NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 99.8%

Time: 12.9s

Alternatives: 16

Speedup: 1.9×

• 37.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -2.450818539647624e+52
  2. eps = -1.1174082052136478e-49

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: 73.5% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[Exp[N[(N[(-1.0 + eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(1.0 * N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * eps$95$m + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 41.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   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

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in eps around inf

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

      1. lower-*.f64 99.0

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 100.0%

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

       1. lift-/.f64 N/A

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

       2. div-inv N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 100.0

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

   13. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;e^{\left(-1 + eps\_m\right) \cdot x} \cdot \left(\frac{1}{eps\_m} + 1\right) - e^{\left(-1 - eps\_m\right) \cdot x} \cdot \left(\frac{1}{eps\_m} - 1\right) \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.375, x, -0.3333333333333333\right), x, 0.5\right), x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (-
       (* (exp (* (+ -1.0 eps_m) x)) (+ (/ 1.0 eps_m) 1.0))
       (* (exp (* (- -1.0 eps_m) x)) (- (/ 1.0 eps_m) 1.0)))
      0.0)
   (/ 1.0 (fma (fma (fma 0.375 x -0.3333333333333333) x 0.5) (* x x) 1.0))
   (fma (* 0.3333333333333333 x) (* x x) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (((exp(((-1.0 + eps_m) * x)) * ((1.0 / eps_m) + 1.0)) - (exp(((-1.0 - eps_m) * x)) * ((1.0 / eps_m) - 1.0))) <= 0.0) {
		tmp = 1.0 / fma(fma(fma(0.375, x, -0.3333333333333333), x, 0.5), (x * x), 1.0);
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(Float64(-1.0 + eps_m) * x)) * Float64(Float64(1.0 / eps_m) + 1.0)) - Float64(exp(Float64(Float64(-1.0 - eps_m) * x)) * Float64(Float64(1.0 / eps_m) - 1.0))) <= 0.0)
		tmp = Float64(1.0 / fma(fma(fma(0.375, x, -0.3333333333333333), x, 0.5), Float64(x * x), 1.0));
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[Exp[N[(N[(-1.0 + eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(N[(N[(0.375 * x + -0.3333333333333333), $MachinePrecision] * x + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 41.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 82.3%

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

   10. Applied rewrites 82.3%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{1}{1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{3}{8} \cdot x - \frac{1}{3}\right)\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 91.7%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.375, x, -0.3333333333333333\right), x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 20.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 38.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 38.8%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(-1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.375, x, -0.3333333333333333\right), x, 0.5\right), x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;e^{\left(-1 + eps\_m\right) \cdot x} \cdot \left(\frac{1}{eps\_m} + 1\right) - e^{\left(-1 - eps\_m\right) \cdot x} \cdot \left(\frac{1}{eps\_m} - 1\right) \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (-
       (* (exp (* (+ -1.0 eps_m) x)) (+ (/ 1.0 eps_m) 1.0))
       (* (exp (* (- -1.0 eps_m) x)) (- (/ 1.0 eps_m) 1.0)))
      0.0)
   (/ 1.0 (fma (fma -0.3333333333333333 x 0.5) (* x x) 1.0))
   (fma (* 0.3333333333333333 x) (* x x) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (((exp(((-1.0 + eps_m) * x)) * ((1.0 / eps_m) + 1.0)) - (exp(((-1.0 - eps_m) * x)) * ((1.0 / eps_m) - 1.0))) <= 0.0) {
		tmp = 1.0 / fma(fma(-0.3333333333333333, x, 0.5), (x * x), 1.0);
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[Exp[N[(N[(-1.0 + eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(N[(-0.3333333333333333 * x + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 41.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 82.3%

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

   10. Applied rewrites 82.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 88.7%

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

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 20.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 38.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 38.8%

