NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.9% → 99.1%

Time: 14.7s

Alternatives: 12

Speedup: 5.6×

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

• could not determine a ground truth

  1. x = -1.699057133428455e+54
  2. eps = -5.705967244562707e-226

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower-+.f64 N/A

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.0%

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(e^{x \cdot \frac{-1 - \varepsilon}{2}}, \color{blue}{e^{x \cdot \frac{-1 - \varepsilon}{2}}}, e^{\varepsilon \cdot x - x}\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.0%

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

10. Taylor expanded in eps around inf

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

12. Applied rewrites 94.3%

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

13. Final simplification 94.3%

    \[\leadsto 0.5 \cdot \mathsf{fma}\left(e^{x \cdot \left(0.5 \cdot \left(-1 - \varepsilon\right)\right)}, e^{x \cdot \left(\varepsilon \cdot -0.5\right)}, e^{\varepsilon \cdot x - x}\right) \]
14. Add Preprocessing

Alternative 2: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 2.0000004:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \left(e^{-x} \cdot \left(1 + x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(1 + eps\_m, 1 + eps\_m, \frac{\mathsf{fma}\left(eps\_m, eps\_m, -1\right) \cdot \left(eps\_m + -1\right)}{1 + eps\_m}\right), \mathsf{fma}\left(eps\_m + \left(-1 - eps\_m\right), 0.5, -0.5\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0000004)
   (* 0.5 (* 2.0 (* (exp (- x)) (+ 1.0 x))))
   (fma
    x
    (fma
     x
     (*
      0.25
      (fma
       (+ 1.0 eps_m)
       (+ 1.0 eps_m)
       (/ (* (fma eps_m eps_m -1.0) (+ eps_m -1.0)) (+ 1.0 eps_m))))
     (fma (+ eps_m (- -1.0 eps_m)) 0.5 -0.5))
    1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 2.0000004) {
		tmp = 0.5 * (2.0 * (exp(-x) * (1.0 + x)));
	} else {
		tmp = fma(x, fma(x, (0.25 * fma((1.0 + eps_m), (1.0 + eps_m), ((fma(eps_m, eps_m, -1.0) * (eps_m + -1.0)) / (1.0 + eps_m)))), fma((eps_m + (-1.0 - eps_m)), 0.5, -0.5)), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) <= 2.0000004)
		tmp = Float64(0.5 * Float64(2.0 * Float64(exp(Float64(-x)) * Float64(1.0 + x))));
	else
		tmp = fma(x, fma(x, Float64(0.25 * fma(Float64(1.0 + eps_m), Float64(1.0 + eps_m), Float64(Float64(fma(eps_m, eps_m, -1.0) * Float64(eps_m + -1.0)) / Float64(1.0 + eps_m)))), fma(Float64(eps_m + Float64(-1.0 - eps_m)), 0.5, -0.5)), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0000004], N[(0.5 * N[(2.0 * N[(N[Exp[(-x)], $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.25 * N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision] + N[(N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps$95$m + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 2.00000039999999979

