NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 98.7%

Time: 10.0s

Alternatives: 10

Speedup: 1.9×

• 38.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -7.461636356150192e+258
  2. eps = 4.7094385344645484e-170

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: 73.6% 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: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 7500000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + x, e^{-x}, \frac{1 + x}{e^{x}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {eps\_m}^{-1}\right) \cdot e^{x \cdot eps\_m} - \left(-e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 7500000000000.0)
   (/ (fma (+ 1.0 x) (exp (- x)) (/ (+ 1.0 x) (exp x))) 2.0)
   (/
    (-
     (* (+ 1.0 (pow eps_m -1.0)) (exp (* x eps_m)))
     (- (exp (- (fma x eps_m x)))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7500000000000.0) {
		tmp = fma((1.0 + x), exp(-x), ((1.0 + x) / exp(x))) / 2.0;
	} else {
		tmp = (((1.0 + pow(eps_m, -1.0)) * exp((x * eps_m))) - -exp(-fma(x, eps_m, x))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 7500000000000.0)
		tmp = Float64(fma(Float64(1.0 + x), exp(Float64(-x)), Float64(Float64(1.0 + x) / exp(x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + (eps_m ^ -1.0)) * exp(Float64(x * eps_m))) - Float64(-exp(Float64(-fma(x, eps_m, x))))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 7500000000000.0], N[(N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[Power[eps$95$m, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - (-N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 7.5e12

   1. Initial program 60.2%

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

   3. Taylor expanded in eps around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. exp-neg N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-exp.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if 7.5e12 < 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-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 7500000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + x, e^{-x}, \frac{1 + x}{e^{x}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{x \cdot \varepsilon} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 82000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + x, e^{-x}, \frac{1 + x}{e^{x}}\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 5.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{eps\_m - -1}{eps\_m} - \left(-e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {eps\_m}^{-1}\right) \cdot e^{x \cdot eps\_m} - -1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 82000000000000.0)
   (/ (fma (+ 1.0 x) (exp (- x)) (/ (+ 1.0 x) (exp x))) 2.0)
   (if (<= eps_m 5.6e+125)
     (/ (- (/ (- eps_m -1.0) eps_m) (- (exp (- (fma x eps_m x))))) 2.0)
     (/ (- (* (+ 1.0 (pow eps_m -1.0)) (exp (* x eps_m))) -1.0) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 82000000000000.0) {
		tmp = fma((1.0 + x), exp(-x), ((1.0 + x) / exp(x))) / 2.0;
	} else if (eps_m <= 5.6e+125) {
		tmp = (((eps_m - -1.0) / eps_m) - -exp(-fma(x, eps_m, x))) / 2.0;
	} else {
		tmp = (((1.0 + pow(eps_m, -1.0)) * exp((x * eps_m))) - -1.0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 82000000000000.0)
		tmp = Float64(fma(Float64(1.0 + x), exp(Float64(-x)), Float64(Float64(1.0 + x) / exp(x))) / 2.0);
	elseif (eps_m <= 5.6e+125)
		tmp = Float64(Float64(Float64(Float64(eps_m - -1.0) / eps_m) - Float64(-exp(Float64(-fma(x, eps_m, x))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + (eps_m ^ -1.0)) * exp(Float64(x * 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, 82000000000000.0], N[(N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 5.6e+125], N[(N[(N[(N[(eps$95$m - -1.0), $MachinePrecision] / eps$95$m), $MachinePrecision] - (-N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[Power[eps$95$m, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 8.2e13

   1. Initial program 60.2%

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

   3. Taylor expanded in eps around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. exp-neg N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-exp.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if 8.2e13 < eps < 5.6000000000000002e125

   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-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-add N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 74.2

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

   10. Applied rewrites 74.2%

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

   if 5.6000000000000002e125 < 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-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 58.8%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 58.8

