NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.2%

Time: 5.6s

Alternatives: 11

Speedup: 2.0×

• 37.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.4883584136732522e+103
  2. eps = 1.7709008435065465e-148

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: 73.0% 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: 99.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 0.00021:\\ \;\;\;\;\left(t\_0 + t\_0\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot eps\_m} + e^{\left(-eps\_m\right) \cdot x}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 0.00021)
     (* (+ t_0 t_0) 0.5)
     (* (+ (exp (* x eps_m)) (exp (* (- eps_m) x))) 0.5))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 0.00021) {
		tmp = (t_0 + t_0) * 0.5;
	} else {
		tmp = (exp((x * eps_m)) + exp((-eps_m * x))) * 0.5;
	}
	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) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 0.00021d0) then
        tmp = (t_0 + t_0) * 0.5d0
    else
        tmp = (exp((x * eps_m)) + exp((-eps_m * x))) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 0.00021) {
		tmp = (t_0 + t_0) * 0.5;
	} else {
		tmp = (Math.exp((x * eps_m)) + Math.exp((-eps_m * x))) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 0.00021:
		tmp = (t_0 + t_0) * 0.5
	else:
		tmp = (math.exp((x * eps_m)) + math.exp((-eps_m * x))) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 0.00021)
		tmp = Float64(Float64(t_0 + t_0) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(Float64(-eps_m) * x))) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 0.00021)
		tmp = (t_0 + t_0) * 0.5;
	else
		tmp = (exp((x * eps_m)) + exp((-eps_m * x))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 0.00021], N[(N[(t$95$0 + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[((-eps$95$m) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.1000000000000001e-4

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 71.6

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

   7. Applied rewrites 71.6%

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

   if 2.1000000000000001e-4 < eps

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 85.3

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

   10. Applied rewrites 85.3%

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

       1. lift--.f64 N/A

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

       2. lift-neg.f64 N/A

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

       3. add-flip-rev N/A

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

       4. lower-+.f64 85.3

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

   12. Applied rewrites 85.3%

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

Alternative 2: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \left(e^{\left(eps\_m - 1\right) \cdot x} - \left(-e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right)\right) \cdot 0.5 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (* (- (exp (* (- eps_m 1.0) x)) (- (exp (- (fma x eps_m x))))) 0.5))

C

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

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(exp(Float64(Float64(eps_m - 1.0) * x)) - Float64(-exp(Float64(-fma(x, eps_m, x))))) * 0.5)
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[Exp[N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - (-N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 73.0%

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

2. Taylor expanded in eps around inf

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

4. Applied rewrites 99.2%

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

Alternative 3: 84.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-225}:\\ \;\;\;\;\left(\left(\left(-x\right) + 1\right) - \left(-e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+108}:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + t\_0\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -7.5e-225)
     (* (- (+ (- x) 1.0) (- (exp (- (fma x eps_m x))))) 0.5)
     (if (<= x 9.5e+108)
       (* (- (exp (* x eps_m)) -1.0) 0.5)
       (* (+ t_0 t_0) 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -7.5e-225) {
		tmp = ((-x + 1.0) - -exp(-fma(x, eps_m, x))) * 0.5;
	} else if (x <= 9.5e+108) {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = (t_0 + t_0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -7.5e-225)
		tmp = Float64(Float64(Float64(Float64(-x) + 1.0) - Float64(-exp(Float64(-fma(x, eps_m, x))))) * 0.5);
	elseif (x <= 9.5e+108)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(t_0 + t_0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -7.5e-225], N[(N[(N[((-x) + 1.0), $MachinePrecision] - (-N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 9.5e+108], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t$95$0 + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.49999999999999954e-225

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 64.5

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 64.0

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

   10. Applied rewrites 64.0%

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

   if -7.49999999999999954e-225 < x < 9.50000000000000097e108

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.0%

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

   if 9.50000000000000097e108 < x

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 71.6

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

   7. Applied rewrites 71.6%

      \[\leadsto \left(e^{-x} + e^{-x}\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-225}:\\ \;\;\;\;\left(\left(\left(-x\right) + 1\right) - \left(-e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+108}:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 + t\_0\right) \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -7.5e-225)
     (* (- (+ (- x) 1.0) (- (exp (- (fma x eps_m x))))) 0.5)
     (if (<= x 9.5e+108)
       (* (- (exp (* x eps_m)) -1.0) 0.5)
       (* (* (+ t_0 t_0) x) 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -7.5e-225) {
		tmp = ((-x + 1.0) - -exp(-fma(x, eps_m, x))) * 0.5;
	} else if (x <= 9.5e+108) {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = ((t_0 + t_0) * x) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -7.5e-225)
		tmp = Float64(Float64(Float64(Float64(-x) + 1.0) - Float64(-exp(Float64(-fma(x, eps_m, x))))) * 0.5);
	elseif (x <= 9.5e+108)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(t_0 + t_0) * x) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -7.5e-225], N[(N[(N[((-x) + 1.0), $MachinePrecision] - (-N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 9.5e+108], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$0 + t$95$0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.49999999999999954e-225

