NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.0% → 99.9%

Time: 15.8s

Alternatives: 16

Speedup: 9.7×

• 38.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -6.852774734665131e+296
  2. eps = 1.1573662062933847e-153

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: 74.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      0.0)
   (* 0.5 (* (exp (- x)) (fma x 2.0 2.0)))
   (* 0.5 (* 2.0 (cosh (* x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = 0.5 * (exp(-x) * fma(x, 2.0, 2.0));
	} else {
		tmp = 0.5 * (2.0 * cosh((x * eps)));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Cosh[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.7%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 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. *-lowering-*.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. +-lowering-+.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. Simplified 99.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. neg-lowering-neg.f64 100.0

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

   11. Simplified 100.0%

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

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

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

       4. cosh-undef N/A

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

       5. *-lowering-*.f64 N/A

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

       6. cosh-lowering-cosh.f64 N/A

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

       7. *-lowering-*.f64 100.0

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

   13. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      0.0)
   (* 0.5 (* (exp (- x)) (+ x (+ x 2.0))))
   (* 0.5 (* 2.0 (cosh (* x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = 0.5 * (exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = 0.5 * (2.0 * cosh((x * eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) <= 0.0d0) then
        tmp = 0.5d0 * (exp(-x) * (x + (x + 2.0d0)))
    else
        tmp = 0.5d0 * (2.0d0 * cosh((x * eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = 0.5 * (Math.exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = 0.5 * (2.0 * Math.cosh((x * eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0:
		tmp = 0.5 * (math.exp(-x) * (x + (x + 2.0)))
	else:
		tmp = 0.5 * (2.0 * math.cosh((x * eps)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Cosh[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.7%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

   5. Simplified 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 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. *-lowering-*.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. +-lowering-+.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. Simplified 99.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. neg-lowering-neg.f64 100.0

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

   11. Simplified 100.0%

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

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

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

       4. cosh-undef N/A

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

       5. *-lowering-*.f64 N/A

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

       6. cosh-lowering-cosh.f64 N/A

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

       7. *-lowering-*.f64 100.0

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

   13. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      0.0)
   (exp (- x))
   (* 0.5 (* 2.0 (cosh (* x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = exp(-x);
	} else {
		tmp = 0.5 * (2.0 * cosh((x * eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) <= 0.0d0) then
        tmp = exp(-x)
    else
        tmp = 0.5d0 * (2.0d0 * cosh((x * eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = Math.exp(-x);
	} else {
		tmp = 0.5 * (2.0 * Math.cosh((x * eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0:
		tmp = math.exp(-x)
	else:
		tmp = 0.5 * (2.0 * math.cosh((x * eps)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[Exp[(-x)], $MachinePrecision], N[(0.5 * N[(2.0 * N[Cosh[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.7%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. *-lowering-*.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. +-lowering-+.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. Simplified 96.5%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. *-lowering-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. exp-lowering-exp.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. neg-lowering-neg.f64 96.5

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

   8. Simplified 96.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.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. +-lowering-+.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. Simplified 99.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. neg-lowering-neg.f64 100.0

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

   11. Simplified 100.0%

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

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

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

       4. cosh-undef N/A

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

       5. *-lowering-*.f64 N/A

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

       6. cosh-lowering-cosh.f64 N/A

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

       7. *-lowering-*.f64 100.0

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

   13. Applied egg-rr 100.0%

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

4. Final simplification 98.7%

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

Alternative 4: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.25, 0.25\right), -0.25\right), -0.25\right)}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      4.0)
   (exp (- x))
   (* x (* x (/ (fma eps (fma eps (fma eps 0.25 0.25) -0.25) -0.25) eps)))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = exp(-x);
	} else {
		tmp = x * (x * (fma(eps, fma(eps, fma(eps, 0.25, 0.25), -0.25), -0.25) / eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 4.0)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(x * Float64(x * Float64(fma(eps, fma(eps, fma(eps, 0.25, 0.25), -0.25), -0.25) / eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[Exp[(-x)], $MachinePrecision], N[(x * N[(x * N[(N[(eps * N[(eps * N[(eps * 0.25 + 0.25), $MachinePrecision] + -0.25), $MachinePrecision] + -0.25), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 58.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. *-lowering-*.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. +-lowering-+.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. Simplified 97.3%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. *-lowering-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. exp-lowering-exp.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. neg-lowering-neg.f64 96.5

