NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.1% → 99.8%

Time: 15.1s

Alternatives: 14

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 = -4.778866022811327e+119
  2. eps = -2.4525030967000727e-152

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: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (+ (/ 1.0 eps) -1.0) (exp (* x (- -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)))) - (((1.0 / eps) + -1.0) * exp((x * (-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))))) - (((1.0d0 / eps) + (-1.0d0)) * exp((x * ((-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)))) - (((1.0 / eps) + -1.0) * Math.exp((x * (-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)))) - (((1.0 / eps) + -1.0) * math.exp((x * (-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(Float64(Float64(1.0 / eps) + -1.0) * exp(Float64(x * 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)))) - (((1.0 / eps) + -1.0) * exp((x * (-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[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $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 35.7%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. 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. lower-*.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. lower-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. lower-neg.f64 N/A

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

   5. Applied rewrites 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 99.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   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. lower-*.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. lower-neg.f64 100.0

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

   11. Applied rewrites 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. lift-*.f64 N/A

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

       2. lift-exp.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-exp.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 100.0

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

       9. lift-+.f64 N/A

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

       10. lift-exp.f64 N/A

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

       11. lift-exp.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-neg.f64 N/A

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

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

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

       15. lift-*.f64 N/A

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

       16. cosh-undef N/A

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

       17. lower-*.f64 N/A

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

       18. lower-cosh.f64 100.0

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

   13. Applied rewrites 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%

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

Alternative 2: 99.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (+ (/ 1.0 eps) -1.0) (exp (* x (- -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)))) - (((1.0 / eps) + -1.0) * exp((x * (-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))))) - (((1.0d0 / eps) + (-1.0d0)) * exp((x * ((-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)))) - (((1.0 / eps) + -1.0) * Math.exp((x * (-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)))) - (((1.0 / eps) + -1.0) * math.exp((x * (-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(Float64(Float64(1.0 / eps) + -1.0) * exp(Float64(x * 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)))) - (((1.0 / eps) + -1.0) * exp((x * (-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[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $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 35.7%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in eps around 0

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

      1. neg-mul-1 N/A

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

      2. lower-exp.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 99.4

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

   8. Applied rewrites 99.4%

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

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

   1. Initial program 99.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   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. lower-*.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. lower-neg.f64 100.0

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

   11. Applied rewrites 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. lift-*.f64 N/A

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

       2. lift-exp.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-exp.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 100.0

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

       9. lift-+.f64 N/A

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

       10. lift-exp.f64 N/A

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

       11. lift-exp.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-neg.f64 N/A

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

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

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

       15. lift-*.f64 N/A

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

       16. cosh-undef N/A

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

       17. lower-*.f64 N/A

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

       18. lower-cosh.f64 100.0

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

   13. Applied rewrites 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 99.7%

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

Alternative 3: 93.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * exp((x * (-1.0 - eps))))) <= 4.0) {
		tmp = exp(-x);
	} else {
		tmp = fma(0.5, x, fma((eps * eps), (x * fma(0.5, (x * eps), -0.5)), 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(Float64(Float64(1.0 / eps) + -1.0) * exp(Float64(x * Float64(-1.0 - eps))))) <= 4.0)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(fma(0.5, x, fma(Float64(eps * eps), Float64(x * fma(0.5, Float64(x * eps), -0.5)), 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[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[Exp[(-x)], $MachinePrecision], N[(N[(0.5 * x + N[(N[(eps * eps), $MachinePrecision] * N[(x * N[(0.5 * N[(x * eps), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision] / eps), $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 50.1%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in eps around 0

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

      1. neg-mul-1 N/A

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

      2. lower-exp.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 97.9

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

   8. Applied rewrites 97.9%

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

   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

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 91.3%

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

4. Final simplification 95.1%

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

Alternative 4: 78.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * exp((x * (-1.0 - eps))))) <= 4.0) {
		tmp = 1.0;
	} else {
		tmp = (x * 0.25) * (x * (eps * 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))))) - (((1.0d0 / eps) + (-1.0d0)) * exp((x * ((-1.0d0) - eps))))) <= 4.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * 0.25d0) * (x * (eps * 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)))) - (((1.0 / eps) + -1.0) * Math.exp((x * (-1.0 - eps))))) <= 4.0) {
		tmp = 1.0;
	} else {
		tmp = (x * 0.25) * (x * (eps * eps));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * math.exp((x * (-1.0 - eps))))) <= 4.0:
		tmp = 1.0
	else:
		tmp = (x * 0.25) * (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(Float64(Float64(1.0 / eps) + -1.0) * exp(Float64(x * Float64(-1.0 - eps))))) <= 4.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * 0.25) * Float64(x * Float64(eps * 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)))) - (((1.0 / eps) + -1.0) * exp((x * (-1.0 - eps))))) <= 4.0)
		tmp = 1.0;
	else
		tmp = (x * 0.25) * (x * (eps * 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[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], 1.0, N[(N[(x * 0.25), $MachinePrecision] * N[(x * N[(eps * 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 50.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.3%

