NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.9% → 99.9%

Time: 14.5s

Alternatives: 17

Speedup: 8.0×

• 37.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.350122532663199e+113
  2. eps = 4.7954716911128096e-74

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: 72.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 - \varepsilon\right)}\\ \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(-1 + \varepsilon\right)} + t\_0 \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_0 + e^{x \cdot \varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x (- -1.0 eps)))))
   (if (<=
        (+
         (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ -1.0 eps))))
         (* t_0 (+ 1.0 (/ -1.0 eps))))
        0.0)
     (* 0.5 (* (exp (- x)) (+ x (+ x 2.0))))
     (* 0.5 (+ t_0 (exp (* x eps)))))))

C

double code(double x, double eps) {
	double t_0 = exp((x * (-1.0 - eps)));
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (-1.0 + eps)))) + (t_0 * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = 0.5 * (exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = 0.5 * (t_0 + exp((x * eps)));
	}
	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 = exp((x * ((-1.0d0) - eps)))
    if ((((1.0d0 + (1.0d0 / eps)) * exp((x * ((-1.0d0) + eps)))) + (t_0 * (1.0d0 + ((-1.0d0) / eps)))) <= 0.0d0) then
        tmp = 0.5d0 * (exp(-x) * (x + (x + 2.0d0)))
    else
        tmp = 0.5d0 * (t_0 + exp((x * eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp((x * (-1.0 - eps)));
	double tmp;
	if ((((1.0 + (1.0 / eps)) * Math.exp((x * (-1.0 + eps)))) + (t_0 * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = 0.5 * (Math.exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = 0.5 * (t_0 + Math.exp((x * eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp((x * (-1.0 - eps)))
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (-1.0 + eps)))) + (t_0 * (1.0 + (-1.0 / eps)))) <= 0.0:
		tmp = 0.5 * (math.exp(-x) * (x + (x + 2.0)))
	else:
		tmp = 0.5 * (t_0 + math.exp((x * eps)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$0 + N[Exp[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 41.2%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   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. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

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

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

      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

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

   5. Simplified 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. 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. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   8. Simplified 100.0%

      \[\leadsto 0.5 \cdot \left(e^{\color{blue}{x \cdot \varepsilon}} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 41.2%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   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. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

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

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

      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

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

   5. Simplified 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. 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. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

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

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

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

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

   11. Simplified 100.0%

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

       1. 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 egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 41.2%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   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. Simplified 99.3%

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

   6. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lower-neg.f64 99.3

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

   8. Simplified 99.3%

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

   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 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. 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. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

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

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

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

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

   11. Simplified 100.0%

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

       1. 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 egg-rr 100.0%

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

4. Final simplification 99.7%

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

Alternative 4: 94.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (-1.0 + eps)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 2.0) {
		tmp = exp(-x);
	} else {
		tmp = fma(0.5, (x * ((eps * fma(eps, fma(x, eps, 0.0), -x)) / 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(-1.0 + eps)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 2.0)
		tmp = exp(Float64(-x));
	else
		tmp = fma(0.5, Float64(x * Float64(Float64(eps * fma(eps, fma(x, eps, 0.0), Float64(-x))) / 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[(-1.0 + eps), $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[Exp[(-x)], $MachinePrecision], N[(0.5 * N[(x * N[(N[(eps * N[(eps * N[(x * eps + 0.0), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision] / eps), $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 55.0%

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

   3. Taylor expanded in eps around inf

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

      1. 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. Simplified 99.4%

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

   6. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lower-neg.f64 99.4

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

   8. Simplified 99.4%

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

   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 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. Simplified 86.0%

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

   7. Applied egg-rr 78.6%

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

   8. Taylor expanded in eps around 0

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

   10. Simplified 89.3%

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

4. Final simplification 94.8%

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

Alternative 5: 79.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ -1.0 eps))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      4.0)
   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 * (-1.0 + eps)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * (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 * ((-1.0d0) + eps)))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) <= 4.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (x * (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 * (-1.0 + eps)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (-1.0 + eps)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0:
		tmp = 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(-1.0 + eps)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 4.0)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(x * 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 * (-1.0 + eps)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0)
		tmp = 1.0;
	else
		tmp = 0.5 * (x * (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[(-1.0 + eps), $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], 1.0, 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 55.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 70.7%