   1. Initial program 63.6%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-out N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 2.00000039999999979 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 86.6%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(\varepsilon + 1, \varepsilon + 1, \left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right), \mathsf{fma}\left(\varepsilon + \left(-1 - \varepsilon\right), 0.5, -0.5\right)\right)}, 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 92.4%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(\varepsilon + 1, \varepsilon + 1, \frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot \left(-1 + \varepsilon\right)}{\varepsilon + 1}\right), \mathsf{fma}\left(\varepsilon + \left(-1 - \varepsilon\right), 0.5, -0.5\right)\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.0000004:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \left(e^{-x} \cdot \left(1 + x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(1 + \varepsilon, 1 + \varepsilon, \frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot \left(\varepsilon + -1\right)}{1 + \varepsilon}\right), \mathsf{fma}\left(\varepsilon + \left(-1 - \varepsilon\right), 0.5, -0.5\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 2.0000004:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(1 + eps\_m, 1 + eps\_m, \frac{\mathsf{fma}\left(eps\_m, eps\_m, -1\right) \cdot \left(eps\_m + -1\right)}{1 + eps\_m}\right), \mathsf{fma}\left(eps\_m + \left(-1 - eps\_m\right), 0.5, -0.5\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0000004)
   (exp (- x))
   (fma
    x
    (fma
     x
     (*
      0.25
      (fma
       (+ 1.0 eps_m)
       (+ 1.0 eps_m)
       (/ (* (fma eps_m eps_m -1.0) (+ eps_m -1.0)) (+ 1.0 eps_m))))
     (fma (+ eps_m (- -1.0 eps_m)) 0.5 -0.5))
    1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 2.0000004) {
		tmp = exp(-x);
	} else {
		tmp = fma(x, fma(x, (0.25 * fma((1.0 + eps_m), (1.0 + eps_m), ((fma(eps_m, eps_m, -1.0) * (eps_m + -1.0)) / (1.0 + eps_m)))), fma((eps_m + (-1.0 - eps_m)), 0.5, -0.5)), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) <= 2.0000004)
		tmp = exp(Float64(-x));
	else
		tmp = fma(x, fma(x, Float64(0.25 * fma(Float64(1.0 + eps_m), Float64(1.0 + eps_m), Float64(Float64(fma(eps_m, eps_m, -1.0) * Float64(eps_m + -1.0)) / Float64(1.0 + eps_m)))), fma(Float64(eps_m + Float64(-1.0 - eps_m)), 0.5, -0.5)), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0000004], N[Exp[(-x)], $MachinePrecision], N[(x * N[(x * N[(0.25 * N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision] + N[(N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps$95$m + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 2.00000039999999979

   1. Initial program 63.6%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 98.6%

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

   if 2.00000039999999979 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 86.6%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(\varepsilon + 1, \varepsilon + 1, \left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right), \mathsf{fma}\left(\varepsilon + \left(-1 - \varepsilon\right), 0.5, -0.5\right)\right)}, 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 92.4%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(\varepsilon + 1, \varepsilon + 1, \frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot \left(-1 + \varepsilon\right)}{\varepsilon + 1}\right), \mathsf{fma}\left(\varepsilon + \left(-1 - \varepsilon\right), 0.5, -0.5\right)\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.0000004:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(1 + \varepsilon, 1 + \varepsilon, \frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot \left(\varepsilon + -1\right)}{1 + \varepsilon}\right), \mathsf{fma}\left(\varepsilon + \left(-1 - \varepsilon\right), 0.5, -0.5\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 2.0000004:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{eps\_m \cdot \left(\mathsf{fma}\left(eps\_m, eps\_m, -1\right) \cdot \left(x \cdot x\right)\right)}{eps\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0000004)
   1.0
   (fma 0.5 (/ (* eps_m (* (fma eps_m eps_m -1.0) (* x x))) eps_m) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 2.0000004) {
		tmp = 1.0;
	} else {
		tmp = fma(0.5, ((eps_m * (fma(eps_m, eps_m, -1.0) * (x * x))) / eps_m), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) <= 2.0000004)
		tmp = 1.0;
	else
		tmp = fma(0.5, Float64(Float64(eps_m * Float64(fma(eps_m, eps_m, -1.0) * Float64(x * x))) / eps_m), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0000004], 1.0, N[(0.5 * N[(N[(eps$95$m * N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 2.00000039999999979

   1. Initial program 63.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.8%

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

   if 2.00000039999999979 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 84.9%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 87.1%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.0000004:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot \left(x \cdot x\right)\right)}{\varepsilon}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 2.0000004:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0000004)
   1.0
   (fma 0.5 (* x (* x (* eps_m eps_m))) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) <= 2.0000004) {
		tmp = 1.0;
	} else {
		tmp = fma(0.5, (x * (x * (eps_m * eps_m))), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) <= 2.0000004)
		tmp = 1.0;
	else
		tmp = fma(0.5, Float64(x * Float64(x * Float64(eps_m * eps_m))), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0000004], 1.0, N[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 2.00000039999999979

   1. Initial program 63.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.8%

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

   if 2.00000039999999979 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 86.6%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.0000004:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower-+.f64 N/A

