NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.3% → 99.8%

Time: 6.3s

Alternatives: 2

Speedup: N/A×

• 38.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -3.2909516262607145e+144
  2. eps = -2.060765254589543e-125

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: 74.3% 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: 99.8% accurate, N/A× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := -1 \cdot \left(x - -1\right)\\ t_1 := {\left(e^{-1}\right)}^{x}\\ \mathbf{if}\;eps\_m \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\frac{t\_0 \cdot \left(-1 \cdot t\_1\right) - t\_0 \cdot t\_1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{-1}\right)}^{\left(\left(1 - eps\_m\right) \cdot x\right)} - \left(-1 \cdot 0 - {\left(e^{-1}\right)}^{\left(\left(eps\_m + 1\right) \cdot x\right)}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* -1.0 (- x -1.0))) (t_1 (pow (exp -1.0) x)))
   (if (<= eps_m 2e-38)
     (/ (- (* t_0 (* -1.0 t_1)) (* t_0 t_1)) 2.0)
     (/
      (-
       (pow (exp -1.0) (* (- 1.0 eps_m) x))
       (- (* -1.0 0.0) (pow (exp -1.0) (* (+ eps_m 1.0) x))))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = -1.0 * (x - -1.0);
	double t_1 = pow(exp(-1.0), x);
	double tmp;
	if (eps_m <= 2e-38) {
		tmp = ((t_0 * (-1.0 * t_1)) - (t_0 * t_1)) / 2.0;
	} else {
		tmp = (pow(exp(-1.0), ((1.0 - eps_m) * x)) - ((-1.0 * 0.0) - pow(exp(-1.0), ((eps_m + 1.0) * x)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) * (x - (-1.0d0))
    t_1 = exp((-1.0d0)) ** x
    if (eps_m <= 2d-38) then
        tmp = ((t_0 * ((-1.0d0) * t_1)) - (t_0 * t_1)) / 2.0d0
    else
        tmp = ((exp((-1.0d0)) ** ((1.0d0 - eps_m) * x)) - (((-1.0d0) * 0.0d0) - (exp((-1.0d0)) ** ((eps_m + 1.0d0) * x)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = -1.0 * (x - -1.0);
	double t_1 = Math.pow(Math.exp(-1.0), x);
	double tmp;
	if (eps_m <= 2e-38) {
		tmp = ((t_0 * (-1.0 * t_1)) - (t_0 * t_1)) / 2.0;
	} else {
		tmp = (Math.pow(Math.exp(-1.0), ((1.0 - eps_m) * x)) - ((-1.0 * 0.0) - Math.pow(Math.exp(-1.0), ((eps_m + 1.0) * x)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = -1.0 * (x - -1.0)
	t_1 = math.pow(math.exp(-1.0), x)
	tmp = 0
	if eps_m <= 2e-38:
		tmp = ((t_0 * (-1.0 * t_1)) - (t_0 * t_1)) / 2.0
	else:
		tmp = (math.pow(math.exp(-1.0), ((1.0 - eps_m) * x)) - ((-1.0 * 0.0) - math.pow(math.exp(-1.0), ((eps_m + 1.0) * x)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(-1.0 * Float64(x - -1.0))
	t_1 = exp(-1.0) ^ x
	tmp = 0.0
	if (eps_m <= 2e-38)
		tmp = Float64(Float64(Float64(t_0 * Float64(-1.0 * t_1)) - Float64(t_0 * t_1)) / 2.0);
	else
		tmp = Float64(Float64((exp(-1.0) ^ Float64(Float64(1.0 - eps_m) * x)) - Float64(Float64(-1.0 * 0.0) - (exp(-1.0) ^ Float64(Float64(eps_m + 1.0) * x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = -1.0 * (x - -1.0);
	t_1 = exp(-1.0) ^ x;
	tmp = 0.0;
	if (eps_m <= 2e-38)
		tmp = ((t_0 * (-1.0 * t_1)) - (t_0 * t_1)) / 2.0;
	else
		tmp = ((exp(-1.0) ^ ((1.0 - eps_m) * x)) - ((-1.0 * 0.0) - (exp(-1.0) ^ ((eps_m + 1.0) * x)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(-1.0 * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[-1.0], $MachinePrecision], x], $MachinePrecision]}, If[LessEqual[eps$95$m, 2e-38], N[(N[(N[(t$95$0 * N[(-1.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Power[N[Exp[-1.0], $MachinePrecision], N[(N[(1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - N[(N[(-1.0 * 0.0), $MachinePrecision] - N[Power[N[Exp[-1.0], $MachinePrecision], N[(N[(eps$95$m + 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.9999999999999999e-38