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

   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. Simplified 86.8%

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

   7. Applied egg-rr 78.4%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 86.8

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

   10. Simplified 86.8%

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

4. Final simplification 77.9%

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

Alternative 6: 99.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

3. Taylor expanded in eps around 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. Simplified 99.7%

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

Alternative 7: 87.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1600000000000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \varepsilon, 0\right), -x\right)}{\varepsilon}, 1\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+147}:\\ \;\;\;\;\left(0.25 \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\varepsilon + 1, \varepsilon + \frac{-1}{\varepsilon}, \left(1 - \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1600000000000.0)
   (fma 0.5 (* x (/ (* eps (fma eps (fma x eps 0.0) (- x))) eps)) 1.0)
   (if (<= x 8e+147)
     (*
      (* 0.25 (* x x))
      (fma
       (+ eps 1.0)
       (+ eps (/ -1.0 eps))
       (* (- 1.0 eps) (- (/ 1.0 eps) eps))))
     (* 0.5 (* x (* x (* eps eps)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1600000000000.0) {
		tmp = fma(0.5, (x * ((eps * fma(eps, fma(x, eps, 0.0), -x)) / eps)), 1.0);
	} else if (x <= 8e+147) {
		tmp = (0.25 * (x * x)) * fma((eps + 1.0), (eps + (-1.0 / eps)), ((1.0 - eps) * ((1.0 / eps) - eps)));
	} else {
		tmp = 0.5 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1600000000000.0)
		tmp = fma(0.5, Float64(x * Float64(Float64(eps * fma(eps, fma(x, eps, 0.0), Float64(-x))) / eps)), 1.0);
	elseif (x <= 8e+147)
		tmp = Float64(Float64(0.25 * Float64(x * x)) * fma(Float64(eps + 1.0), Float64(eps + Float64(-1.0 / eps)), Float64(Float64(1.0 - eps) * Float64(Float64(1.0 / eps) - eps))));
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1600000000000.0], N[(0.5 * N[(x * N[(N[(eps * N[(eps * N[(x * eps + 0.0), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 8e+147], N[(N[(0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(eps + 1.0), $MachinePrecision] * N[(eps + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - eps), $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 1.6e12

   1. Initial program 64.8%

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

   3. Taylor expanded in 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. Simplified 87.5%

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

   7. Applied egg-rr 87.5%

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

   8. Taylor expanded in eps around 0

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

   10. Simplified 90.8%

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

   if 1.6e12 < x < 7.9999999999999998e147

   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. Simplified 33.8%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

   8. Simplified 85.3%

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

   if 7.9999999999999998e147 < 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. Simplified 50.5%

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

   7. Applied egg-rr 49.7%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 80.4

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

   10. Simplified 80.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1600000000000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \varepsilon, 0\right), -x\right)}{\varepsilon}, 1\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+147}:\\ \;\;\;\;\left(0.25 \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\varepsilon + 1, \varepsilon + \frac{-1}{\varepsilon}, \left(1 - \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_0, 1\right)\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+108}:\\ \;\;\;\;\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x (* eps eps)))))
   (if (<= x -4.6e-223)
     (fma 0.5 t_0 1.0)
     (if (<= x 2.25e-27)
       (fma 0.5 (* x (* eps (* x eps))) 1.0)
       (if (<= x 8.2e+108)
         (* (* 0.08333333333333333 (* eps (* eps eps))) (* x (* x x)))
         (* 0.5 t_0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * (eps * eps));
	double tmp;
	if (x <= -4.6e-223) {
		tmp = fma(0.5, t_0, 1.0);
	} else if (x <= 2.25e-27) {
		tmp = fma(0.5, (x * (eps * (x * eps))), 1.0);
	} else if (x <= 8.2e+108) {
		tmp = (0.08333333333333333 * (eps * (eps * eps))) * (x * (x * x));
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * Float64(eps * eps)))
	tmp = 0.0
	if (x <= -4.6e-223)
		tmp = fma(0.5, t_0, 1.0);
	elseif (x <= 2.25e-27)
		tmp = fma(0.5, Float64(x * Float64(eps * Float64(x * eps))), 1.0);
	elseif (x <= 8.2e+108)
		tmp = Float64(Float64(0.08333333333333333 * Float64(eps * Float64(eps * eps))) * Float64(x * Float64(x * x)));
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.6e-223], N[(0.5 * t$95$0 + 1.0), $MachinePrecision], If[LessEqual[x, 2.25e-27], N[(0.5 * N[(x * N[(eps * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 8.2e+108], N[(N[(0.08333333333333333 * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.5999999999999999e-223

