NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 99.5%

Time: 9.5s

Alternatives: 11

Speedup: 8.0×

• 38.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -3.3480925953118086e+239
  2. eps = 8.752771930348391e-234

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-1 - \varepsilon\right) \cdot x}\\ t_1 := 1 + {\varepsilon}^{-1}\\ t_2 := e^{-x}\\ \mathbf{if}\;\frac{t\_1 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot t\_0}{2} \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \left(1 + x\right) - -1, t\_2 \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot e^{x \cdot \varepsilon} - -1 \cdot t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* (- -1.0 eps) x)))
        (t_1 (+ 1.0 (pow eps -1.0)))
        (t_2 (exp (- x))))
   (if (<=
        (/
         (- (* t_1 (exp (* (+ -1.0 eps) x))) (* (- (pow eps -1.0) 1.0) t_0))
         2.0)
        1.1)
     (* (fma t_2 (- (+ 1.0 x) -1.0) (* t_2 x)) 0.5)
     (/ (- (* t_1 (exp (* x eps))) (* -1.0 t_0)) 2.0))))

C

double code(double x, double eps) {
	double t_0 = exp(((-1.0 - eps) * x));
	double t_1 = 1.0 + pow(eps, -1.0);
	double t_2 = exp(-x);
	double tmp;
	if ((((t_1 * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * t_0)) / 2.0) <= 1.1) {
		tmp = fma(t_2, ((1.0 + x) - -1.0), (t_2 * x)) * 0.5;
	} else {
		tmp = ((t_1 * exp((x * eps))) - (-1.0 * t_0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = exp(Float64(Float64(-1.0 - eps) * x))
	t_1 = Float64(1.0 + (eps ^ -1.0))
	t_2 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * t_0)) / 2.0) <= 1.1)
		tmp = Float64(fma(t_2, Float64(Float64(1.0 + x) - -1.0), Float64(t_2 * x)) * 0.5);
	else
		tmp = Float64(Float64(Float64(t_1 * exp(Float64(x * eps))) - Float64(-1.0 * t_0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$1 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 1.1], N[(N[(t$95$2 * N[(N[(1.0 + x), $MachinePrecision] - -1.0), $MachinePrecision] + N[(t$95$2 * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$1 * N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(-1.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.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))))) #s(literal 2 binary64)) < 1.1000000000000001

   1. Initial program 54.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

   if 1.1000000000000001 < (/.f64 (-.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))))) #s(literal 2 binary64))

   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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(e^{-x}, \left(1 + x\right) - -1, e^{-x} \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{x \cdot \varepsilon} - -1 \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.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))))) #s(literal 2 binary64)) < 2

   1. Initial program 54.3%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.4%

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

   if 2 < (/.f64 (-.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))))) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 11.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.8%

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

   10. Applied rewrites 85.6%

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

4. Final simplification 93.3%

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

Alternative 3: 78.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.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))))) #s(literal 2 binary64)) < 2

   1. Initial program 54.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.7%

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

   if 2 < (/.f64 (-.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))))) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 11.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.8%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 84.2%

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

4. Final simplification 78.4%

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

Alternative 4: 71.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.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))))) #s(literal 2 binary64)) < 2

   1. Initial program 54.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.7%

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

   if 2 < (/.f64 (-.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))))) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 11.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.8%

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

   10. Applied rewrites 84.2%

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

   11. Taylor expanded in eps around inf

       \[\leadsto \frac{{\varepsilon}^{2} \cdot {x}^{\color{blue}{2}}}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 75.0%