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

4. Final simplification 57.3%

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

Alternative 4: 77.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\frac{1 \cdot e^{\left(-1 + eps\_m\right) \cdot x} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{1}{eps\_m} + 1\right) - e^{\left(-1 - eps\_m\right) \cdot x} \cdot \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5.6e-20)
   (/ (+ x 1.0) (exp x))
   (if (<= eps_m 2e+58)
     (/ (- (* 1.0 (exp (* (+ -1.0 eps_m) x))) -1.0) 2.0)
     (/
      (-
       (* 1.0 (+ (/ 1.0 eps_m) 1.0))
       (* (exp (* (- -1.0 eps_m) x)) (- (/ 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 <= 5.6e-20) {
		tmp = (x + 1.0) / exp(x);
	} else if (eps_m <= 2e+58) {
		tmp = ((1.0 * exp(((-1.0 + eps_m) * x))) - -1.0) / 2.0;
	} else {
		tmp = ((1.0 * ((1.0 / eps_m) + 1.0)) - (exp(((-1.0 - eps_m) * x)) * ((1.0 / eps_m) - 1.0))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 5.6d-20) then
        tmp = (x + 1.0d0) / exp(x)
    else if (eps_m <= 2d+58) then
        tmp = ((1.0d0 * exp((((-1.0d0) + eps_m) * x))) - (-1.0d0)) / 2.0d0
    else
        tmp = ((1.0d0 * ((1.0d0 / eps_m) + 1.0d0)) - (exp((((-1.0d0) - eps_m) * x)) * ((1.0d0 / eps_m) - 1.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5.6e-20) {
		tmp = (x + 1.0) / Math.exp(x);
	} else if (eps_m <= 2e+58) {
		tmp = ((1.0 * Math.exp(((-1.0 + eps_m) * x))) - -1.0) / 2.0;
	} else {
		tmp = ((1.0 * ((1.0 / eps_m) + 1.0)) - (Math.exp(((-1.0 - eps_m) * x)) * ((1.0 / eps_m) - 1.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 5.6e-20:
		tmp = (x + 1.0) / math.exp(x)
	elif eps_m <= 2e+58:
		tmp = ((1.0 * math.exp(((-1.0 + eps_m) * x))) - -1.0) / 2.0
	else:
		tmp = ((1.0 * ((1.0 / eps_m) + 1.0)) - (math.exp(((-1.0 - eps_m) * x)) * ((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 <= 5.6e-20)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	elseif (eps_m <= 2e+58)
		tmp = Float64(Float64(Float64(1.0 * exp(Float64(Float64(-1.0 + eps_m) * x))) - -1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 * Float64(Float64(1.0 / eps_m) + 1.0)) - Float64(exp(Float64(Float64(-1.0 - eps_m) * x)) * Float64(Float64(1.0 / eps_m) - 1.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 5.6e-20)
		tmp = (x + 1.0) / exp(x);
	elseif (eps_m <= 2e+58)
		tmp = ((1.0 * exp(((-1.0 + eps_m) * x))) - -1.0) / 2.0;
	else
		tmp = ((1.0 * ((1.0 / eps_m) + 1.0)) - (exp(((-1.0 - eps_m) * x)) * ((1.0 / eps_m) - 1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5.6e-20], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 2e+58], N[(N[(N[(1.0 * N[Exp[N[(N[(-1.0 + eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 * N[(N[(1.0 / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 5.6000000000000005e-20