   1. Initial program 59.8%

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

   3. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. mul-1-neg N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. mul-1-neg N/A

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

      15. exp-prod N/A

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

      16. lower-pow.f64 N/A

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

   5. Applied rewrites 70.6%

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

   if 1.9999999999999999e-38 < eps

   1. Initial program 97.6%

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

   3. Taylor expanded in eps around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. exp-neg N/A

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

      10. *-commutative N/A

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

      11. exp-neg N/A

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

      12. lower-*.f64 N/A

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

      13. exp-neg N/A

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

      14. *-commutative N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\frac{\left(-1 \cdot \left(x - -1\right)\right) \cdot \left(-1 \cdot {\left(e^{-1}\right)}^{x}\right) - \left(-1 \cdot \left(x - -1\right)\right) \cdot {\left(e^{-1}\right)}^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{-1}\right)}^{\left(\left(1 - \varepsilon\right) \cdot x\right)} - \left(-1 \cdot 0 - {\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.9% accurate, N/A× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := -1 \cdot \left(x - -1\right)\\ t_1 := {\left(e^{-1}\right)}^{x}\\ \frac{t\_0 \cdot \left(-1 \cdot t\_1\right) - t\_0 \cdot t\_1}{2} \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* -1.0 (- x -1.0))) (t_1 (pow (exp -1.0) x)))
   (/ (- (* t_0 (* -1.0 t_1)) (* t_0 t_1)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = -1.0 * (x - -1.0);
	double t_1 = pow(exp(-1.0), x);
	return ((t_0 * (-1.0 * t_1)) - (t_0 * t_1)) / 2.0;
}

Fortran

eps_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (-1.0d0) * (x - (-1.0d0))
    t_1 = exp((-1.0d0)) ** x
    code = ((t_0 * ((-1.0d0) * t_1)) - (t_0 * t_1)) / 2.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = -1.0 * (x - -1.0);
	double t_1 = Math.pow(Math.exp(-1.0), x);
	return ((t_0 * (-1.0 * t_1)) - (t_0 * t_1)) / 2.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = -1.0 * (x - -1.0)
	t_1 = math.pow(math.exp(-1.0), x)
	return ((t_0 * (-1.0 * t_1)) - (t_0 * t_1)) / 2.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(-1.0 * Float64(x - -1.0))
	t_1 = exp(-1.0) ^ x
	return Float64(Float64(Float64(t_0 * Float64(-1.0 * t_1)) - Float64(t_0 * t_1)) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	t_0 = -1.0 * (x - -1.0);
	t_1 = exp(-1.0) ^ x;
	tmp = ((t_0 * (-1.0 * t_1)) - (t_0 * t_1)) / 2.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(-1.0 * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[-1.0], $MachinePrecision], x], $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(-1.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Initial program 71.9%

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

3. Taylor expanded in eps around 0

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

   1. lower--.f64 N/A

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

   2. distribute-rgt1-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. mul-1-neg N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. lower-*.f64 N/A

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

   11. distribute-rgt1-in N/A

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

   12. lower-*.f64 N/A

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

   13. lower-+.f64 N/A

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

   14. mul-1-neg N/A

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

   15. exp-prod N/A

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

   16. lower-pow.f64 N/A

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

5. Applied rewrites 58.4%

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

6. Final simplification 58.4%

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

Reproduce

herbie shell --seed 2025065 
(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))