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

4. Final simplification 74.3%

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

Alternative 5: 84.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.08333333333333333, \varepsilon \cdot \varepsilon, 0.16666666666666666\right)\\ \mathbf{if}\;x \leq -2.35 \cdot 10^{-132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.25, x, \mathsf{fma}\left(-0.6666666666666666, \varepsilon \cdot \varepsilon, 0.6666666666666666\right)\right), x, \varepsilon \cdot \varepsilon - 1\right), x \cdot x, 2\right)}{2}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\varepsilon + 1\right), \varepsilon - 1, 2\right)}{2}\\ \mathbf{elif}\;x \leq 125000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot t\_0 - 0.6666666666666666, \varepsilon \cdot \varepsilon, \mathsf{fma}\left(-0.125, x, 0.3333333333333333\right)\right), x, \varepsilon \cdot \varepsilon - 0.5\right), x \cdot x, 1\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma 0.08333333333333333 (* eps eps) 0.16666666666666666)))
   (if (<= x -2.35e-132)
     (/
      (fma
       (fma
        (fma
         (- (* t_0 (* eps eps)) 0.25)
         x
         (fma -0.6666666666666666 (* eps eps) 0.6666666666666666))
        x
        (- (* eps eps) 1.0))
       (* x x)
       2.0)
      2.0)
     (if (<= x 1.02e-156)
       (/ (fma (* (* x x) (+ eps 1.0)) (- eps 1.0) 2.0) 2.0)
       (if (<= x 125000000000.0)
         (/
          (-
           (fma
            (fma
             (fma
              (- (* x t_0) 0.6666666666666666)
              (* eps eps)
              (fma -0.125 x 0.3333333333333333))
             x
             (- (* eps eps) 0.5))
            (* x x)
            1.0)
           (/ (fma -1.0 x -1.0) (exp x)))
          2.0)
         (/ (* (* (* eps eps) x) x) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = fma(0.08333333333333333, (eps * eps), 0.16666666666666666);
	double tmp;
	if (x <= -2.35e-132) {
		tmp = fma(fma(fma(((t_0 * (eps * eps)) - 0.25), x, fma(-0.6666666666666666, (eps * eps), 0.6666666666666666)), x, ((eps * eps) - 1.0)), (x * x), 2.0) / 2.0;
	} else if (x <= 1.02e-156) {
		tmp = fma(((x * x) * (eps + 1.0)), (eps - 1.0), 2.0) / 2.0;
	} else if (x <= 125000000000.0) {
		tmp = (fma(fma(fma(((x * t_0) - 0.6666666666666666), (eps * eps), fma(-0.125, x, 0.3333333333333333)), x, ((eps * eps) - 0.5)), (x * x), 1.0) - (fma(-1.0, x, -1.0) / exp(x))) / 2.0;
	} else {
		tmp = (((eps * eps) * x) * x) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(0.08333333333333333, Float64(eps * eps), 0.16666666666666666)
	tmp = 0.0
	if (x <= -2.35e-132)
		tmp = Float64(fma(fma(fma(Float64(Float64(t_0 * Float64(eps * eps)) - 0.25), x, fma(-0.6666666666666666, Float64(eps * eps), 0.6666666666666666)), x, Float64(Float64(eps * eps) - 1.0)), Float64(x * x), 2.0) / 2.0);
	elseif (x <= 1.02e-156)
		tmp = Float64(fma(Float64(Float64(x * x) * Float64(eps + 1.0)), Float64(eps - 1.0), 2.0) / 2.0);
	elseif (x <= 125000000000.0)
		tmp = Float64(Float64(fma(fma(fma(Float64(Float64(x * t_0) - 0.6666666666666666), Float64(eps * eps), fma(-0.125, x, 0.3333333333333333)), x, Float64(Float64(eps * eps) - 0.5)), Float64(x * x), 1.0) - Float64(fma(-1.0, x, -1.0) / exp(x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(eps * eps) * x) * x) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(0.08333333333333333 * N[(eps * eps), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]}, If[LessEqual[x, -2.35e-132], N[(N[(N[(N[(N[(N[(t$95$0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * x + N[(-0.6666666666666666 * N[(eps * eps), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.02e-156], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision] * N[(eps - 1.0), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 125000000000.0], N[(N[(N[(N[(N[(N[(N[(x * t$95$0), $MachinePrecision] - 0.6666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + N[(-0.125 * x + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(-1.0 * x + -1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(eps * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.3500000000000001e-132

   1. Initial program 79.8%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 29.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{{x}^{2} \cdot \left(\left(x \cdot \left(\frac{2}{3} + \left(\frac{-2}{3} \cdot {\varepsilon}^{2} + x \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{6} + \frac{1}{12} \cdot {\varepsilon}^{2}\right) - \frac{1}{4}\right)\right)\right) + {\varepsilon}^{2}\right) - 1\right)}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, \varepsilon \cdot \varepsilon, 0.16666666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.25, x, \mathsf{fma}\left(-0.6666666666666666, \varepsilon \cdot \varepsilon, 0.6666666666666666\right)\right), x, \varepsilon \cdot \varepsilon - 1\right), \color{blue}{x \cdot x}, 2\right)}{2} \]

