}}{2} \]

      12. associate--l+ N/A

          \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \left(\color{blue}{\left(x + \left(1 - -1\right)\right)} + x\right)}{2} \]

      13. metadata-eval N/A

          \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(x + \color{blue}{2}\right) + x\right)}{2} \]

      14. lower-+.f64 0.0

          \[\leadsto \frac{e^{-x} \cdot \left(\color{blue}{\left(x + 2\right)} + x\right)}{2} \]

   5. Simplified 0.0%

      \[\leadsto \frac{\color{blue}{e^{-x} \cdot \left(\left(x + 2\right) + x\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \color{blue}{2}}{2} \]
   7. Step-by-step derivation

   8. Simplified 97.0%

      \[\leadsto \frac{e^{-x} \cdot \color{blue}{2}}{2} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot x - 1\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot x - 1\right) + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot x - 1, 1\right)} \]

       3. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

       6. lower-fma.f64 51.0

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, -1\right)}, 1\right) \]

   11. Simplified 51.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)} \]

   if -460 < x

   1. Initial program 66.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}\right)}{2} \]

      2. unsub-neg N/A

         \[\leadsto \frac{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \color{blue}{\left(-1 \cdot e^{\mathsf{neg}\left(x\right)} - x \cdot e^{\mathsf{neg}\left(x\right)}\right)}}{2} \]

      3. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) + x \cdot e^{\mathsf{neg}\left(x\right)}}}{2} \]

      4. distribute-rgt1-in N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x + 1\right) \cdot e^{\mathsf{neg}\left(x\right)}} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) + x \cdot e^{\mathsf{neg}\left(x\right)}}{2} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(x + 1\right) - -1\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}}{2} \]

      6. *-commutative N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(x + 1\right) - -1\right) + \color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot x}}{2} \]

      7. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}}{2} \]

      9. lower-exp.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}{2} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(x\right)}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}{2} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \color{blue}{\left(\left(\left(x + 1\right) - -1\right) + x\right)}}{2} \]

      12. associate--l+ N/A

          \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \left(\color{blue}{\left(x + \left(1 - -1\right)\right)} + x\right)}{2} \]

      13. metadata-eval N/A

          \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(x + \color{blue}{2}\right) + x\right)}{2} \]

      14. lower-+.f64 67.4

          \[\leadsto \frac{e^{-x} \cdot \left(\color{blue}{\left(x + 2\right)} + x\right)}{2} \]

   5. Simplified 67.4%

      \[\leadsto \frac{\color{blue}{e^{-x} \cdot \left(\left(x + 2\right) + x\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + {x}^{2} \cdot \left(\frac{2}{3} \cdot x - 1\right)}}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{2}{3} \cdot x - 1\right) + 2}}{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{3} \cdot x - 1, 2\right)}}{2} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} \cdot x - 1, 2\right)}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} \cdot x - 1, 2\right)}{2} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{3} \cdot x + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{2} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \frac{2}{3}} + \left(\mathsf{neg}\left(1\right)\right), 2\right)}{2} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, x \cdot \frac{2}{3} + \color{blue}{-1}, 2\right)}{2} \]

      8. lower-fma.f64 61.4

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.6666666666666666, -1\right)}, 2\right)}{2} \]

   8. Simplified 61.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.6666666666666666, -1\right), 2\right)}}{2} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1} \]

       2. sub-neg N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} + 1 \]

       3. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \left(\color{blue}{x \cdot \frac{1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) + 1 \]

       4. metadata-eval N/A

          \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot 1}\right)\right)\right) + 1 \]

       5. *-inverses N/A

          \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{\frac{x}{x}}\right)\right)\right) + 1 \]

       6. associate-/l* N/A

          \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot x}{x}}\right)\right)\right) + 1 \]

       7. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\frac{\color{blue}{x \cdot \frac{1}{2}}}{x}\right)\right)\right) + 1 \]

       8. associate-/l* N/A

          \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\frac{1}{2}}{x}}\right)\right)\right) + 1 \]

       9. metadata-eval N/A

          \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(x \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{x}\right)\right)\right) + 1 \]

       10. associate-*r/ N/A

           \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x}\right)}\right)\right)\right) + 1 \]

       11. distribute-rgt-neg-out N/A

           \[\leadsto {x}^{2} \cdot \left(x \cdot \frac{1}{3} + \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right) + 1 \]

       12. distribute-lft-in N/A

           \[\leadsto {x}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{3} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)} + 1 \]

       13. sub-neg N/A

           \[\leadsto {x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{1}{3} - \frac{1}{2} \cdot \frac{1}{x}\right)}\right) + 1 \]

       14. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, x \cdot \left(\frac{1}{3} - \frac{1}{2} \cdot \frac{1}{x}\right), 1\right)} \]

   11. Simplified 61.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 56.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (fma x (fma x 0.5 -1.0) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return fma(x, fma(x, 0.5, -1.0), 1.0);
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return fma(x, fma(x, 0.5, -1.0), 1.0)
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 70.8%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}\right)}{2} \]