   1. Initial program 76.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. Simplified 90.5%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 90.5

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

   8. Simplified 90.5%

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

   if -4.5999999999999999e-223 < x < 2.2500000000000001e-27

   1. Initial program 49.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. Simplified 86.7%

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

   7. Applied egg-rr 92.8%

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

   8. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \frac{x}{\varepsilon} \cdot \color{blue}{0}\right)\right)\right), 1\right) \]

      6. mul0-rgt N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \color{blue}{0}\right)\right)\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(x + 0\right)\right)}\right), 1\right) \]

      8. lower-+.f64 92.8

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

   10. Simplified 92.8%

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

   if 2.2500000000000001e-27 < x < 8.1999999999999998e108

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\mathsf{neg}\left(\left(1 + \varepsilon\right) \cdot x\right)}}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right) + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\mathsf{neg}\left(\left(1 + \varepsilon\right) \cdot x\right)}}{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(x, \left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\mathsf{neg}\left(\left(1 + \varepsilon\right) \cdot x\right)}}{2} \]

   5. Simplified 27.6%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right) \cdot \mathsf{fma}\left(x \cdot 0.16666666666666666, -1 + \varepsilon, 0.5\right), -1 + \varepsilon\right), 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. cube-mult N/A

         \[\leadsto \left(\frac{1}{12} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \]

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

          \[\leadsto \left(\frac{1}{12} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

      13. lower-*.f64 56.6

          \[\leadsto \left(0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Simplified 56.6%

      \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if 8.1999999999999998e108 < 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. Simplified 50.6%

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

   7. Applied egg-rr 49.8%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 80.8

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

   10. Simplified 80.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+108}:\\ \;\;\;\;\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \varepsilon, 0\right), -x\right)}{\varepsilon}, 1\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+108}:\\ \;\;\;\;\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3.3e-6)
   (fma 0.5 (* x (/ (* eps (fma eps (fma x eps 0.0) (- x))) eps)) 1.0)
   (if (<= x 8.2e+108)
     (* (* 0.08333333333333333 (* eps (* eps eps))) (* x (* x x)))
     (* 0.5 (* x (* x (* eps eps)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.3e-6) {
		tmp = fma(0.5, (x * ((eps * fma(eps, fma(x, eps, 0.0), -x)) / eps)), 1.0);
	} else if (x <= 8.2e+108) {
		tmp = (0.08333333333333333 * (eps * (eps * eps))) * (x * (x * x));
	} else {
		tmp = 0.5 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 3.3e-6)
		tmp = fma(0.5, Float64(x * Float64(Float64(eps * fma(eps, fma(x, eps, 0.0), Float64(-x))) / eps)), 1.0);
	elseif (x <= 8.2e+108)
		tmp = Float64(Float64(0.08333333333333333 * Float64(eps * Float64(eps * eps))) * Float64(x * Float64(x * x)));
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 3.3e-6], N[(0.5 * N[(x * N[(N[(eps * N[(eps * N[(x * eps + 0.0), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 8.2e+108], N[(N[(0.08333333333333333 * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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 < 3.30000000000000017e-6

   1. Initial program 64.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. Simplified 87.7%

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

   7. Applied egg-rr 87.8%

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

   8. Taylor expanded in eps around 0

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

   10. Simplified 91.1%

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

   if 3.30000000000000017e-6 < x < 8.1999999999999998e108

   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

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\mathsf{neg}\left(\left(1 + \varepsilon\right) \cdot x\right)}}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right) + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\mathsf{neg}\left(\left(1 + \varepsilon\right) \cdot x\right)}}{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(x, \left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\mathsf{neg}\left(\left(1 + \varepsilon\right) \cdot x\right)}}{2} \]

   5. Simplified 24.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right) \cdot \mathsf{fma}\left(x \cdot 0.16666666666666666, -1 + \varepsilon, 0.5\right), -1 + \varepsilon\right), 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. cube-mult N/A

         \[\leadsto \left(\frac{1}{12} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \]