   if -2.3500000000000001e-132 < x < 1.02e-156

   1. Initial program 51.6%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.2%

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

   10. Applied rewrites 92.0%

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

   if 1.02e-156 < x < 1.25e11

   1. Initial program 63.4%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 76.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + {x}^{2} \cdot \left(\left(x \cdot \left(\frac{1}{3} + \left(\frac{-2}{3} \cdot {\varepsilon}^{2} + x \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{6} + \frac{1}{12} \cdot {\varepsilon}^{2}\right) - \frac{1}{8}\right)\right)\right) + {\varepsilon}^{2}\right) - \frac{1}{2}\right)\right) - \frac{\color{blue}{\mathsf{fma}\left(-1, x, -1\right)}}{e^{x}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, \varepsilon \cdot \varepsilon, 0.16666666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.125, x, \mathsf{fma}\left(-0.6666666666666666, \varepsilon \cdot \varepsilon, 0.3333333333333333\right)\right), x, \varepsilon \cdot \varepsilon - 0.5\right), x \cdot x, 1\right) - \frac{\color{blue}{\mathsf{fma}\left(-1, x, -1\right)}}{e^{x}}}{2} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + \left(\frac{-1}{8} \cdot x + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{12} \cdot \left({\varepsilon}^{2} \cdot x\right) + \frac{1}{6} \cdot x\right) - \frac{2}{3}\right)\right), x, \varepsilon \cdot \varepsilon - \frac{1}{2}\right), x \cdot x, 1\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 93.3%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.08333333333333333, \varepsilon \cdot \varepsilon, 0.16666666666666666\right) - 0.6666666666666666, \varepsilon \cdot \varepsilon, \mathsf{fma}\left(-0.125, x, 0.3333333333333333\right)\right), x, \varepsilon \cdot \varepsilon - 0.5\right), x \cdot x, 1\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}{2} \]

   if 1.25e11 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 8.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.0%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 67.7%

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

Alternative 6: 84.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, \varepsilon \cdot \varepsilon, 0.16666666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.25, x, \mathsf{fma}\left(-0.6666666666666666, \varepsilon \cdot \varepsilon, 0.6666666666666666\right)\right), x, \varepsilon \cdot \varepsilon - 1\right), x \cdot x, 2\right)}{2}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\varepsilon + 1\right), \varepsilon - 1, 2\right)}{2}\\ \mathbf{elif}\;x \leq 38:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\varepsilon + 1\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon + 1}, x \cdot x, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.35e-132)
   (/
    (fma
     (fma
      (fma
       (-
        (*
         (fma 0.08333333333333333 (* eps eps) 0.16666666666666666)
         (* eps eps))
        0.25)
       x
       (fma -0.6666666666666666 (* eps eps) 0.6666666666666666))
      x
      (- (* eps eps) 1.0))
     (* x x)
     2.0)
    2.0)
   (if (<= x 1.02e-156)
     (/ (fma (* (* x x) (+ eps 1.0)) (- eps 1.0) 2.0) 2.0)
     (if (<= x 38.0)
       (/
        (fma (/ (* (+ eps 1.0) (fma eps eps -1.0)) (+ eps 1.0)) (* x x) 2.0)
        2.0)
       (/ (* (* (* eps eps) x) x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.35e-132) {
		tmp = fma(fma(fma(((fma(0.08333333333333333, (eps * eps), 0.16666666666666666) * (eps * eps)) - 0.25), x, fma(-0.6666666666666666, (eps * eps), 0.6666666666666666)), x, ((eps * eps) - 1.0)), (x * x), 2.0) / 2.0;
	} else if (x <= 1.02e-156) {
		tmp = fma(((x * x) * (eps + 1.0)), (eps - 1.0), 2.0) / 2.0;
	} else if (x <= 38.0) {
		tmp = fma((((eps + 1.0) * fma(eps, eps, -1.0)) / (eps + 1.0)), (x * x), 2.0) / 2.0;
	} else {
		tmp = (((eps * eps) * x) * x) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.35e-132)
		tmp = Float64(fma(fma(fma(Float64(Float64(fma(0.08333333333333333, Float64(eps * eps), 0.16666666666666666) * Float64(eps * eps)) - 0.25), x, fma(-0.6666666666666666, Float64(eps * eps), 0.6666666666666666)), x, Float64(Float64(eps * eps) - 1.0)), Float64(x * x), 2.0) / 2.0);
	elseif (x <= 1.02e-156)
		tmp = Float64(fma(Float64(Float64(x * x) * Float64(eps + 1.0)), Float64(eps - 1.0), 2.0) / 2.0);
	elseif (x <= 38.0)
		tmp = Float64(fma(Float64(Float64(Float64(eps + 1.0) * fma(eps, eps, -1.0)) / Float64(eps + 1.0)), Float64(x * x), 2.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(eps * eps) * x) * x) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.35e-132], N[(N[(N[(N[(N[(N[(N[(0.08333333333333333 * N[(eps * eps), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * x + N[(-0.6666666666666666 * N[(eps * eps), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.02e-156], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision] * N[(eps - 1.0), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 38.0], N[(N[(N[(N[(N[(eps + 1.0), $MachinePrecision] * N[(eps * eps + -1.0), $MachinePrecision]), $MachinePrecision] / N[(eps + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(eps * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.3500000000000001e-132