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

   8. Simplified 96.5%

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

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. distribute-rgt1-in N/A

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

      14. *-lowering-*.f64 N/A

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

   5. Simplified 42.7%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 71.8%

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

   9. Taylor expanded in eps around 0

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

       1. /-lowering-/.f64 N/A

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

   11. Simplified 86.8%

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

   12. Taylor expanded in x around -inf

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

   14. Simplified 87.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.25, 0.25\right), -0.25\right), -0.25\right)}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      4.0)
   (fma (* x x) (fma x 0.3333333333333333 -0.5) 1.0)
   (* 0.25 (* x (* x (* eps eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = fma((x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	} else {
		tmp = 0.25 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 4.0)
		tmp = fma(Float64(x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	else
		tmp = Float64(0.25 * Float64(x * Float64(x * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.25 * N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 58.1%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 72.0

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

   8. Simplified 72.0%

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

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. distribute-rgt1-in N/A

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

      14. *-lowering-*.f64 N/A

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

   5. Simplified 42.7%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 71.8%

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

   9. Taylor expanded in eps around inf

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

       1. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

          \[\leadsto \frac{1}{4} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       10. *-lowering-*.f64 81.9

           \[\leadsto 0.25 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   11. Simplified 81.9%

       \[\leadsto \color{blue}{0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(e^{x \cdot \varepsilon - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.5d0 * (exp(((x * eps) - x)) + exp((x * ((-1.0d0) - eps))))
end function

Java

public static double code(double x, double eps) {
	return 0.5 * (Math.exp(((x * eps) - x)) + Math.exp((x * (-1.0 - eps))));
}

Python

def code(x, eps):
	return 0.5 * (math.exp(((x * eps) - x)) + math.exp((x * (-1.0 - eps))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = 0.5 * (exp(((x * eps) - x)) + exp((x * (-1.0 - eps))));
end

Wolfram

code[x_, eps_] := N[(0.5 * N[(N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in eps around inf

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

   1. *-lowering-*.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. +-lowering-+.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. Simplified 98.5%

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

Alternative 7: 73.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-257}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right) \cdot \mathsf{fma}\left(x \cdot -0.08333333333333333, \varepsilon + 1, 0.25\right), \mathsf{fma}\left(\varepsilon, -0.5, -0.5\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \varepsilon \cdot \left(x \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 720:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.25, \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot 0.08333333333333333\right), 0.5 \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.8e-257)
   (fma
    x
    (fma
     x
     (*
      (* (+ eps 1.0) (+ eps 1.0))
      (fma (* x -0.08333333333333333) (+ eps 1.0) 0.25))
     (fma eps -0.5 -0.5))
    1.0)
   (if (<= x 8.8e-187)
     (fma 0.5 (* eps (* x (* x eps))) 1.0)
     (if (<= x 720.0)
       (fma
        x
        (fma
         x
         (fma
          eps
          (* eps 0.25)
          (* (* x (* eps (* eps eps))) 0.08333333333333333))
         (* 0.5 eps))
        1.0)
       (* 0.25 (* x (* x (* eps eps))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.8e-257) {
		tmp = fma(x, fma(x, (((eps + 1.0) * (eps + 1.0)) * fma((x * -0.08333333333333333), (eps + 1.0), 0.25)), fma(eps, -0.5, -0.5)), 1.0);
	} else if (x <= 8.8e-187) {
		tmp = fma(0.5, (eps * (x * (x * eps))), 1.0);
	} else if (x <= 720.0) {
		tmp = fma(x, fma(x, fma(eps, (eps * 0.25), ((x * (eps * (eps * eps))) * 0.08333333333333333)), (0.5 * eps)), 1.0);
	} else {
		tmp = 0.25 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.8e-257)
		tmp = fma(x, fma(x, Float64(Float64(Float64(eps + 1.0) * Float64(eps + 1.0)) * fma(Float64(x * -0.08333333333333333), Float64(eps + 1.0), 0.25)), fma(eps, -0.5, -0.5)), 1.0);
	elseif (x <= 8.8e-187)
		tmp = fma(0.5, Float64(eps * Float64(x * Float64(x * eps))), 1.0);
	elseif (x <= 720.0)
		tmp = fma(x, fma(x, fma(eps, Float64(eps * 0.25), Float64(Float64(x * Float64(eps * Float64(eps * eps))) * 0.08333333333333333)), Float64(0.5 * eps)), 1.0);
	else
		tmp = Float64(0.25 * Float64(x * Float64(x * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.8e-257], N[(x * N[(x * N[(N[(N[(eps + 1.0), $MachinePrecision] * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x * -0.08333333333333333), $MachinePrecision] * N[(eps + 1.0), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision] + N[(eps * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 8.8e-187], N[(0.5 * N[(eps * N[(x * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 720.0], N[(x * N[(x * N[(eps * N[(eps * 0.25), $MachinePrecision] + N[(N[(x * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + N[(0.5 * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.25 * N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.7999999999999996e-257