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

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 81.0

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

   8. Applied rewrites 81.0%

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

   9. Taylor expanded in eps around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 81.0

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

   11. Applied rewrites 81.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 84.7

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

   13. Applied rewrites 84.7%

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

4. Final simplification 78.2%

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

Alternative 5: 75.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * exp((x * (-1.0 - eps))))) <= 4.0) {
		tmp = 1.0;
	} else {
		tmp = (0.5 * (eps * eps)) * (x * x);
	}
	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))))) - (((1.0d0 / eps) + (-1.0d0)) * exp((x * ((-1.0d0) - eps))))) <= 4.0d0) then
        tmp = 1.0d0
    else
        tmp = (0.5d0 * (eps * eps)) * (x * x)
    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)))) - (((1.0 / eps) + -1.0) * Math.exp((x * (-1.0 - eps))))) <= 4.0) {
		tmp = 1.0;
	} else {
		tmp = (0.5 * (eps * eps)) * (x * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * math.exp((x * (-1.0 - eps))))) <= 4.0:
		tmp = 1.0
	else:
		tmp = (0.5 * (eps * eps)) * (x * x)
	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(Float64(Float64(1.0 / eps) + -1.0) * exp(Float64(x * Float64(-1.0 - eps))))) <= 4.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.5 * Float64(eps * eps)) * Float64(x * x));
	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)))) - (((1.0 / eps) + -1.0) * exp((x * (-1.0 - eps))))) <= 4.0)
		tmp = 1.0;
	else
		tmp = (0.5 * (eps * eps)) * (x * x);
	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[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], 1.0, N[(N[(0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(x * x), $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 50.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.3%

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

   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

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.0

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

   8. Applied rewrites 81.0%

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

4. Final simplification 76.6%

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

Alternative 6: 72.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * exp((x * (-1.0 - eps))))) <= 4.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (eps * (x * (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))))) - (((1.0d0 / eps) + (-1.0d0)) * exp((x * ((-1.0d0) - eps))))) <= 4.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (eps * (x * (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)))) - (((1.0 / eps) + -1.0) * Math.exp((x * (-1.0 - eps))))) <= 4.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (eps * (x * (x * eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * math.exp((x * (-1.0 - eps))))) <= 4.0:
		tmp = 1.0
	else:
		tmp = 0.5 * (eps * (x * (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(Float64(Float64(1.0 / eps) + -1.0) * exp(Float64(x * Float64(-1.0 - eps))))) <= 4.0)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(eps * Float64(x * 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)))) - (((1.0 / eps) + -1.0) * exp((x * (-1.0 - eps))))) <= 4.0)
		tmp = 1.0;
	else
		tmp = 0.5 * (eps * (x * (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[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], 1.0, N[(0.5 * N[(eps * N[(x * 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))))) < 4

   1. Initial program 50.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.3%

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

   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

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

   9. Taylor expanded in eps around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. lower-fma.f64 N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-frac N/A

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

       10. metadata-eval N/A

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

       11. lower-/.f64 79.4

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

   11. Applied rewrites 79.4%

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

   12. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 74.4

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

   14. Applied rewrites 74.4%

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

4. Final simplification 73.8%

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

Alternative 7: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower-+.f64 N/A