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

5. Applied rewrites 99.0%

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

Alternative 7: 80.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(eps\_m, \mathsf{fma}\left(x, -0.5 \cdot \left(x + x\right), eps\_m \cdot \mathsf{fma}\left(eps\_m \cdot x, x, x \cdot \left(x - x\right)\right)\right), x - x \cdot \left(1 + \left(x - x\right)\right)\right)}{eps\_m}, 1\right)\\ \mathbf{elif}\;x \leq 4200:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(1 + eps\_m, 1 + eps\_m, \frac{\mathsf{fma}\left(eps\_m, eps\_m, -1\right) \cdot \left(eps\_m + -1\right)}{1 + eps\_m}\right), \mathsf{fma}\left(eps\_m + \left(-1 - eps\_m\right), 0.5, -0.5\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(-1 + \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.85e-147)
   (fma
    0.5
    (/
     (fma
      eps_m
      (fma x (* -0.5 (+ x x)) (* eps_m (fma (* eps_m x) x (* x (- x x)))))
      (- x (* x (+ 1.0 (- x x)))))
     eps_m)
    1.0)
   (if (<= x 4200.0)
     (fma
      x
      (fma
       x
       (*
        0.25
        (fma
         (+ 1.0 eps_m)
         (+ 1.0 eps_m)
         (/ (* (fma eps_m eps_m -1.0) (+ eps_m -1.0)) (+ 1.0 eps_m))))
       (fma (+ eps_m (- -1.0 eps_m)) 0.5 -0.5))
      1.0)
     (/ (- (+ 1.0 (/ 1.0 eps_m)) (+ -1.0 (/ 1.0 eps_m))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.85e-147) {
		tmp = fma(0.5, (fma(eps_m, fma(x, (-0.5 * (x + x)), (eps_m * fma((eps_m * x), x, (x * (x - x))))), (x - (x * (1.0 + (x - x))))) / eps_m), 1.0);
	} else if (x <= 4200.0) {
		tmp = fma(x, fma(x, (0.25 * fma((1.0 + eps_m), (1.0 + eps_m), ((fma(eps_m, eps_m, -1.0) * (eps_m + -1.0)) / (1.0 + eps_m)))), fma((eps_m + (-1.0 - eps_m)), 0.5, -0.5)), 1.0);
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) - (-1.0 + (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.85e-147)
		tmp = fma(0.5, Float64(fma(eps_m, fma(x, Float64(-0.5 * Float64(x + x)), Float64(eps_m * fma(Float64(eps_m * x), x, Float64(x * Float64(x - x))))), Float64(x - Float64(x * Float64(1.0 + Float64(x - x))))) / eps_m), 1.0);
	elseif (x <= 4200.0)
		tmp = fma(x, fma(x, Float64(0.25 * fma(Float64(1.0 + eps_m), Float64(1.0 + eps_m), Float64(Float64(fma(eps_m, eps_m, -1.0) * Float64(eps_m + -1.0)) / Float64(1.0 + eps_m)))), fma(Float64(eps_m + Float64(-1.0 - eps_m)), 0.5, -0.5)), 1.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(-1.0 + Float64(1.0 / eps_m))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.85e-147], N[(0.5 * N[(N[(eps$95$m * N[(x * N[(-0.5 * N[(x + x), $MachinePrecision]), $MachinePrecision] + N[(eps$95$m * N[(N[(eps$95$m * x), $MachinePrecision] * x + N[(x * N[(x - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - N[(x * N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 4200.0], N[(x * N[(x * N[(0.25 * N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision] + N[(N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps$95$m + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.8500000000000001e-147