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. *-lowering-*.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. +-lowering-+.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. Simplified 96.6%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 63.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + 1} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right), 1\right)} \]

   11. Simplified 61.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \mathsf{fma}\left(x \cdot -0.08333333333333333, 1 + \varepsilon, 0.25\right), \mathsf{fma}\left(\varepsilon, -0.5, -0.5\right)\right), 1\right)} \]

   if -6.7999999999999996e-257 < x < 8.80000000000000032e-187

   1. Initial program 54.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.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. +-lowering-+.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. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. neg-lowering-neg.f64 100.0

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0

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

   14. Simplified 97.6%

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

   if 8.80000000000000032e-187 < x < 720

   1. Initial program 60.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.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. +-lowering-+.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. Simplified 98.2%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 98.1

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

   8. Simplified 98.1%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 72.8%

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

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{12} \cdot \left({\varepsilon}^{3} \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{12} \cdot \left({\varepsilon}^{3} \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) + 1} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{12} \cdot \left({\varepsilon}^{3} \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right), 1\right)} \]

   14. Simplified 65.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.25, \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot 0.08333333333333333\right), 0.5 \cdot \varepsilon\right), 1\right)} \]

   if 720 < 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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. distribute-rgt1-in N/A

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

      14. *-lowering-*.f64 N/A

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

   5. Simplified 24.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 54.4%

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

   9. Taylor expanded in eps around inf

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

       1. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

          \[\leadsto \frac{1}{4} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       10. *-lowering-*.f64 75.7

           \[\leadsto 0.25 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   11. Simplified 75.7%

       \[\leadsto \color{blue}{0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-257}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right) \cdot \mathsf{fma}\left(x \cdot -0.08333333333333333, \varepsilon + 1, 0.25\right), \mathsf{fma}\left(\varepsilon, -0.5, -0.5\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \varepsilon \cdot \left(x \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 720:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.25, \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot 0.08333333333333333\right), 0.5 \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right) \cdot \mathsf{fma}\left(x \cdot -0.08333333333333333, \varepsilon + 1, 0.25\right), \mathsf{fma}\left(\varepsilon, -0.5, -0.5\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \varepsilon \cdot \left(x \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 800:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\varepsilon \cdot \left(x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-262)
   (fma
    x
    (fma
     x
     (*
      (* (+ eps 1.0) (+ eps 1.0))
      (fma (* x -0.08333333333333333) (+ eps 1.0) 0.25))
     (fma eps -0.5 -0.5))
    1.0)
   (if (<= x 5.2e-187)
     (fma 0.5 (* eps (* x (* x eps))) 1.0)
     (if (<= x 800.0)
       (fma x (/ (* eps (* x (* 0.25 (* eps eps)))) eps) 1.0)
       (* 0.25 (* x (* x (* eps eps))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-262) {
		tmp = fma(x, fma(x, (((eps + 1.0) * (eps + 1.0)) * fma((x * -0.08333333333333333), (eps + 1.0), 0.25)), fma(eps, -0.5, -0.5)), 1.0);
	} else if (x <= 5.2e-187) {
		tmp = fma(0.5, (eps * (x * (x * eps))), 1.0);
	} else if (x <= 800.0) {
		tmp = fma(x, ((eps * (x * (0.25 * (eps * eps)))) / eps), 1.0);
	} else {
		tmp = 0.25 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-262)
		tmp = fma(x, fma(x, Float64(Float64(Float64(eps + 1.0) * Float64(eps + 1.0)) * fma(Float64(x * -0.08333333333333333), Float64(eps + 1.0), 0.25)), fma(eps, -0.5, -0.5)), 1.0);
	elseif (x <= 5.2e-187)
		tmp = fma(0.5, Float64(eps * Float64(x * Float64(x * eps))), 1.0);
	elseif (x <= 800.0)
		tmp = fma(x, Float64(Float64(eps * Float64(x * Float64(0.25 * Float64(eps * eps)))) / eps), 1.0);
	else
		tmp = Float64(0.25 * Float64(x * Float64(x * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-262], N[(x * N[(x * N[(N[(N[(eps + 1.0), $MachinePrecision] * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x * -0.08333333333333333), $MachinePrecision] * N[(eps + 1.0), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision] + N[(eps * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 5.2e-187], N[(0.5 * N[(eps * N[(x * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 800.0], N[(x * N[(N[(eps * N[(x * N[(0.25 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.25 * N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.2e-262

