NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 99.8%

Time: 14.1s

Alternatives: 10

Speedup: 9.7×

• 38.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -5.319753076643332e+175
  2. eps = 4.176355856298032e-252

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\\ \mathbf{if}\;t\_0 \leq 2.000000005:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (+
          (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
          (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))))
   (if (<= t_0 2.000000005)
     (* 0.5 (* (exp (- x)) (+ x (+ x 2.0))))
     (/ t_0 2.0))))

C

double code(double x, double eps) {
	double t_0 = ((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)));
	double tmp;
	if (t_0 <= 2.000000005) {
		tmp = 0.5 * (exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))
    if (t_0 <= 2.000000005d0) then
        tmp = 0.5d0 * (exp(-x) * (x + (x + 2.0d0)))
    else
        tmp = t_0 / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)));
	double tmp;
	if (t_0 <= 2.000000005) {
		tmp = 0.5 * (Math.exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))
	tmp = 0
	if t_0 <= 2.000000005:
		tmp = 0.5 * (math.exp(-x) * (x + (x + 2.0)))
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps))))
	tmp = 0.0
	if (t_0 <= 2.000000005)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) * Float64(x + Float64(x + 2.0))));
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)));
	tmp = 0.0;
	if (t_0 <= 2.000000005)
		tmp = 0.5 * (exp(-x) * (x + (x + 2.0)));
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.000000005], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 2.00000000499999997

   1. Initial program 56.3%

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

   3. Taylor expanded in eps around 0

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

      1. 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 2.00000000499999997 < (-.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. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.000000005:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.1%

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

   3. Taylor expanded in eps around 0

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

      1. 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 2 < (-.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 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 87.3%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 89.2%

      \[\leadsto \mathsf{fma}\left(x, 0.5 \cdot \mathsf{fma}\left(\varepsilon + 1, -1 + \frac{1}{\varepsilon}, \frac{\mathsf{fma}\left(\varepsilon, \left(-x\right) + \mathsf{fma}\left(\varepsilon, \left(1 - x\right) + \mathsf{fma}\left(x, \varepsilon, x\right), 0\right), -1\right)}{\varepsilon}\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

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

Alternative 3: 93.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.1%

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

   3. Taylor expanded in eps around 0

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

      1. 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%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.0%

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

   if 2 < (-.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 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 87.3%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 89.2%

      \[\leadsto \mathsf{fma}\left(x, 0.5 \cdot \mathsf{fma}\left(\varepsilon + 1, -1 + \frac{1}{\varepsilon}, \frac{\mathsf{fma}\left(\varepsilon, \left(-x\right) + \mathsf{fma}\left(\varepsilon, \left(1 - x\right) + \mathsf{fma}\left(x, \varepsilon, x\right), 0\right), -1\right)}{\varepsilon}\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.7%

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

Alternative 4: 78.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.6%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. 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 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 87.1%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 87.1%

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

4. Final simplification 76.8%

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

Alternative 5: 82.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot \mathsf{fma}\left(1 + \varepsilon, \frac{1}{\varepsilon}, \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot x\right), -1\right)}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - \frac{1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.32e+60)
   (fma
    x
    (*
     0.5
     (fma (+ 1.0 eps) (/ 1.0 eps) (/ (fma eps (* eps (* eps x)) -1.0) eps)))
    1.0)
   (if (<= x 1.45e+128)
     (/ (- (/ 1.0 eps) (/ 1.0 eps)) 2.0)
     (* 0.5 (* x (* x (* eps eps)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.32e+60) {
		tmp = fma(x, (0.5 * fma((1.0 + eps), (1.0 / eps), (fma(eps, (eps * (eps * x)), -1.0) / eps))), 1.0);
	} else if (x <= 1.45e+128) {
		tmp = ((1.0 / eps) - (1.0 / eps)) / 2.0;
	} else {
		tmp = 0.5 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.32e+60)
		tmp = fma(x, Float64(0.5 * fma(Float64(1.0 + eps), Float64(1.0 / eps), Float64(fma(eps, Float64(eps * Float64(eps * x)), -1.0) / eps))), 1.0);
	elseif (x <= 1.45e+128)
		tmp = Float64(Float64(Float64(1.0 / eps) - Float64(1.0 / eps)) / 2.0);
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.32e+60], N[(x * N[(0.5 * N[(N[(1.0 + eps), $MachinePrecision] * N[(1.0 / eps), $MachinePrecision] + N[(N[(eps * N[(eps * N[(eps * x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 1.45e+128], N[(N[(N[(1.0 / eps), $MachinePrecision] - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.5 * N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.32e60