   1. Initial program 70.7%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.3%

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

   if 5.6000000000000005e-20 < eps < 1.99999999999999989e58

   1. Initial program 80.6%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 100.0%

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

   if 1.99999999999999989e58 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\frac{1 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{\left(-1 + eps\_m\right) \cdot x}\\ \mathbf{if}\;x \leq -215:\\ \;\;\;\;\left(\frac{e^{-x}}{eps\_m} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{1 \cdot t\_0 - -1}{2}\\ \mathbf{elif}\;x \leq 10^{+192}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \left(\frac{1}{eps\_m} + 1\right) - -1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* (+ -1.0 eps_m) x))))
   (if (<= x -215.0)
     (* (- (/ (exp (- x)) eps_m) -1.0) 0.5)
     (if (<= x 4.8e-20)
       (/ (- (* 1.0 t_0) -1.0) 2.0)
       (if (<= x 1e+192)
         (/ (+ x 1.0) (exp x))
         (/ (- (* t_0 (+ (/ 1.0 eps_m) 1.0)) -1.0) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(((-1.0 + eps_m) * x));
	double tmp;
	if (x <= -215.0) {
		tmp = ((exp(-x) / eps_m) - -1.0) * 0.5;
	} else if (x <= 4.8e-20) {
		tmp = ((1.0 * t_0) - -1.0) / 2.0;
	} else if (x <= 1e+192) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = ((t_0 * ((1.0 / eps_m) + 1.0)) - -1.0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((((-1.0d0) + eps_m) * x))
    if (x <= (-215.0d0)) then
        tmp = ((exp(-x) / eps_m) - (-1.0d0)) * 0.5d0
    else if (x <= 4.8d-20) then
        tmp = ((1.0d0 * t_0) - (-1.0d0)) / 2.0d0
    else if (x <= 1d+192) then
        tmp = (x + 1.0d0) / exp(x)
    else
        tmp = ((t_0 * ((1.0d0 / eps_m) + 1.0d0)) - (-1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(((-1.0 + eps_m) * x));
	double tmp;
	if (x <= -215.0) {
		tmp = ((Math.exp(-x) / eps_m) - -1.0) * 0.5;
	} else if (x <= 4.8e-20) {
		tmp = ((1.0 * t_0) - -1.0) / 2.0;
	} else if (x <= 1e+192) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = ((t_0 * ((1.0 / eps_m) + 1.0)) - -1.0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(((-1.0 + eps_m) * x))
	tmp = 0
	if x <= -215.0:
		tmp = ((math.exp(-x) / eps_m) - -1.0) * 0.5
	elif x <= 4.8e-20:
		tmp = ((1.0 * t_0) - -1.0) / 2.0
	elif x <= 1e+192:
		tmp = (x + 1.0) / math.exp(x)
	else:
		tmp = ((t_0 * ((1.0 / eps_m) + 1.0)) - -1.0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(Float64(-1.0 + eps_m) * x))
	tmp = 0.0
	if (x <= -215.0)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps_m) - -1.0) * 0.5);
	elseif (x <= 4.8e-20)
		tmp = Float64(Float64(Float64(1.0 * t_0) - -1.0) / 2.0);
	elseif (x <= 1e+192)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(Float64(Float64(t_0 * Float64(Float64(1.0 / eps_m) + 1.0)) - -1.0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(((-1.0 + eps_m) * x));
	tmp = 0.0;
	if (x <= -215.0)
		tmp = ((exp(-x) / eps_m) - -1.0) * 0.5;
	elseif (x <= 4.8e-20)
		tmp = ((1.0 * t_0) - -1.0) / 2.0;
	elseif (x <= 1e+192)
		tmp = (x + 1.0) / exp(x);
	else
		tmp = ((t_0 * ((1.0 / eps_m) + 1.0)) - -1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(N[(-1.0 + eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -215.0], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4.8e-20], N[(N[(N[(1.0 * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+192], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[(N[(1.0 / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -215

   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

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. neg-mul-1 N/A

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

       3. lower-exp.f64 N/A

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

       4. neg-mul-1 N/A

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

       5. lower-neg.f64 37.5

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

   11. Applied rewrites 37.5%

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

       1. lift-/.f64 N/A

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

       2. div-inv N/A

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

       3. metadata-eval N/A

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

   13. Applied rewrites 37.5%

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

   if -215 < x < 4.79999999999999986e-20

   1. Initial program 59.9%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 58.5

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 38.9%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 80.1%

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

   if 4.79999999999999986e-20 < x < 1.00000000000000004e192

   1. Initial program 90.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.4%

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

   if 1.00000000000000004e192 < 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