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. *-lowering-*.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. +-lowering-+.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. Simplified 96.6%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 63.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + 1} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right), 1\right)} \]

   11. Simplified 61.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \mathsf{fma}\left(x \cdot -0.08333333333333333, 1 + \varepsilon, 0.25\right), \mathsf{fma}\left(\varepsilon, -0.5, -0.5\right)\right), 1\right)} \]

   if -3.2e-262 < x < 5.1999999999999999e-187

   1. Initial program 54.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.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. +-lowering-+.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. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. neg-lowering-neg.f64 100.0

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0

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

   14. Simplified 97.6%

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

   if 5.1999999999999999e-187 < x < 800

   1. Initial program 60.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. distribute-rgt1-in N/A

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

      14. *-lowering-*.f64 N/A

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

   5. Simplified 39.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 61.0%

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

   9. Taylor expanded in eps around 0

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

       1. /-lowering-/.f64 N/A

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

   11. Simplified 77.7%

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

   12. Taylor expanded in eps around inf

       \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{\frac{1}{4} \cdot \left({\varepsilon}^{3} \cdot x\right)}}{\varepsilon}, 1\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 N/A

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 N/A

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 85.5

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

   14. Simplified 85.5%

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

   if 800 < 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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. distribute-rgt1-in N/A

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

      14. *-lowering-*.f64 N/A

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

   5. Simplified 24.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 54.4%

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

   9. Taylor expanded in eps around inf

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

       1. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

          \[\leadsto \frac{1}{4} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       10. *-lowering-*.f64 75.7

           \[\leadsto 0.25 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   11. Simplified 75.7%

       \[\leadsto \color{blue}{0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right) \cdot \mathsf{fma}\left(x \cdot -0.08333333333333333, \varepsilon + 1, 0.25\right), \mathsf{fma}\left(\varepsilon, -0.5, -0.5\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \varepsilon \cdot \left(x \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 800:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\varepsilon \cdot \left(x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \frac{\varepsilon \cdot \left(x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}{\varepsilon}, 1\right)\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{-262}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \varepsilon \cdot \left(x \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 1050:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma x (/ (* eps (* x (* 0.25 (* eps eps)))) eps) 1.0)))
   (if (<= x -2.05e-262)
     t_0
     (if (<= x 4.2e-187)
       (fma 0.5 (* eps (* x (* x eps))) 1.0)
       (if (<= x 1050.0) t_0 (* 0.25 (* x (* x (* eps eps)))))))))

C

double code(double x, double eps) {
	double t_0 = fma(x, ((eps * (x * (0.25 * (eps * eps)))) / eps), 1.0);
	double tmp;
	if (x <= -2.05e-262) {
		tmp = t_0;
	} else if (x <= 4.2e-187) {
		tmp = fma(0.5, (eps * (x * (x * eps))), 1.0);
	} else if (x <= 1050.0) {
		tmp = t_0;
	} else {
		tmp = 0.25 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(x, Float64(Float64(eps * Float64(x * Float64(0.25 * Float64(eps * eps)))) / eps), 1.0)
	tmp = 0.0
	if (x <= -2.05e-262)
		tmp = t_0;
	elseif (x <= 4.2e-187)
		tmp = fma(0.5, Float64(eps * Float64(x * Float64(x * eps))), 1.0);
	elseif (x <= 1050.0)
		tmp = t_0;
	else
		tmp = Float64(0.25 * Float64(x * Float64(x * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(N[(eps * N[(x * N[(0.25 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -2.05e-262], t$95$0, If[LessEqual[x, 4.2e-187], N[(0.5 * N[(eps * N[(x * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 1050.0], t$95$0, N[(0.25 * N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.05000000000000013e-262 or 4.19999999999999985e-187 < x < 1050