   1. Initial program 79.8%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 29.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{{x}^{2} \cdot \left(\left(x \cdot \left(\frac{2}{3} + \left(\frac{-2}{3} \cdot {\varepsilon}^{2} + x \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{6} + \frac{1}{12} \cdot {\varepsilon}^{2}\right) - \frac{1}{4}\right)\right)\right) + {\varepsilon}^{2}\right) - 1\right)}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, \varepsilon \cdot \varepsilon, 0.16666666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.25, x, \mathsf{fma}\left(-0.6666666666666666, \varepsilon \cdot \varepsilon, 0.6666666666666666\right)\right), x, \varepsilon \cdot \varepsilon - 1\right), \color{blue}{x \cdot x}, 2\right)}{2} \]

   if -2.3500000000000001e-132 < x < 1.02e-156

   1. Initial program 51.6%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.2%

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

   10. Applied rewrites 92.0%

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

   if 1.02e-156 < x < 38

   1. Initial program 61.7%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 79.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.0%

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

   10. Applied rewrites 94.5%

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

   if 38 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 8.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.9%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 66.0%

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

Alternative 7: 83.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{\left(\varepsilon + 1\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon + 1}, x \cdot x, 2\right)}{2}\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\varepsilon + 1\right), \varepsilon - 1, 2\right)}{2}\\ \mathbf{elif}\;x \leq 38:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (/
          (fma (/ (* (+ eps 1.0) (fma eps eps -1.0)) (+ eps 1.0)) (* x x) 2.0)
          2.0)))
   (if (<= x -2.3e-132)
     t_0
     (if (<= x 1.02e-156)
       (/ (fma (* (* x x) (+ eps 1.0)) (- eps 1.0) 2.0) 2.0)
       (if (<= x 38.0) t_0 (/ (* (* (* eps eps) x) x) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = fma((((eps + 1.0) * fma(eps, eps, -1.0)) / (eps + 1.0)), (x * x), 2.0) / 2.0;
	double tmp;
	if (x <= -2.3e-132) {
		tmp = t_0;
	} else if (x <= 1.02e-156) {
		tmp = fma(((x * x) * (eps + 1.0)), (eps - 1.0), 2.0) / 2.0;
	} else if (x <= 38.0) {
		tmp = t_0;
	} else {
		tmp = (((eps * eps) * x) * x) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(fma(Float64(Float64(Float64(eps + 1.0) * fma(eps, eps, -1.0)) / Float64(eps + 1.0)), Float64(x * x), 2.0) / 2.0)
	tmp = 0.0
	if (x <= -2.3e-132)
		tmp = t_0;
	elseif (x <= 1.02e-156)
		tmp = Float64(fma(Float64(Float64(x * x) * Float64(eps + 1.0)), Float64(eps - 1.0), 2.0) / 2.0);
	elseif (x <= 38.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(eps * eps) * x) * x) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(N[(eps + 1.0), $MachinePrecision] * N[(eps * eps + -1.0), $MachinePrecision]), $MachinePrecision] / N[(eps + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2.3e-132], t$95$0, If[LessEqual[x, 1.02e-156], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision] * N[(eps - 1.0), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 38.0], t$95$0, N[(N[(N[(N[(eps * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.30000000000000003e-132 or 1.02e-156 < x < 38

   1. Initial program 72.7%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 48.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.1%

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

   10. Applied rewrites 91.8%

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

   if -2.30000000000000003e-132 < x < 1.02e-156

   1. Initial program 51.6%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.2%

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

   10. Applied rewrites 92.0%

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

   if 38 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 8.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.9%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 66.0%