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

   11. Applied rewrites 58.8%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 82000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + x, e^{-x}, \frac{1 + x}{e^{x}}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 5.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{\varepsilon - -1}{\varepsilon} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{x \cdot \varepsilon} - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 82000000000000:\\ \;\;\;\;\frac{e^{-x} \cdot \left(\left(x + 1\right) + \left(x + 1\right)\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 5.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{eps\_m - -1}{eps\_m} - \left(-e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {eps\_m}^{-1}\right) \cdot e^{x \cdot eps\_m} - -1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 82000000000000.0)
   (/ (* (exp (- x)) (+ (+ x 1.0) (+ x 1.0))) 2.0)
   (if (<= eps_m 5.6e+125)
     (/ (- (/ (- eps_m -1.0) eps_m) (- (exp (- (fma x eps_m x))))) 2.0)
     (/ (- (* (+ 1.0 (pow eps_m -1.0)) (exp (* x eps_m))) -1.0) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 82000000000000.0) {
		tmp = (exp(-x) * ((x + 1.0) + (x + 1.0))) / 2.0;
	} else if (eps_m <= 5.6e+125) {
		tmp = (((eps_m - -1.0) / eps_m) - -exp(-fma(x, eps_m, x))) / 2.0;
	} else {
		tmp = (((1.0 + pow(eps_m, -1.0)) * exp((x * eps_m))) - -1.0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 82000000000000.0)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(Float64(x + 1.0) + Float64(x + 1.0))) / 2.0);
	elseif (eps_m <= 5.6e+125)
		tmp = Float64(Float64(Float64(Float64(eps_m - -1.0) / eps_m) - Float64(-exp(Float64(-fma(x, eps_m, x))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + (eps_m ^ -1.0)) * exp(Float64(x * 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, 82000000000000.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 5.6e+125], N[(N[(N[(N[(eps$95$m - -1.0), $MachinePrecision] / eps$95$m), $MachinePrecision] - (-N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[Power[eps$95$m, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 8.2e13

   1. Initial program 60.2%

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

   3. Taylor expanded in eps around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. exp-neg N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-exp.f64 71.9

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

   5. Applied rewrites 71.9%

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

   7. Applied rewrites 71.9%

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

   if 8.2e13 < eps < 5.6000000000000002e125

   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-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-add N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 74.2

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

   10. Applied rewrites 74.2%

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

   if 5.6000000000000002e125 < 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-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 58.8%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 58.8

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

   11. Applied rewrites 58.8%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 82000000000000:\\ \;\;\;\;\frac{e^{-x} \cdot \left(\left(x + 1\right) + \left(x + 1\right)\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 5.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{\varepsilon - -1}{\varepsilon} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{x \cdot \varepsilon} - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.48 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - \left|x\right| \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left({eps\_m}^{-1} + 1\right) - \left({eps\_m}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -1\right), x \cdot x, 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.48e-5)
   (/ (- 2.0 (* (fabs x) x)) 2.0)
   (if (<= x 2.3e+153)
     (/ (- (+ (pow eps_m -1.0) 1.0) (- (pow eps_m -1.0) 1.0)) 2.0)
     (/ (fma (fma 0.6666666666666666 x -1.0) (* x x) 2.0) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.48e-5) {
		tmp = (2.0 - (fabs(x) * x)) / 2.0;
	} else if (x <= 2.3e+153) {
		tmp = ((pow(eps_m, -1.0) + 1.0) - (pow(eps_m, -1.0) - 1.0)) / 2.0;
	} else {
		tmp = fma(fma(0.6666666666666666, x, -1.0), (x * x), 2.0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.48e-5)
		tmp = Float64(Float64(2.0 - Float64(abs(x) * x)) / 2.0);
	elseif (x <= 2.3e+153)
		tmp = Float64(Float64(Float64((eps_m ^ -1.0) + 1.0) - Float64((eps_m ^ -1.0) - 1.0)) / 2.0);
	else
		tmp = Float64(fma(fma(0.6666666666666666, x, -1.0), Float64(x * x), 2.0) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.48e-5], N[(N[(2.0 - N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.3e+153], N[(N[(N[(N[Power[eps$95$m, -1.0], $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[Power[eps$95$m, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.6666666666666666 * x + -1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.4800000000000001e-5