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

          \[\leadsto \left(\frac{1}{12} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

      13. lower-*.f64 57.4

          \[\leadsto \left(0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Simplified 57.4%

      \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if 8.1999999999999998e108 < 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. Simplified 50.6%

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

   7. Applied egg-rr 49.8%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 80.8

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

   10. Simplified 80.8%

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

Alternative 10: 83.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ t_1 := \mathsf{fma}\left(0.5, t\_0, 1\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot \varepsilon\right) \cdot \left(0.5 \cdot x\right), \varepsilon, 1\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x (* eps eps)))) (t_1 (fma 0.5 t_0 1.0)))
   (if (<= x -1.45e-223)
     t_1
     (if (<= x 1.7e-163)
       (fma (* (* x eps) (* 0.5 x)) eps 1.0)
       (if (<= x 3.3e-6) t_1 (* 0.5 t_0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * (eps * eps));
	double t_1 = fma(0.5, t_0, 1.0);
	double tmp;
	if (x <= -1.45e-223) {
		tmp = t_1;
	} else if (x <= 1.7e-163) {
		tmp = fma(((x * eps) * (0.5 * x)), eps, 1.0);
	} else if (x <= 3.3e-6) {
		tmp = t_1;
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * Float64(eps * eps)))
	t_1 = fma(0.5, t_0, 1.0)
	tmp = 0.0
	if (x <= -1.45e-223)
		tmp = t_1;
	elseif (x <= 1.7e-163)
		tmp = fma(Float64(Float64(x * eps) * Float64(0.5 * x)), eps, 1.0);
	elseif (x <= 3.3e-6)
		tmp = t_1;
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * t$95$0 + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.45e-223], t$95$1, If[LessEqual[x, 1.7e-163], N[(N[(N[(x * eps), $MachinePrecision] * N[(0.5 * x), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision], If[LessEqual[x, 3.3e-6], t$95$1, N[(0.5 * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e-223 or 1.70000000000000007e-163 < x < 3.30000000000000017e-6

   1. Initial program 68.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. Simplified 90.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 90.8

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

   8. Simplified 90.8%

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

   if -1.45e-223 < x < 1.70000000000000007e-163

   1. Initial program 50.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. Simplified 79.4%

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

   7. Applied egg-rr 97.9%

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

   8. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \frac{x}{\varepsilon} \cdot \color{blue}{0}\right)\right)\right), 1\right) \]

      6. mul0-rgt N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \color{blue}{0}\right)\right)\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(x + 0\right)\right)}\right), 1\right) \]

      8. lower-+.f64 97.9

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

   10. Simplified 97.9%

       \[\leadsto \mathsf{fma}\left(0.5, x \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + 0\right)\right)\right)}, 1\right) \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(x + 0\right)}\right)\right)\right) + 1 \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(x \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(x + 0\right)\right)}\right)\right) + 1 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + 0\right)\right)\right)}\right) + 1 \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + 0\right)\right)\right)\right)} + 1 \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + 0\right)\right)\right)\right)} + 1 \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + 0\right)\right)\right)} + 1 \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + 0\right)\right)\right)} + 1 \]

       8. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(x + 0\right)\right) \cdot \varepsilon\right)} + 1 \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \left(\color{blue}{\left(\varepsilon \cdot \left(x + 0\right)\right)} \cdot \varepsilon\right) + 1 \]

       10. lift-+.f64 N/A

           \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \left(\left(\varepsilon \cdot \color{blue}{\left(x + 0\right)}\right) \cdot \varepsilon\right) + 1 \]

       11. +-rgt-identity N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. lower-fma.f64 N/A

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

   12. Applied egg-rr 97.9%

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

   if 3.30000000000000017e-6 < 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. Simplified 45.1%