   1. Initial program 69.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. distribute-rgt1-in N/A

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

      14. *-lowering-*.f64 N/A

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

   5. Simplified 39.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 64.3%

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

   9. Taylor expanded in eps around 0

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

       1. /-lowering-/.f64 N/A

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

   11. Simplified 76.7%

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

   12. Taylor expanded in eps around inf

       \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{\frac{1}{4} \cdot \left({\varepsilon}^{3} \cdot x\right)}}{\varepsilon}, 1\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 N/A

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 N/A

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 87.2

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

   14. Simplified 87.2%

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

   if -2.05000000000000013e-262 < x < 4.19999999999999985e-187

   1. Initial program 54.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.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. +-lowering-+.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. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. neg-lowering-neg.f64 100.0

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0

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

   14. Simplified 97.6%

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

   if 1050 < 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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. distribute-rgt1-in N/A

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

      14. *-lowering-*.f64 N/A

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

   5. Simplified 24.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 54.4%

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

   9. Taylor expanded in eps around inf

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

       1. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

          \[\leadsto \frac{1}{4} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       10. *-lowering-*.f64 75.7

           \[\leadsto 0.25 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   11. Simplified 75.7%

       \[\leadsto \color{blue}{0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\varepsilon \cdot \left(x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}{\varepsilon}, 1\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \varepsilon \cdot \left(x \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 1050:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\varepsilon \cdot \left(x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{-256}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \varepsilon \cdot \left(x \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 240:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma x (* x (* 0.25 (* eps eps))) 1.0)))
   (if (<= x -2e-256)
     t_0
     (if (<= x 4.5e-187)
       (fma 0.5 (* eps (* x (* x eps))) 1.0)
       (if (<= x 240.0) t_0 (* 0.25 (* x (* x (* eps eps)))))))))

C

double code(double x, double eps) {
	double t_0 = fma(x, (x * (0.25 * (eps * eps))), 1.0);
	double tmp;
	if (x <= -2e-256) {
		tmp = t_0;
	} else if (x <= 4.5e-187) {
		tmp = fma(0.5, (eps * (x * (x * eps))), 1.0);
	} else if (x <= 240.0) {
		tmp = t_0;
	} else {
		tmp = 0.25 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(x, Float64(x * Float64(0.25 * Float64(eps * eps))), 1.0)
	tmp = 0.0
	if (x <= -2e-256)
		tmp = t_0;
	elseif (x <= 4.5e-187)
		tmp = fma(0.5, Float64(eps * Float64(x * Float64(x * eps))), 1.0);
	elseif (x <= 240.0)
		tmp = t_0;
	else
		tmp = Float64(0.25 * Float64(x * Float64(x * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * N[(0.25 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -2e-256], t$95$0, If[LessEqual[x, 4.5e-187], N[(0.5 * N[(eps * N[(x * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 240.0], t$95$0, N[(0.25 * N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999995e-256 or 4.4999999999999998e-187 < x < 240

   1. Initial program 69.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. distribute-rgt1-in N/A

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

      14. *-lowering-*.f64 N/A

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

   5. Simplified 39.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 64.3%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. metadata-eval N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 82.9

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

   11. Simplified 82.9%

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

   if -1.99999999999999995e-256 < x < 4.4999999999999998e-187

   1. Initial program 54.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.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. +-lowering-+.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. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. neg-lowering-neg.f64 100.0

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0

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

   14. Simplified 97.6%

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

   if 240 < 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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. distribute-rgt1-in N/A

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

      14. *-lowering-*.f64 N/A

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

   5. Simplified 24.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 54.4%

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

   9. Taylor expanded in eps around inf

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

       1. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

          \[\leadsto \frac{1}{4} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       10. *-lowering-*.f64 75.7

           \[\leadsto 0.25 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   11. Simplified 75.7%

       \[\leadsto \color{blue}{0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-256}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \varepsilon \cdot \left(x \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 240:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 220:\\ \;\;\;\;\mathsf{fma}\left(0.5, \varepsilon \cdot \left(x \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 220.0)
   (fma 0.5 (* eps (* x (* x eps))) 1.0)
   (* 0.25 (* x (* x (* eps eps))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 220.0) {
		tmp = fma(0.5, (eps * (x * (x * eps))), 1.0);
	} else {
		tmp = 0.25 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 220.0)
		tmp = fma(0.5, Float64(eps * Float64(x * Float64(x * eps))), 1.0);
	else
		tmp = Float64(0.25 * Float64(x * Float64(x * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 220.0], N[(0.5 * N[(eps * N[(x * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.25 * N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 220