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 64.5

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 64.0

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

   10. Applied rewrites 64.0%

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

   if -7.49999999999999954e-225 < x < 9.50000000000000097e108

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.0%

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

   if 9.50000000000000097e108 < x

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. add-flip N/A

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

      6. mul-1-neg N/A

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

      7. lower-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-exp.f64 16.6

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

   7. Applied rewrites 16.6%

      \[\leadsto \left(\left(e^{-x} + e^{-x}\right) \cdot x\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-225}:\\ \;\;\;\;\left(\left(\left(-x\right) + 1\right) - \left(-e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+104}:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{eps\_m} - \frac{1 - eps\_m}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -7.5e-225)
   (* (- (+ (- x) 1.0) (- (exp (- (fma x eps_m x))))) 0.5)
   (if (<= x 2.2e+104)
     (* (- (exp (* x eps_m)) -1.0) 0.5)
     (/ (- (/ 1.0 eps_m) (/ (- 1.0 eps_m) eps_m)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -7.5e-225) {
		tmp = ((-x + 1.0) - -exp(-fma(x, eps_m, x))) * 0.5;
	} else if (x <= 2.2e+104) {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = ((1.0 / eps_m) - ((1.0 - eps_m) / eps_m)) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -7.5e-225)
		tmp = Float64(Float64(Float64(Float64(-x) + 1.0) - Float64(-exp(Float64(-fma(x, eps_m, x))))) * 0.5);
	elseif (x <= 2.2e+104)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 / eps_m) - Float64(Float64(1.0 - eps_m) / eps_m)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -7.5e-225], N[(N[(N[((-x) + 1.0), $MachinePrecision] - (-N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.2e+104], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - N[(N[(1.0 - eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.49999999999999954e-225

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 64.5

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 64.0

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

   10. Applied rewrites 64.0%

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

   if -7.49999999999999954e-225 < x < 2.2e104

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.0%

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

   if 2.2e104 < x

   1. Initial program 73.0%

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

   2. Taylor expanded in x around 0

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

      1. sub-to-mult N/A

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

      2. inv-pow N/A

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

      3. pow-neg N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. sub-flip N/A

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

      7. mul-1-neg N/A

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

      8. mult-flip N/A

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

      9. mul-1-neg N/A

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

      10. sub-flip N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]

      3. lift-exp.f64 19.2

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 18.9

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

   10. Applied rewrites 18.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{\varepsilon}} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1850:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps\_m} - \frac{-eps\_m}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(\left(eps\_m - \frac{\mathsf{fma}\left(eps\_m, eps\_m, -1\right)}{eps\_m - 1}\right) - 1, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+104}:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{eps\_m} - \frac{1 - eps\_m}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1850.0)
   (/ (- (/ (exp (- x)) eps_m) (/ (- eps_m) eps_m)) 2.0)
   (if (<= x -4.5e-150)
     (*
      (fma (- (- eps_m (/ (fma eps_m eps_m -1.0) (- eps_m 1.0))) 1.0) x 2.0)
      0.5)
     (if (<= x 2.2e+104)
       (* (- (exp (* x eps_m)) -1.0) 0.5)
       (/ (- (/ 1.0 eps_m) (/ (- 1.0 eps_m) eps_m)) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1850.0) {
		tmp = ((exp(-x) / eps_m) - (-eps_m / eps_m)) / 2.0;
	} else if (x <= -4.5e-150) {
		tmp = fma(((eps_m - (fma(eps_m, eps_m, -1.0) / (eps_m - 1.0))) - 1.0), x, 2.0) * 0.5;
	} else if (x <= 2.2e+104) {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = ((1.0 / eps_m) - ((1.0 - eps_m) / eps_m)) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1850.0)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps_m) - Float64(Float64(-eps_m) / eps_m)) / 2.0);
	elseif (x <= -4.5e-150)
		tmp = Float64(fma(Float64(Float64(eps_m - Float64(fma(eps_m, eps_m, -1.0) / Float64(eps_m - 1.0))) - 1.0), x, 2.0) * 0.5);
	elseif (x <= 2.2e+104)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 / eps_m) - Float64(Float64(1.0 - eps_m) / eps_m)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1850.0], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] - N[((-eps$95$m) / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -4.5e-150], N[(N[(N[(N[(eps$95$m - N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] / N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.2e+104], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - N[(N[(1.0 - eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1850