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

   7. Applied egg-rr 44.4%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 72.5

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

   10. Simplified 72.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot \varepsilon\right) \cdot \left(0.5 \cdot x\right), \varepsilon, 1\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ t_1 := \mathsf{fma}\left(0.5, t\_0, 1\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-239}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x (* eps eps)))) (t_1 (fma 0.5 t_0 1.0)))
   (if (<= x -1.45e-223)
     t_1
     (if (<= x 1.85e-239) 1.0 (if (<= x 3.3e-6) t_1 (* 0.5 t_0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * (eps * eps));
	double t_1 = fma(0.5, t_0, 1.0);
	double tmp;
	if (x <= -1.45e-223) {
		tmp = t_1;
	} else if (x <= 1.85e-239) {
		tmp = 1.0;
	} else if (x <= 3.3e-6) {
		tmp = t_1;
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * Float64(eps * eps)))
	t_1 = fma(0.5, t_0, 1.0)
	tmp = 0.0
	if (x <= -1.45e-223)
		tmp = t_1;
	elseif (x <= 1.85e-239)
		tmp = 1.0;
	elseif (x <= 3.3e-6)
		tmp = t_1;
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * t$95$0 + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.45e-223], t$95$1, If[LessEqual[x, 1.85e-239], 1.0, If[LessEqual[x, 3.3e-6], t$95$1, N[(0.5 * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e-223 or 1.85000000000000008e-239 < x < 3.30000000000000017e-6

   1. Initial program 65.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 91.1%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 91.1

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

   8. Simplified 91.1%

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

   if -1.45e-223 < x < 1.85000000000000008e-239

   1. Initial program 57.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

   if 3.30000000000000017e-6 < 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. Simplified 45.1%

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

   7. Applied egg-rr 44.4%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 72.5

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

   10. Simplified 72.5%

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

Alternative 12: 83.4% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_0, 1\right)\\ \mathbf{elif}\;x \leq 1600000000000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x (* eps eps)))))
   (if (<= x -4.6e-223)
     (fma 0.5 t_0 1.0)
     (if (<= x 1600000000000.0)
       (fma 0.5 (* x (* eps (* x eps))) 1.0)
       (* 0.5 t_0)))))

C

double code(double x, double eps) {
	double t_0 = x * (x * (eps * eps));
	double tmp;
	if (x <= -4.6e-223) {
		tmp = fma(0.5, t_0, 1.0);
	} else if (x <= 1600000000000.0) {
		tmp = fma(0.5, (x * (eps * (x * eps))), 1.0);
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * Float64(eps * eps)))
	tmp = 0.0
	if (x <= -4.6e-223)
		tmp = fma(0.5, t_0, 1.0);
	elseif (x <= 1600000000000.0)
		tmp = fma(0.5, Float64(x * Float64(eps * Float64(x * eps))), 1.0);
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.6e-223], N[(0.5 * t$95$0 + 1.0), $MachinePrecision], If[LessEqual[x, 1600000000000.0], N[(0.5 * N[(x * N[(eps * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5999999999999999e-223

   1. Initial program 76.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. Simplified 90.5%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 90.5

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

   8. Simplified 90.5%

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

   if -4.5999999999999999e-223 < x < 1.6e12

   1. Initial program 52.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 84.4%

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

   7. Applied egg-rr 89.9%

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

   8. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \frac{x}{\varepsilon} \cdot \color{blue}{0}\right)\right)\right), 1\right) \]

      6. mul0-rgt N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \color{blue}{0}\right)\right)\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(x + 0\right)\right)}\right), 1\right) \]

      8. lower-+.f64 89.9

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

   10. Simplified 89.9%

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

   if 1.6e12 < 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. Simplified 43.5%

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

   7. Applied egg-rr 42.8%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 72.4

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

   10. Simplified 72.4%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 1600000000000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.4% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.6) (fma x (fma x (fma x -0.16666666666666666 0.5) -1.0) 1.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.6) {
		tmp = fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.6)
		tmp = fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.6], N[(x * N[(x * N[(x * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 64.2%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   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. Simplified 99.5%

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

   6. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lower-neg.f64 79.9

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

   8. Simplified 79.9%

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

   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 74.0

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

   11. Simplified 74.0%

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

   if 1.6000000000000001 < 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. Simplified 44.5%

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

   7. Applied egg-rr 43.7%

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

   8. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{0}\right)}{\varepsilon} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \left(x \cdot 0\right)}{\varepsilon} \]

      8. mul0-rgt N/A

         \[\leadsto \frac{\left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{0}}{\varepsilon} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot 0\right)}}{\varepsilon} \]

      10. metadata-eval N/A

          \[\leadsto \frac{x \cdot \color{blue}{0}}{\varepsilon} \]