   1. Initial program 65.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. *-lowering-*.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. +-lowering-+.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. Simplified 97.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 97.8

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

   8. Simplified 97.8%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. neg-lowering-neg.f64 98.4

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

   11. Simplified 98.4%

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

   12. Taylor expanded in x around 0

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

   14. Simplified 77.6%

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

   if 220 < 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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. distribute-rgt1-in N/A

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

      14. *-lowering-*.f64 N/A

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

   5. Simplified 24.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 54.4%

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

   9. Taylor expanded in eps around inf

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

       1. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

          \[\leadsto \frac{1}{4} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       10. *-lowering-*.f64 75.7

           \[\leadsto 0.25 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   11. Simplified 75.7%

       \[\leadsto \color{blue}{0.25 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 56.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -90:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, \varepsilon, x\right), -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -90.0)
   (fma (fma x eps x) -0.5 1.0)
   (fma (* x x) (fma x 0.3333333333333333 -0.5) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -90.0) {
		tmp = fma(fma(x, eps, x), -0.5, 1.0);
	} else {
		tmp = fma((x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -90.0)
		tmp = fma(fma(x, eps, x), -0.5, 1.0);
	else
		tmp = fma(Float64(x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -90.0], N[(N[(x * eps + x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -90

   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

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

      1. *-lowering-*.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. +-lowering-+.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. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 41.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. *-rgt-identity N/A

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

       7. accelerator-lowering-fma.f64 29.4

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

   11. Simplified 29.4%

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

   if -90 < x

   1. Initial program 74.1%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

   5. Simplified 62.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 59.4

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

   8. Simplified 59.4%

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

Alternative 13: 49.8% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, \varepsilon, x\right), -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \varepsilon, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.75) (fma (fma x eps x) -0.5 1.0) (fma 0.5 (* x eps) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.75) {
		tmp = fma(fma(x, eps, x), -0.5, 1.0);
	} else {
		tmp = fma(0.5, (x * eps), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.75)
		tmp = fma(fma(x, eps, x), -0.5, 1.0);
	else
		tmp = fma(0.5, Float64(x * eps), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.75], N[(N[(x * eps + x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision], N[(0.5 * N[(x * eps), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75

   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

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

      1. *-lowering-*.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. +-lowering-+.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. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 41.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. *-rgt-identity N/A

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

       7. accelerator-lowering-fma.f64 29.4

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

   11. Simplified 29.4%

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

   if -1.75 < x

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. *-lowering-*.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. +-lowering-+.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. Simplified 98.4%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 86.0

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

   8. Simplified 86.0%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 63.0%

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

   12. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 49.6

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

   14. Simplified 49.6%

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

Alternative 14: 46.1% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 4e-16) 1.0 (* 0.5 (* x eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 4e-16) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 4d-16) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (x * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 4e-16) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 4e-16:
		tmp = 1.0
	else:
		tmp = 0.5 * (x * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 4e-16)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(x * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 4e-16)
		tmp = 1.0;
	else
		tmp = 0.5 * (x * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 4e-16], 1.0, N[(0.5 * N[(x * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999999e-16

   1. Initial program 65.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 61.1%

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

   if 3.9999999999999999e-16 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. distribute-rgt1-in N/A

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

      14. *-lowering-*.f64 N/A

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

   5. Simplified 25.2%

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

   6. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 15.8

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

   8. Simplified 15.8%

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

Alternative 15: 49.5% accurate, 22.8× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (fma 0.5 (* x eps) 1.0))

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(0.5 * N[(x * eps), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in eps around inf

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

   1. *-lowering-*.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. +-lowering-+.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. Simplified 98.5%

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

6. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 87.5

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

8. Simplified 87.5%

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

9. Taylor expanded in x around 0

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

11. Simplified 62.9%

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

12. Taylor expanded in x around 0

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

    1. associate-*r* N/A

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

    2. +-commutative N/A

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

    3. associate-*r* N/A

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

    4. accelerator-lowering-fma.f64 N/A

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

    5. *-commutative N/A

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

    6. *-lowering-*.f64 46.7

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

14. Simplified 46.7%

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

Alternative 16: 43.2% accurate, 273.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

double code(double x, double eps) {
	return 1.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 41.0%

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

Reproduce

herbie shell --seed 2024199 
(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))