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - -1}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -215:\\ \;\;\;\;\left(\frac{e^{-x}}{\varepsilon} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{1 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - -1}{2}\\ \mathbf{elif}\;x \leq 10^{+192}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+67}:\\ \;\;\;\;\frac{1 \cdot e^{\left(-1 + eps\_m\right) \cdot x} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} + 1\right) - \frac{-1}{e^{\mathsf{fma}\left(eps\_m, x, x\right)}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5.6e-20)
   (/ (+ x 1.0) (exp x))
   (if (<= eps_m 2e+67)
     (/ (- (* 1.0 (exp (* (+ -1.0 eps_m) x))) -1.0) 2.0)
     (/ (- (+ (/ 1.0 eps_m) 1.0) (/ -1.0 (exp (fma eps_m x x)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5.6e-20) {
		tmp = (x + 1.0) / exp(x);
	} else if (eps_m <= 2e+67) {
		tmp = ((1.0 * exp(((-1.0 + eps_m) * x))) - -1.0) / 2.0;
	} else {
		tmp = (((1.0 / eps_m) + 1.0) - (-1.0 / exp(fma(eps_m, x, x)))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5.6e-20)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	elseif (eps_m <= 2e+67)
		tmp = Float64(Float64(Float64(1.0 * exp(Float64(Float64(-1.0 + eps_m) * x))) - -1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) + 1.0) - Float64(-1.0 / exp(fma(eps_m, x, x)))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5.6e-20], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 2e+67], N[(N[(N[(1.0 * N[Exp[N[(N[(-1.0 + eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision] - N[(-1.0 / N[Exp[N[(eps$95$m * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 5.6000000000000005e-20

   1. Initial program 70.7%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.3%

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

   if 5.6000000000000005e-20 < eps < 1.99999999999999997e67

   1. Initial program 83.9%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.2%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 91.9%

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

   if 1.99999999999999997e67 < 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

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 65.5

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

   8. Applied rewrites 65.5%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+67}:\\ \;\;\;\;\frac{1 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \frac{-1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1 \cdot e^{\left(-1 + eps\_m\right) \cdot x} - -1}{2}\\ \mathbf{if}\;x \leq -215:\\ \;\;\;\;\left(\frac{e^{-x}}{eps\_m} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{+192}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (- (* 1.0 (exp (* (+ -1.0 eps_m) x))) -1.0) 2.0)))
   (if (<= x -215.0)
     (* (- (/ (exp (- x)) eps_m) -1.0) 0.5)
     (if (<= x 4.8e-20) t_0 (if (<= x 1e+192) (/ (+ x 1.0) (exp x)) t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = ((1.0 * exp(((-1.0 + eps_m) * x))) - -1.0) / 2.0;
	double tmp;
	if (x <= -215.0) {
		tmp = ((exp(-x) / eps_m) - -1.0) * 0.5;
	} else if (x <= 4.8e-20) {
		tmp = t_0;
	} else if (x <= 1e+192) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 * exp((((-1.0d0) + eps_m) * x))) - (-1.0d0)) / 2.0d0
    if (x <= (-215.0d0)) then
        tmp = ((exp(-x) / eps_m) - (-1.0d0)) * 0.5d0
    else if (x <= 4.8d-20) then
        tmp = t_0
    else if (x <= 1d+192) then
        tmp = (x + 1.0d0) / exp(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = ((1.0 * Math.exp(((-1.0 + eps_m) * x))) - -1.0) / 2.0;
	double tmp;
	if (x <= -215.0) {
		tmp = ((Math.exp(-x) / eps_m) - -1.0) * 0.5;
	} else if (x <= 4.8e-20) {
		tmp = t_0;
	} else if (x <= 1e+192) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = ((1.0 * math.exp(((-1.0 + eps_m) * x))) - -1.0) / 2.0
	tmp = 0
	if x <= -215.0:
		tmp = ((math.exp(-x) / eps_m) - -1.0) * 0.5
	elif x <= 4.8e-20:
		tmp = t_0
	elif x <= 1e+192:
		tmp = (x + 1.0) / math.exp(x)
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(1.0 * exp(Float64(Float64(-1.0 + eps_m) * x))) - -1.0) / 2.0)
	tmp = 0.0
	if (x <= -215.0)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps_m) - -1.0) * 0.5);
	elseif (x <= 4.8e-20)
		tmp = t_0;
	elseif (x <= 1e+192)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = ((1.0 * exp(((-1.0 + eps_m) * x))) - -1.0) / 2.0;
	tmp = 0.0;
	if (x <= -215.0)
		tmp = ((exp(-x) / eps_m) - -1.0) * 0.5;
	elseif (x <= 4.8e-20)
		tmp = t_0;
	elseif (x <= 1e+192)
		tmp = (x + 1.0) / exp(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[(1.0 * N[Exp[N[(N[(-1.0 + eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -215.0], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4.8e-20], t$95$0, If[LessEqual[x, 1e+192], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -215