      11. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x}{\varepsilon} \cdot 0} \]

      12. mul0-rgt 53.2

          \[\leadsto \color{blue}{0} \]

   10. Simplified 53.2%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 64.7% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1600000000000:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1600000000000.0) (fma x (fma 0.5 x -1.0) 1.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1600000000000.0) {
		tmp = fma(x, fma(0.5, x, -1.0), 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1600000000000.0)
		tmp = fma(x, fma(0.5, x, -1.0), 1.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1600000000000.0], N[(x * N[(0.5 * x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.6e12

   1. Initial program 64.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. Simplified 99.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Simplified 87.2%

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

   9. Taylor expanded in eps around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. distribute-lft-in N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 69.7

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

   11. Simplified 69.7%

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

   if 1.6e12 < 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. Simplified 43.5%

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

   7. Applied egg-rr 42.8%

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

   8. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{0}\right)}{\varepsilon} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \left(x \cdot 0\right)}{\varepsilon} \]

      8. mul0-rgt N/A

         \[\leadsto \frac{\left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{0}}{\varepsilon} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot 0\right)}}{\varepsilon} \]

      10. metadata-eval N/A

          \[\leadsto \frac{x \cdot \color{blue}{0}}{\varepsilon} \]

      11. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x}{\varepsilon} \cdot 0} \]

      12. mul0-rgt 55.3

          \[\leadsto \color{blue}{0} \]

   10. Simplified 55.3%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 57.9% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 1.0) (- 1.0 x) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 - x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 1.0d0 - x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 - x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 - x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 - x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 - x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.0], N[(1.0 - x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 64.2%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   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. Simplified 99.5%

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

   6. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lower-neg.f64 79.9

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

   8. Simplified 79.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + -1 \cdot x} \]
   10. Step-by-step derivation

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 58.0

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

   11. Simplified 58.0%

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

   if 1 < 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. Simplified 44.5%

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

   7. Applied egg-rr 43.7%

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

   8. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{0}\right)}{\varepsilon} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \left(x \cdot 0\right)}{\varepsilon} \]

      8. mul0-rgt N/A

         \[\leadsto \frac{\left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{0}}{\varepsilon} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot 0\right)}}{\varepsilon} \]

      10. metadata-eval N/A

          \[\leadsto \frac{x \cdot \color{blue}{0}}{\varepsilon} \]

      11. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x}{\varepsilon} \cdot 0} \]

      12. mul0-rgt 53.2

          \[\leadsto \color{blue}{0} \]

   10. Simplified 53.2%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 57.5% accurate, 38.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1600000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 1600000000000.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1600000000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1600000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1600000000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1600000000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1600000000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1600000000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1600000000000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.6e12

   1. Initial program 64.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 56.8%

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

   if 1.6e12 < 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. Simplified 43.5%

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

   7. Applied egg-rr 42.8%

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

   8. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{0}\right)}{\varepsilon} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \left(x \cdot 0\right)}{\varepsilon} \]

      8. mul0-rgt N/A

         \[\leadsto \frac{\left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{0}}{\varepsilon} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot 0\right)}}{\varepsilon} \]

      10. metadata-eval N/A

          \[\leadsto \frac{x \cdot \color{blue}{0}}{\varepsilon} \]

      11. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x}{\varepsilon} \cdot 0} \]

      12. mul0-rgt 55.3

          \[\leadsto \color{blue}{0} \]

   10. Simplified 55.3%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 16.0% accurate, 273.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 75.4%

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

3. Taylor expanded in x around 0

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

5. Simplified 74.3%

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

7. Applied egg-rr 74.1%

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

8. Taylor expanded in eps around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{0}\right)}{\varepsilon} \]

   7. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \left(x \cdot 0\right)}{\varepsilon} \]

   8. mul0-rgt N/A

      \[\leadsto \frac{\left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{0}}{\varepsilon} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot 0\right)}}{\varepsilon} \]

   10. metadata-eval N/A

       \[\leadsto \frac{x \cdot \color{blue}{0}}{\varepsilon} \]

   11. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{x}{\varepsilon} \cdot 0} \]

   12. mul0-rgt 18.3

       \[\leadsto \color{blue}{0} \]

10. Simplified 18.3%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))