   1. Initial program 60.8%

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

   3. Taylor expanded in eps around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. exp-neg N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-exp.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 61.4%

      \[\leadsto \frac{2 - \color{blue}{x \cdot x}}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.9%

       \[\leadsto \frac{2 - \left|\left(-x\right) \cdot -1\right| \cdot x}{2} \]

   if 1.4800000000000001e-5 < x < 2.3000000000000001e153

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 30.8

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 51.6

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

   8. Applied rewrites 51.6%

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

   if 2.3000000000000001e153 < 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

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

      1. sub-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. exp-neg N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-exp.f64 40.9

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

   5. Applied rewrites 40.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -1\right), \color{blue}{x \cdot x}, 2\right)}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.48 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - \left|x\right| \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -1\right), x \cdot x, 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{2 - \left|x\right| \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{{eps\_m}^{-1} - \left({eps\_m}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -1\right), x \cdot x, 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.4)
   (/ (- 2.0 (* (fabs x) x)) 2.0)
   (if (<= x 2.3e+153)
     (/ (- (pow eps_m -1.0) (- (pow eps_m -1.0) 1.0)) 2.0)
     (/ (fma (fma 0.6666666666666666 x -1.0) (* x x) 2.0) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.4) {
		tmp = (2.0 - (fabs(x) * x)) / 2.0;
	} else if (x <= 2.3e+153) {
		tmp = (pow(eps_m, -1.0) - (pow(eps_m, -1.0) - 1.0)) / 2.0;
	} else {
		tmp = fma(fma(0.6666666666666666, x, -1.0), (x * x), 2.0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64(2.0 - Float64(abs(x) * x)) / 2.0);
	elseif (x <= 2.3e+153)
		tmp = Float64(Float64((eps_m ^ -1.0) - Float64((eps_m ^ -1.0) - 1.0)) / 2.0);
	else
		tmp = Float64(fma(fma(0.6666666666666666, x, -1.0), Float64(x * x), 2.0) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.4], N[(N[(2.0 - N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.3e+153], N[(N[(N[Power[eps$95$m, -1.0], $MachinePrecision] - N[(N[Power[eps$95$m, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.6666666666666666 * x + -1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.3999999999999999

   1. Initial program 61.0%

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

   3. Taylor expanded in eps around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. exp-neg N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-exp.f64 61.6

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

   5. Applied rewrites 61.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 61.1%

      \[\leadsto \frac{2 - \color{blue}{x \cdot x}}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.5%

       \[\leadsto \frac{2 - \left|\left(-x\right) \cdot -1\right| \cdot x}{2} \]

   if 1.3999999999999999 < x < 2.3000000000000001e153

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 31.6

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

   5. Applied rewrites 31.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 52.9

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

   8. Applied rewrites 52.9%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 52.9%

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

   if 2.3000000000000001e153 < 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

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

      1. sub-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. exp-neg N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-exp.f64 40.9

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

   5. Applied rewrites 40.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -1\right), \color{blue}{x \cdot x}, 2\right)}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{2 - \left|x\right| \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -1\right), x \cdot x, 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 82000000000000:\\ \;\;\;\;\frac{e^{-x} \cdot \left(\left(x + 1\right) + \left(x + 1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{eps\_m - -1}{eps\_m} - \left(-e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 82000000000000.0)
   (/ (* (exp (- x)) (+ (+ x 1.0) (+ x 1.0))) 2.0)
   (/ (- (/ (- eps_m -1.0) eps_m) (- (exp (- (fma x eps_m x))))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 82000000000000.0) {
		tmp = (exp(-x) * ((x + 1.0) + (x + 1.0))) / 2.0;
	} else {
		tmp = (((eps_m - -1.0) / eps_m) - -exp(-fma(x, eps_m, x))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 82000000000000.0)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(Float64(x + 1.0) + Float64(x + 1.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(eps_m - -1.0) / eps_m) - Float64(-exp(Float64(-fma(x, eps_m, x))))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 82000000000000.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(eps$95$m - -1.0), $MachinePrecision] / eps$95$m), $MachinePrecision] - (-N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.2e13

   1. Initial program 60.2%

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

   3. Taylor expanded in eps around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. exp-neg N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-exp.f64 71.9