   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

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. neg-mul-1 N/A

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

       3. lower-exp.f64 N/A

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

       4. neg-mul-1 N/A

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

       5. lower-neg.f64 37.5

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

   11. Applied rewrites 37.5%

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

       1. lift-/.f64 N/A

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

       2. div-inv N/A

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

       3. metadata-eval N/A

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

   13. Applied rewrites 37.5%

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

   if -215 < x < 4.79999999999999986e-20 or 1.00000000000000004e192 < x

   1. Initial program 69.2%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 38.7%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 70.2%

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

   if 4.79999999999999986e-20 < x < 1.00000000000000004e192

   1. Initial program 90.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.4%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -215:\\ \;\;\;\;\left(\frac{e^{-x}}{\varepsilon} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{1 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - -1}{2}\\ \mathbf{elif}\;x \leq 10^{+192}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -800:\\ \;\;\;\;\left(\frac{e^{-x}}{eps\_m} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 10^{+192}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -800.0)
   (* (- (/ (exp (- x)) eps_m) -1.0) 0.5)
   (if (<= x 1e+192)
     (/ (+ x 1.0) (exp x))
     (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -800.0) {
		tmp = ((exp(-x) / eps_m) - -1.0) * 0.5;
	} else if (x <= 1e+192) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -800.0)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps_m) - -1.0) * 0.5);
	elseif (x <= 1e+192)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -800.0], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1e+192], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -800

   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

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. neg-mul-1 N/A

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

       3. lower-exp.f64 N/A

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

       4. neg-mul-1 N/A

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

       5. lower-neg.f64 37.5

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

   11. Applied rewrites 37.5%

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

       1. lift-/.f64 N/A

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

       2. div-inv N/A

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

       3. metadata-eval N/A

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

   13. Applied rewrites 37.5%

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

   if -800 < x < 1.00000000000000004e192

   1. Initial program 67.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.9%

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

   if 1.00000000000000004e192 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.1%

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

Alternative 9: 67.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{elif}\;eps\_m \leq 1.8 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \mathsf{fma}\left(-1 - eps\_m, x, 1\right) \cdot \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5.6e-20)
   (/ (+ x 1.0) (exp x))
   (if (<= eps_m 1.8e+186)
     (fma (* 0.3333333333333333 x) (* x x) 1.0)
     (/ (- 1.0 (* (fma (- -1.0 eps_m) x 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 <= 5.6e-20) {
		tmp = (x + 1.0) / exp(x);
	} else if (eps_m <= 1.8e+186) {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	} else {
		tmp = (1.0 - (fma((-1.0 - eps_m), x, 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 <= 5.6e-20)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	elseif (eps_m <= 1.8e+186)
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(1.0 - Float64(fma(Float64(-1.0 - eps_m), x, 1.0) * Float64(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, 5.6e-20], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 1.8e+186], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 5.6000000000000005e-20