   1. Initial program 63.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 89.4%

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

   9. Taylor expanded in eps around 0

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{2} \cdot \mathsf{fma}\left(\varepsilon + 1, \frac{1}{\color{blue}{\varepsilon}}, \frac{\mathsf{fma}\left(\varepsilon, \left(\mathsf{neg}\left(x\right)\right) + \mathsf{fma}\left(\varepsilon, \left(1 - x\right) + \mathsf{fma}\left(x, \varepsilon, x\right), 0\right), -1\right)}{\varepsilon}\right), 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 89.0%

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

   12. Taylor expanded in eps around inf

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

   14. Applied rewrites 89.9%

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

   if 1.32e60 < x < 1.45e128

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 8.7

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

   5. Applied rewrites 8.7%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 63.7

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 63.7%

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

   12. Taylor expanded in eps around 0

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

   14. Applied rewrites 81.5%

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

   if 1.45e128 < 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 42.9%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 43.5%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 70.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot \mathsf{fma}\left(1 + \varepsilon, \frac{1}{\varepsilon}, \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot x\right), -1\right)}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - \frac{1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq 21000:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot t\_0, 1\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - \frac{1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* eps eps))))
   (if (<= x 21000.0)
     (fma x (* 0.5 t_0) 1.0)
     (if (<= x 1.45e+128)
       (/ (- (/ 1.0 eps) (/ 1.0 eps)) 2.0)
       (* 0.5 (* x t_0))))))

C

double code(double x, double eps) {
	double t_0 = x * (eps * eps);
	double tmp;
	if (x <= 21000.0) {
		tmp = fma(x, (0.5 * t_0), 1.0);
	} else if (x <= 1.45e+128) {
		tmp = ((1.0 / eps) - (1.0 / eps)) / 2.0;
	} else {
		tmp = 0.5 * (x * t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(eps * eps))
	tmp = 0.0
	if (x <= 21000.0)
		tmp = fma(x, Float64(0.5 * t_0), 1.0);
	elseif (x <= 1.45e+128)
		tmp = Float64(Float64(Float64(1.0 / eps) - Float64(1.0 / eps)) / 2.0);
	else
		tmp = Float64(0.5 * Float64(x * t_0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 21000.0], N[(x * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 1.45e+128], N[(N[(N[(1.0 / eps), $MachinePrecision] - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.5 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 21000

   1. Initial program 61.0%

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

   3. Taylor expanded in 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 90.6%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 91.5%

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

   if 21000 < x < 1.45e128

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 9.8

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 56.9

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

   8. Applied rewrites 56.9%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 56.9%

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

   12. Taylor expanded in eps around 0

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

   14. Applied rewrites 67.2%

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

   if 1.45e128 < 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 42.9%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 43.5%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 70.6%

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

Alternative 7: 81.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq 0.034:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot t\_0, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* eps eps))))
   (if (<= x 0.034) (fma x (* 0.5 t_0) 1.0) (* 0.5 (* x t_0)))))

C

double code(double x, double eps) {
	double t_0 = x * (eps * eps);
	double tmp;
	if (x <= 0.034) {
		tmp = fma(x, (0.5 * t_0), 1.0);
	} else {
		tmp = 0.5 * (x * t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(eps * eps))
	tmp = 0.0
	if (x <= 0.034)
		tmp = fma(x, Float64(0.5 * t_0), 1.0);
	else
		tmp = Float64(0.5 * Float64(x * t_0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.034], N[(x * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.034000000000000002

   1. Initial program 60.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 92.5%

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

   if 0.034000000000000002 < 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 35.6%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 36.2%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 59.6%

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

Alternative 8: 52.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+99}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 8.5e+99) 1.0 (* x (fma x (fma x 0.5 -1.0) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 8.5e+99) {
		tmp = 1.0;
	} else {
		tmp = x * fma(x, fma(x, 0.5, -1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 8.5e+99)
		tmp = 1.0;
	else
		tmp = Float64(x * fma(x, fma(x, 0.5, -1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 8.5e+99], 1.0, N[(x * N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.49999999999999984e99

   1. Initial program 64.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.1%

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

   if 8.49999999999999984e99 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. 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 61.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 61.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 39.9%

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

Alternative 9: 52.6% accurate, 15.2× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (* x x) (fma x 0.3333333333333333 -0.5) 1.0))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.7%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
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 63.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 54.1%

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

Alternative 10: 43.8% 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 72.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 45.4%

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

Reproduce

herbie shell --seed 2024221 
(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))