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

5. Applied rewrites 99.6%

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

Alternative 8: 82.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(0.5, x \cdot \varepsilon, -0.5\right), \varepsilon\right)\right)}{\varepsilon}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot \mathsf{fma}\left(\varepsilon + 1, \frac{1}{\varepsilon} + -1, \frac{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, -0.5, \varepsilon \cdot \left(\left(1 + \mathsf{fma}\left(x, -0.5, x \cdot \varepsilon\right)\right) + 0.5 \cdot x\right)\right) + x \cdot -0.5, -1\right)}{\varepsilon}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.25\right) \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5e+67)
   (/ (fma 0.5 x (fma (* eps eps) (* x (fma 0.5 (* x eps) -0.5)) eps)) eps)
   (if (<= x 2.1e-8)
     (fma
      x
      (*
       0.5
       (fma
        (+ eps 1.0)
        (+ (/ 1.0 eps) -1.0)
        (/
         (fma
          eps
          (+
           (fma x -0.5 (* eps (+ (+ 1.0 (fma x -0.5 (* x eps))) (* 0.5 x))))
           (* x -0.5))
          -1.0)
         eps)))
      1.0)
     (* (* x 0.25) (* x (* eps eps))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5e+67) {
		tmp = fma(0.5, x, fma((eps * eps), (x * fma(0.5, (x * eps), -0.5)), eps)) / eps;
	} else if (x <= 2.1e-8) {
		tmp = fma(x, (0.5 * fma((eps + 1.0), ((1.0 / eps) + -1.0), (fma(eps, (fma(x, -0.5, (eps * ((1.0 + fma(x, -0.5, (x * eps))) + (0.5 * x)))) + (x * -0.5)), -1.0) / eps))), 1.0);
	} else {
		tmp = (x * 0.25) * (x * (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5e+67)
		tmp = Float64(fma(0.5, x, fma(Float64(eps * eps), Float64(x * fma(0.5, Float64(x * eps), -0.5)), eps)) / eps);
	elseif (x <= 2.1e-8)
		tmp = fma(x, Float64(0.5 * fma(Float64(eps + 1.0), Float64(Float64(1.0 / eps) + -1.0), Float64(fma(eps, Float64(fma(x, -0.5, Float64(eps * Float64(Float64(1.0 + fma(x, -0.5, Float64(x * eps))) + Float64(0.5 * x)))) + Float64(x * -0.5)), -1.0) / eps))), 1.0);
	else
		tmp = Float64(Float64(x * 0.25) * Float64(x * Float64(eps * eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5e+67], N[(N[(0.5 * x + N[(N[(eps * eps), $MachinePrecision] * N[(x * N[(0.5 * N[(x * eps), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision], If[LessEqual[x, 2.1e-8], N[(x * N[(0.5 * N[(N[(eps + 1.0), $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(eps * N[(N[(x * -0.5 + N[(eps * N[(N[(1.0 + N[(x * -0.5 + N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * 0.25), $MachinePrecision] * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.99999999999999976e67

   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

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 85.8

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 100.0%

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

   if -4.99999999999999976e67 < x < 2.09999999999999994e-8

   1. Initial program 50.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 92.3%

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

   if 2.09999999999999994e-8 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

   9. Taylor expanded in eps around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 56.0

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

   11. Applied rewrites 56.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 72.7

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

   13. Applied rewrites 72.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(0.5, x \cdot \varepsilon, -0.5\right), \varepsilon\right)\right)}{\varepsilon}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot \mathsf{fma}\left(\varepsilon + 1, \frac{1}{\varepsilon} + -1, \frac{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, -0.5, \varepsilon \cdot \left(\left(1 + \mathsf{fma}\left(x, -0.5, x \cdot \varepsilon\right)\right) + 0.5 \cdot x\right)\right) + x \cdot -0.5, -1\right)}{\varepsilon}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.25\right) \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(0.5, x \cdot \varepsilon, -0.5\right), \varepsilon\right)\right)}{\varepsilon}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot t\_0, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.25\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* eps eps))))
   (if (<= x -5e+67)
     (/ (fma 0.5 x (fma (* eps eps) (* x (fma 0.5 (* x eps) -0.5)) eps)) eps)
     (if (<= x 2.1e-8) (fma x (* 0.5 t_0) 1.0) (* (* x 0.25) t_0)))))

C

double code(double x, double eps) {
	double t_0 = x * (eps * eps);
	double tmp;
	if (x <= -5e+67) {
		tmp = fma(0.5, x, fma((eps * eps), (x * fma(0.5, (x * eps), -0.5)), eps)) / eps;
	} else if (x <= 2.1e-8) {
		tmp = fma(x, (0.5 * t_0), 1.0);
	} else {
		tmp = (x * 0.25) * t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(eps * eps))
	tmp = 0.0
	if (x <= -5e+67)
		tmp = Float64(fma(0.5, x, fma(Float64(eps * eps), Float64(x * fma(0.5, Float64(x * eps), -0.5)), eps)) / eps);
	elseif (x <= 2.1e-8)
		tmp = fma(x, Float64(0.5 * t_0), 1.0);
	else
		tmp = Float64(Float64(x * 0.25) * t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+67], N[(N[(0.5 * x + N[(N[(eps * eps), $MachinePrecision] * N[(x * N[(0.5 * N[(x * eps), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision], If[LessEqual[x, 2.1e-8], N[(x * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * 0.25), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.99999999999999976e67