   1. Initial program 70.7%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.3%

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

   if 5.6000000000000005e-20 < eps < 1.8000000000000001e186

   1. Initial program 95.2%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 36.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 65.9%

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

   if 1.8000000000000001e186 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. neg-mul-1 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 57.2

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 45.2

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

   8. Applied rewrites 45.2%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 45.2%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \mathsf{fma}\left(-1 - \varepsilon, x, 1\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5.6 \cdot 10^{-20}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;eps\_m \leq 1.8 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \mathsf{fma}\left(-1 - eps\_m, x, 1\right) \cdot \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5.6e-20)
   (* (exp (- x)) (+ x 1.0))
   (if (<= eps_m 1.8e+186)
     (fma (* 0.3333333333333333 x) (* x x) 1.0)
     (/ (- 1.0 (* (fma (- -1.0 eps_m) x 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 <= 5.6e-20) {
		tmp = exp(-x) * (x + 1.0);
	} else if (eps_m <= 1.8e+186) {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	} else {
		tmp = (1.0 - (fma((-1.0 - eps_m), x, 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 <= 5.6e-20)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	elseif (eps_m <= 1.8e+186)
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(1.0 - Float64(fma(Float64(-1.0 - eps_m), x, 1.0) * Float64(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, 5.6e-20], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 1.8e+186], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 5.6000000000000005e-20

   1. Initial program 70.7%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.3%

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

   if 5.6000000000000005e-20 < eps < 1.8000000000000001e186

   1. Initial program 95.2%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 36.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 65.9%

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

   if 1.8000000000000001e186 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. neg-mul-1 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 57.2

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 45.2

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

   8. Applied rewrites 45.2%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 45.2%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.6 \cdot 10^{-20}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \mathsf{fma}\left(-1 - \varepsilon, x, 1\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1}{eps\_m} - 1\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 - \mathsf{fma}\left(-1 - eps\_m, x, 1\right) \cdot t\_0}{2}\\ \mathbf{elif}\;x \leq 1.82:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+192}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} + 1\right) - t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps_m) 1.0)))
   (if (<= x -6.2e-5)
     (/ (- 1.0 (* (fma (- -1.0 eps_m) x 1.0) t_0)) 2.0)
     (if (<= x 1.82)
       (fma (fma (fma -0.125 x 0.3333333333333333) x -0.5) (* x x) 1.0)
       (if (<= x 1.05e+192)
         (/ (- (+ (/ 1.0 eps_m) 1.0) t_0) 2.0)
         (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 / eps_m) - 1.0;
	double tmp;
	if (x <= -6.2e-5) {
		tmp = (1.0 - (fma((-1.0 - eps_m), x, 1.0) * t_0)) / 2.0;
	} else if (x <= 1.82) {
		tmp = fma(fma(fma(-0.125, x, 0.3333333333333333), x, -0.5), (x * x), 1.0);
	} else if (x <= 1.05e+192) {
		tmp = (((1.0 / eps_m) + 1.0) - t_0) / 2.0;
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 / eps_m) - 1.0)
	tmp = 0.0
	if (x <= -6.2e-5)
		tmp = Float64(Float64(1.0 - Float64(fma(Float64(-1.0 - eps_m), x, 1.0) * t_0)) / 2.0);
	elseif (x <= 1.82)
		tmp = fma(fma(fma(-0.125, x, 0.3333333333333333), x, -0.5), Float64(x * x), 1.0);
	elseif (x <= 1.05e+192)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) + 1.0) - t_0) / 2.0);
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -6.2e-5], N[(N[(1.0 - N[(N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.82], N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 1.05e+192], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.20000000000000027e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. neg-mul-1 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 29.1

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

   8. Applied rewrites 29.1%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 29.1%

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

   if -6.20000000000000027e-5 < x < 1.82000000000000006

   1. Initial program 58.2%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   if 1.82000000000000006 < x < 1.04999999999999997e192