   2. unsub-neg N/A

      \[\leadsto \frac{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \color{blue}{\left(-1 \cdot e^{\mathsf{neg}\left(x\right)} - x \cdot e^{\mathsf{neg}\left(x\right)}\right)}}{2} \]

   3. associate-+l- N/A

      \[\leadsto \frac{\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) + x \cdot e^{\mathsf{neg}\left(x\right)}}}{2} \]

   4. distribute-rgt1-in N/A

      \[\leadsto \frac{\left(\color{blue}{\left(x + 1\right) \cdot e^{\mathsf{neg}\left(x\right)}} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) + x \cdot e^{\mathsf{neg}\left(x\right)}}{2} \]

   5. distribute-rgt-out-- N/A

      \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(x + 1\right) - -1\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}}{2} \]

   6. *-commutative N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(x + 1\right) - -1\right) + \color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot x}}{2} \]

   7. distribute-lft-out N/A

      \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}}{2} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}}{2} \]

   9. lower-exp.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}{2} \]

   10. lower-neg.f64 N/A

       \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(x\right)}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}{2} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \color{blue}{\left(\left(\left(x + 1\right) - -1\right) + x\right)}}{2} \]

   12. associate--l+ N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \left(\color{blue}{\left(x + \left(1 - -1\right)\right)} + x\right)}{2} \]

   13. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(x + \color{blue}{2}\right) + x\right)}{2} \]

   14. lower-+.f64 59.2

       \[\leadsto \frac{e^{-x} \cdot \left(\color{blue}{\left(x + 2\right)} + x\right)}{2} \]

5. Simplified 59.2%

   \[\leadsto \frac{\color{blue}{e^{-x} \cdot \left(\left(x + 2\right) + x\right)}}{2} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \color{blue}{2}}{2} \]
7. Step-by-step derivation

8. Simplified 69.7%

   \[\leadsto \frac{e^{-x} \cdot \color{blue}{2}}{2} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot x - 1\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot x - 1\right) + 1} \]

    2. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot x - 1, 1\right)} \]

    3. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

    6. lower-fma.f64 57.9

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, -1\right)}, 1\right) \]

11. Simplified 57.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)} \]
12. Add Preprocessing

Alternative 19: 43.8% accurate, 273.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 1.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 1.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 1.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 70.8%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}\right)}{2} \]

   2. unsub-neg N/A

      \[\leadsto \frac{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \color{blue}{\left(-1 \cdot e^{\mathsf{neg}\left(x\right)} - x \cdot e^{\mathsf{neg}\left(x\right)}\right)}}{2} \]

   3. associate-+l- N/A

      \[\leadsto \frac{\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) + x \cdot e^{\mathsf{neg}\left(x\right)}}}{2} \]

   4. distribute-rgt1-in N/A

      \[\leadsto \frac{\left(\color{blue}{\left(x + 1\right) \cdot e^{\mathsf{neg}\left(x\right)}} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) + x \cdot e^{\mathsf{neg}\left(x\right)}}{2} \]

   5. distribute-rgt-out-- N/A

      \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(x + 1\right) - -1\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}}{2} \]

   6. *-commutative N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(x + 1\right) - -1\right) + \color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot x}}{2} \]

   7. distribute-lft-out N/A

      \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}}{2} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}}{2} \]

   9. lower-exp.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(x\right)}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}{2} \]

   10. lower-neg.f64 N/A

       \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(x\right)}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)}{2} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \color{blue}{\left(\left(\left(x + 1\right) - -1\right) + x\right)}}{2} \]

   12. associate--l+ N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \left(\color{blue}{\left(x + \left(1 - -1\right)\right)} + x\right)}{2} \]

   13. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot \left(\left(x + \color{blue}{2}\right) + x\right)}{2} \]

   14. lower-+.f64 59.2

       \[\leadsto \frac{e^{-x} \cdot \left(\color{blue}{\left(x + 2\right)} + x\right)}{2} \]

5. Simplified 59.2%

   \[\leadsto \frac{\color{blue}{e^{-x} \cdot \left(\left(x + 2\right) + x\right)}}{2} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + -1 \cdot {x}^{2}}}{2} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{2 + \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)}}{2} \]

   2. unsub-neg N/A

      \[\leadsto \frac{\color{blue}{2 - {x}^{2}}}{2} \]

   3. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{2 - {x}^{2}}}{2} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 - \color{blue}{x \cdot x}}{2} \]

   5. lower-*.f64 43.4

      \[\leadsto \frac{2 - \color{blue}{x \cdot x}}{2} \]

8. Simplified 43.4%

   \[\leadsto \frac{\color{blue}{2 - x \cdot x}}{2} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
10. Step-by-step derivation

11. Simplified 44.2%

    \[\leadsto \color{blue}{1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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))