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

   5. Applied rewrites 71.9%

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

   7. Applied rewrites 71.9%

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

   if 8.2e13 < 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-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-add N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 61.3

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

   10. Applied rewrites 61.3%

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

Alternative 7: 71.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -1\right), x \cdot x, 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 4e+153)
   (/ (* (exp (- x)) 2.0) 2.0)
   (/ (fma (fma 0.6666666666666666 x -1.0) (* x x) 2.0) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 4e+153) {
		tmp = (exp(-x) * 2.0) / 2.0;
	} else {
		tmp = fma(fma(0.6666666666666666, x, -1.0), (x * x), 2.0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 4e+153)
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) / 2.0);
	else
		tmp = Float64(fma(fma(0.6666666666666666, x, -1.0), Float64(x * x), 2.0) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 4e+153], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.6666666666666666 * x + -1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e153

   1. Initial program 67.4%

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

   3. Taylor expanded in eps around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. exp-neg N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-exp.f64 60.5

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

   5. Applied rewrites 60.5%

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

   7. Applied rewrites 60.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{e^{-x} \cdot 2}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.7%

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

   if 4e153 < 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

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

      1. sub-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. exp-neg N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-exp.f64 40.9

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

   5. Applied rewrites 40.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -1\right), \color{blue}{x \cdot x}, 2\right)}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3500:\\ \;\;\;\;\frac{2 - \left|x\right| \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -1\right), x \cdot x, 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -3500.0)
   (/ (- 2.0 (* (fabs x) x)) 2.0)
   (/ (fma (fma 0.6666666666666666 x -1.0) (* x x) 2.0) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -3500.0) {
		tmp = (2.0 - (fabs(x) * x)) / 2.0;
	} else {
		tmp = fma(fma(0.6666666666666666, x, -1.0), (x * x), 2.0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -3500.0)
		tmp = Float64(Float64(2.0 - Float64(abs(x) * x)) / 2.0);
	else
		tmp = Float64(fma(fma(0.6666666666666666, x, -1.0), Float64(x * x), 2.0) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -3500.0], N[(N[(2.0 - N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.6666666666666666 * x + -1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3500

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. exp-neg N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-exp.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.1%

      \[\leadsto \frac{2 - \color{blue}{x \cdot x}}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.5%

       \[\leadsto \frac{2 - \left|\left(-x\right) \cdot -1\right| \cdot x}{2} \]

   if -3500 < x

   1. Initial program 67.1%

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

   3. Taylor expanded in eps around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. exp-neg N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-exp.f64 66.5

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 61.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -1\right), \color{blue}{x \cdot x}, 2\right)}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3500:\\ \;\;\;\;\frac{2 - \left|x\right| \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -1\right), x \cdot x, 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% accurate, 12.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{2 - \left|x\right| \cdot x}{2} \end{array} \]

FPCore

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

C

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

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = (2.0d0 - (abs(x) * x)) / 2.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return (2.0 - (Math.abs(x) * x)) / 2.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (2.0 - (math.fabs(x) * x)) / 2.0

Julia

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

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (2.0 - (abs(x) * x)) / 2.0;
end

Wolfram

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

Derivation

1. Initial program 71.2%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. distribute-rgt1-in N/A

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

   3. distribute-lft-out N/A

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. distribute-rgt1-in N/A

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

   12. exp-neg N/A

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

   13. associate-*r/ N/A

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

   14. *-rgt-identity N/A

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

   15. lower-/.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. lower-exp.f64 58.2

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

5. Applied rewrites 58.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 45.4%

   \[\leadsto \frac{2 - \color{blue}{x \cdot x}}{2} \]
9. Step-by-step derivation

10. Applied rewrites 53.8%

    \[\leadsto \frac{2 - \left|\left(-x\right) \cdot -1\right| \cdot x}{2} \]

11. Final simplification 53.8%

    \[\leadsto \frac{2 - \left|x\right| \cdot x}{2} \]
12. Add Preprocessing

Alternative 10: 43.9% 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 71.2%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
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-lft-in N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 69.3

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

5. Applied rewrites 69.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 46.3%

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

Reproduce

herbie shell --seed 2024305 
(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))