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

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

      1. neg-mul-1 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 10.3

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 5.1

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

   8. Applied rewrites 5.1%

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

   9. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-/.f64 61.8

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

   11. Applied rewrites 61.8%

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

   if 1.04999999999999997e192 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.1%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 - \mathsf{fma}\left(-1 - \varepsilon, x, 1\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.82:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+192}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1}{eps\_m} - 1\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 - \mathsf{fma}\left(-1 - eps\_m, x, 1\right) \cdot t\_0}{2}\\ \mathbf{elif}\;x \leq 1.82:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{1}{eps\_m} - t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps_m) 1.0)))
   (if (<= x -6.2e-5)
     (/ (- 1.0 (* (fma (- -1.0 eps_m) x 1.0) t_0)) 2.0)
     (if (<= x 1.82)
       (fma (fma (fma -0.125 x 0.3333333333333333) x -0.5) (* x x) 1.0)
       (if (<= x 1.05e+192)
         (/ (- (/ 1.0 eps_m) t_0) 2.0)
         (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 / eps_m) - 1.0;
	double tmp;
	if (x <= -6.2e-5) {
		tmp = (1.0 - (fma((-1.0 - eps_m), x, 1.0) * t_0)) / 2.0;
	} else if (x <= 1.82) {
		tmp = fma(fma(fma(-0.125, x, 0.3333333333333333), x, -0.5), (x * x), 1.0);
	} else if (x <= 1.05e+192) {
		tmp = ((1.0 / eps_m) - t_0) / 2.0;
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 / eps_m) - 1.0)
	tmp = 0.0
	if (x <= -6.2e-5)
		tmp = Float64(Float64(1.0 - Float64(fma(Float64(-1.0 - eps_m), x, 1.0) * t_0)) / 2.0);
	elseif (x <= 1.82)
		tmp = fma(fma(fma(-0.125, x, 0.3333333333333333), x, -0.5), Float64(x * x), 1.0);
	elseif (x <= 1.05e+192)
		tmp = Float64(Float64(Float64(1.0 / eps_m) - t_0) / 2.0);
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -6.2e-5], N[(N[(1.0 - N[(N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.82], N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 1.05e+192], N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.20000000000000027e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. neg-mul-1 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 29.1

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

   8. Applied rewrites 29.1%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 29.1%

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

   if -6.20000000000000027e-5 < x < 1.82000000000000006

   1. Initial program 58.2%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   if 1.82000000000000006 < x < 1.04999999999999997e192

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

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

      1. neg-mul-1 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 10.3

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 5.1

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

   8. Applied rewrites 5.1%

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

   9. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-/.f64 61.8

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

   11. Applied rewrites 61.8%

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

   12. Taylor expanded in eps around 0

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

   14. Applied rewrites 61.8%

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

   if 1.04999999999999997e192 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.1%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 - \mathsf{fma}\left(-1 - \varepsilon, x, 1\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.82:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2.25 \cdot 10^{-42}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2.25e-42)
   (/ 1.0 (fma (* x x) 0.5 1.0))
   (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.25e-42) {
		tmp = 1.0 / fma((x * x), 0.5, 1.0);
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2.25e-42)
		tmp = Float64(1.0 / fma(Float64(x * x), 0.5, 1.0));
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 2.25e-42], N[(1.0 / N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.25e-42

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.7%

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

   10. Applied rewrites 51.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 52.0%

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

   if 2.25e-42 < eps

   1. Initial program 92.7%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 28.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.3%

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

Alternative 14: 53.5% accurate, 15.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right) \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0)
end

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 50.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 46.4%

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

Alternative 15: 53.4% accurate, 16.1× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (fma (* 0.3333333333333333 x) (* x x) 1.0))

C

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

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 50.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 46.4%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 46.3%

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

Alternative 16: 44.7% 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 78.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 36.0%

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

Reproduce

herbie shell --seed 2024254 
(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))