   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

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 85.8

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 100.0%

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

   if -4.99999999999999976e67 < x < 2.09999999999999994e-8

   1. Initial program 50.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 90.3

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

   8. Applied rewrites 90.3%

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

   if 2.09999999999999994e-8 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

   9. Taylor expanded in eps around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 56.0

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

   11. Applied rewrites 56.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 72.7

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

   13. Applied rewrites 72.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(0.5, x \cdot \varepsilon, -0.5\right), \varepsilon\right)\right)}{\varepsilon}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.25\right) \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot t\_0, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.25\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* eps eps))))
   (if (<= x 2.1e-8) (fma x (* 0.5 t_0) 1.0) (* (* x 0.25) t_0))))

C

double code(double x, double eps) {
	double t_0 = x * (eps * eps);
	double tmp;
	if (x <= 2.1e-8) {
		tmp = fma(x, (0.5 * t_0), 1.0);
	} else {
		tmp = (x * 0.25) * t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(eps * eps))
	tmp = 0.0
	if (x <= 2.1e-8)
		tmp = fma(x, Float64(0.5 * t_0), 1.0);
	else
		tmp = Float64(Float64(x * 0.25) * t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.1e-8], N[(x * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * 0.25), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.09999999999999994e-8

   1. Initial program 59.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 89.5

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

   8. Applied rewrites 89.5%

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

   if 2.09999999999999994e-8 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

   9. Taylor expanded in eps around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 56.0

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

   11. Applied rewrites 56.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 72.7

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

   13. Applied rewrites 72.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.25\right) \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.2% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, -1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-7)
   (fma x (fma x (fma x -0.16666666666666666 0.5) -1.0) 1.0)
   (fma x (fma 0.5 x -1.0) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2e-7) {
		tmp = fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = fma(x, fma(0.5, x, -1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2e-7)
		tmp = fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	else
		tmp = fma(x, fma(0.5, x, -1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2e-7], N[(x * N[(x * N[(x * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(0.5 * x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999999e-7

   1. Initial program 97.8%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in eps around 0

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

      1. neg-mul-1 N/A

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

      2. lower-exp.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 98.9

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, 1\right)} \]

       3. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \color{blue}{-1}, 1\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{-1}{6} \cdot x, -1\right)}, 1\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot x + \frac{1}{2}}, -1\right), 1\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

       8. lower-fma.f64 60.2

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   11. Applied rewrites 60.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   if -1.9999999999999999e-7 < x

   1. Initial program 66.1%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in eps around 0

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

      1. neg-mul-1 N/A

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

      2. lower-exp.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 67.9

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 61.7

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

   11. Applied rewrites 61.7%

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

Alternative 12: 57.2% accurate, 21.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (fma x (fma 0.5 x -1.0) 1.0))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower-+.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in eps around 0

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

   1. neg-mul-1 N/A

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

   2. lower-exp.f64 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-neg.f64 73.2

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

8. Applied rewrites 73.2%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. lower-fma.f64 59.2

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

11. Applied rewrites 59.2%

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

Alternative 13: 49.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 76.6%

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

6. Taylor expanded in eps around inf

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

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

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

   2. metadata-eval N/A

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

   3. *-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 59.2

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

8. Applied rewrites 59.2%

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

9. Taylor expanded in eps around inf

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

    1. unpow2 N/A

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

    2. associate-*l* N/A

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

    3. lower-*.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. sub-neg N/A

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

    6. lower-fma.f64 N/A

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

    7. associate-*r/ N/A

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

    8. metadata-eval N/A

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

    9. distribute-neg-frac N/A

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

    10. metadata-eval N/A

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

    11. lower-/.f64 75.4

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

11. Applied rewrites 75.4%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 52.7%

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

Alternative 14: 42.9% 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 71.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 43.2%

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

Reproduce

herbie shell --seed 2024214 
(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))