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

Alternative 8: 80.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-231}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\varepsilon + 1\right), \varepsilon - 1, 2\right)}{2}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot x, x, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5e-231)
   (/ (fma (* (* x x) (+ eps 1.0)) (- eps 1.0) 2.0) 2.0)
   (if (<= x 1.4)
     (/ (fma (* (fma eps eps -1.0) x) x 2.0) 2.0)
     (/ (* (* (* eps eps) x) x) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5e-231) {
		tmp = fma(((x * x) * (eps + 1.0)), (eps - 1.0), 2.0) / 2.0;
	} else if (x <= 1.4) {
		tmp = fma((fma(eps, eps, -1.0) * x), x, 2.0) / 2.0;
	} else {
		tmp = (((eps * eps) * x) * x) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5e-231)
		tmp = Float64(fma(Float64(Float64(x * x) * Float64(eps + 1.0)), Float64(eps - 1.0), 2.0) / 2.0);
	elseif (x <= 1.4)
		tmp = Float64(fma(Float64(fma(eps, eps, -1.0) * x), x, 2.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(eps * eps) * x) * x) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5e-231], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision] * N[(eps - 1.0), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.4], N[(N[(N[(N[(eps * eps + -1.0), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(eps * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.00000000000000023e-231

   1. Initial program 65.7%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 52.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.3%

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

   10. Applied rewrites 89.0%

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

   if 5.00000000000000023e-231 < x < 1.3999999999999999

   1. Initial program 61.3%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 76.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.3%

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

   10. Applied rewrites 88.8%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 8.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.9%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 66.0%

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

Alternative 9: 81.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot x, x, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.4)
   (/ (fma (* (fma eps eps -1.0) x) x 2.0) 2.0)
   (/ (* (* (* eps eps) x) x) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.4) {
		tmp = fma((fma(eps, eps, -1.0) * x), x, 2.0) / 2.0;
	} else {
		tmp = (((eps * eps) * x) * x) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(fma(Float64(fma(eps, eps, -1.0) * x), x, 2.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(eps * eps) * x) * x) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.4], N[(N[(N[(N[(eps * eps + -1.0), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(eps * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 64.3%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 59.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.3%

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

   10. Applied rewrites 86.3%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 8.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.9%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 66.0%

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

Alternative 10: 81.5% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\varepsilon \cdot \varepsilon\right) \cdot x\\ \mathbf{if}\;x \leq 44:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, x, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* eps eps) x)))
   (if (<= x 44.0) (/ (fma t_0 x 2.0) 2.0) (/ (* t_0 x) 2.0))))

C

double code(double x, double eps) {
	double t_0 = (eps * eps) * x;
	double tmp;
	if (x <= 44.0) {
		tmp = fma(t_0, x, 2.0) / 2.0;
	} else {
		tmp = (t_0 * x) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(eps * eps) * x)
	tmp = 0.0
	if (x <= 44.0)
		tmp = Float64(fma(t_0, x, 2.0) / 2.0);
	else
		tmp = Float64(Float64(t_0 * x) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps * eps), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 44.0], N[(N[(t$95$0 * x + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 44

   1. Initial program 64.3%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 59.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.3%

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

   10. Applied rewrites 86.3%

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

   11. Taylor expanded in eps around inf

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

   13. Applied rewrites 86.0%

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

   if 44 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + \left(x \cdot e^{\mathsf{neg}\left(x\right)} + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + {\varepsilon}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right) - \left(\frac{-1}{24} \cdot \left({x}^{4} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{120} \cdot \left({x}^{5} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot e^{\mathsf{neg}\left(x\right)}\right) + \frac{-1}{6} \cdot \left({x}^{3} \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]

   5. Applied rewrites 8.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(e^{-x} \cdot \mathsf{fma}\left({x}^{4}, 0.041666666666666664, {x}^{5} \cdot 0.008333333333333333\right) - e^{-x} \cdot \mathsf{fma}\left(-0.008333333333333333, {x}^{5}, -0.041666666666666664 \cdot {x}^{4}\right), \varepsilon \cdot \varepsilon, e^{-x} \cdot \mathsf{fma}\left(0.5 \cdot x, x, {x}^{3} \cdot 0.16666666666666666\right)\right) - e^{-x} \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, -0.5 \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \varepsilon, \frac{1 + x}{e^{x}}\right) - \frac{\mathsf{fma}\left(-1, x, -1\right)}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.9%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 66.0%

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

Alternative 11: 43.7% accurate, 273.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 74.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 42.3%

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

Reproduce

herbie shell --seed 2024351 
(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))