   1. Initial program 70.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 91.3%

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

   10. Applied rewrites 92.9%

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

   if 1.8500000000000001e-147 < x < 4200

   1. Initial program 67.7%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 81.4%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(\varepsilon + 1, \varepsilon + 1, \left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right), \mathsf{fma}\left(\varepsilon + \left(-1 - \varepsilon\right), 0.5, -0.5\right)\right)}, 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 94.2%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(\varepsilon + 1, \varepsilon + 1, \frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot \left(-1 + \varepsilon\right)}{\varepsilon + 1}\right), \mathsf{fma}\left(\varepsilon + \left(-1 - \varepsilon\right), 0.5, -0.5\right)\right), 1\right) \]

   if 4200 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 24.1

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

   5. Applied rewrites 24.1%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 58.1

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

   8. Applied rewrites 58.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, -0.5 \cdot \left(x + x\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot x, x, x \cdot \left(x - x\right)\right)\right), x - x \cdot \left(1 + \left(x - x\right)\right)\right)}{\varepsilon}, 1\right)\\ \mathbf{elif}\;x \leq 4200:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(1 + \varepsilon, 1 + \varepsilon, \frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot \left(\varepsilon + -1\right)}{1 + \varepsilon}\right), \mathsf{fma}\left(\varepsilon + \left(-1 - \varepsilon\right), 0.5, -0.5\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{1 + \mathsf{fma}\left(eps\_m, \mathsf{fma}\left(eps\_m, \mathsf{fma}\left(x, eps\_m, 0\right), -x\right), -1\right)}{eps\_m}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(-1 + \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 9.5e+14)
   (fma
    0.5
    (*
     x
     (/ (+ 1.0 (fma eps_m (fma eps_m (fma x eps_m 0.0) (- x)) -1.0)) eps_m))
    1.0)
   (/ (- (+ 1.0 (/ 1.0 eps_m)) (+ -1.0 (/ 1.0 eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 9.5e+14) {
		tmp = fma(0.5, (x * ((1.0 + fma(eps_m, fma(eps_m, fma(x, eps_m, 0.0), -x), -1.0)) / eps_m)), 1.0);
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) - (-1.0 + (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 9.5e+14)
		tmp = fma(0.5, Float64(x * Float64(Float64(1.0 + fma(eps_m, fma(eps_m, fma(x, eps_m, 0.0), Float64(-x)), -1.0)) / eps_m)), 1.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(-1.0 + Float64(1.0 / eps_m))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 9.5e+14], N[(0.5 * N[(x * N[(N[(1.0 + N[(eps$95$m * N[(eps$95$m * N[(x * eps$95$m + 0.0), $MachinePrecision] + (-x)), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5e14

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 72.3%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 89.4%

       \[\leadsto \mathsf{fma}\left(0.5, x \cdot \frac{1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \varepsilon, 0\right), -x\right), -1\right)}{\color{blue}{\varepsilon}}, 1\right) \]

   if 9.5e14 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 58.3

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

   8. Applied rewrites 58.3%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \varepsilon, 0\right), -x\right), -1\right)}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 5.3 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(-1 + \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 5.3e+14)
   (fma 0.5 (* x (* x (* eps_m eps_m))) 1.0)
   (/ (- (+ 1.0 (/ 1.0 eps_m)) (+ -1.0 (/ 1.0 eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 5.3e+14) {
		tmp = fma(0.5, (x * (x * (eps_m * eps_m))), 1.0);
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) - (-1.0 + (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 5.3e+14)
		tmp = fma(0.5, Float64(x * Float64(x * Float64(eps_m * eps_m))), 1.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(-1.0 + Float64(1.0 / eps_m))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 5.3e+14], N[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.3e14

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 86.2%

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

   if 5.3e14 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 58.3

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

   8. Applied rewrites 58.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.3 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 71.8%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 37.0%

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

9. Taylor expanded in eps around inf

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

11. Applied rewrites 69.2%

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

Alternative 11: 57.1% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower-+.f64 N/A

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 72.0%

    \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.25 \cdot \mathsf{fma}\left(\varepsilon + 1, \varepsilon + 1, \left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right), \mathsf{fma}\left(\varepsilon + \left(-1 - \varepsilon\right), 0.5, -0.5\right)\right)}, 1\right) \]

11. Taylor expanded in eps around 0

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

13. Applied rewrites 55.2%

    \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]
14. Add Preprocessing

Alternative 12: 43.6% 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 79.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 38.1%

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

Reproduce

herbie shell --seed 2024219 
(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))