   1. Initial program 73.0%

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

   2. Taylor expanded in x around 0

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

      1. sub-to-mult N/A

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

      2. inv-pow N/A

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

      3. pow-neg N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. sub-flip N/A

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

      7. mul-1-neg N/A

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

      8. mult-flip N/A

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

      9. mul-1-neg N/A

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

      10. sub-flip N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]

      3. lift-exp.f64 19.2

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{-1 \cdot \varepsilon}{\varepsilon}}{2} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 19.7

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{-\varepsilon}{\varepsilon}}{2} \]

   10. Applied rewrites 19.7%

       \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{-\varepsilon}{\varepsilon}}{2} \]

   if -1850 < x < -4.5000000000000002e-150

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. add-flip N/A

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

      6. mul-1-neg N/A

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

      7. add-flip N/A

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

      8. mul-1-neg N/A

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

      9. sub-negate-rev N/A

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

      10. sub-negate-rev N/A

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

      11. mul-1-neg N/A

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

      12. add-flip N/A

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

      13. lower--.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 43.9

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

   7. Applied rewrites 43.9%

      \[\leadsto \mathsf{fma}\left(\left(\varepsilon - \left(\varepsilon + 1\right)\right) - 1, x, 2\right) \cdot 0.5 \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 52.0

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

   9. Applied rewrites 52.0%

      \[\leadsto \mathsf{fma}\left(\left(\varepsilon - \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}\right) - 1, x, 2\right) \cdot 0.5 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. pow2 N/A

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

       3. lower--.f64 N/A

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

       4. sub-flip N/A

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

       5. metadata-eval N/A

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

       6. pow2 N/A

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

       7. lower-fma.f64 52.0

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

   11. Applied rewrites 52.0%

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

   if -4.5000000000000002e-150 < x < 2.2e104

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.0%

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

   if 2.2e104 < x

   1. Initial program 73.0%

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

   2. Taylor expanded in x around 0

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

      1. sub-to-mult N/A

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

      2. inv-pow N/A

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

      3. pow-neg N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. sub-flip N/A

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

      7. mul-1-neg N/A

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

      8. mult-flip N/A

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

      9. mul-1-neg N/A

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

      10. sub-flip N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]

      3. lift-exp.f64 19.2

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 18.9

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

   10. Applied rewrites 18.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{\varepsilon}} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(\left(eps\_m - \frac{\mathsf{fma}\left(eps\_m, eps\_m, -1\right)}{eps\_m - 1}\right) - 1, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+104}:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{eps\_m} - \frac{1 - eps\_m}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4.5e-150)
   (*
    (fma (- (- eps_m (/ (fma eps_m eps_m -1.0) (- eps_m 1.0))) 1.0) x 2.0)
    0.5)
   (if (<= x 2.2e+104)
     (* (- (exp (* x eps_m)) -1.0) 0.5)
     (/ (- (/ 1.0 eps_m) (/ (- 1.0 eps_m) eps_m)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4.5e-150) {
		tmp = fma(((eps_m - (fma(eps_m, eps_m, -1.0) / (eps_m - 1.0))) - 1.0), x, 2.0) * 0.5;
	} else if (x <= 2.2e+104) {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = ((1.0 / eps_m) - ((1.0 - eps_m) / eps_m)) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4.5e-150)
		tmp = Float64(fma(Float64(Float64(eps_m - Float64(fma(eps_m, eps_m, -1.0) / Float64(eps_m - 1.0))) - 1.0), x, 2.0) * 0.5);
	elseif (x <= 2.2e+104)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 / eps_m) - Float64(Float64(1.0 - eps_m) / eps_m)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -4.5e-150], N[(N[(N[(N[(eps$95$m - N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] / N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.2e+104], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - N[(N[(1.0 - eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5000000000000002e-150

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. add-flip N/A

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

      6. mul-1-neg N/A

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

      7. add-flip N/A

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

      8. mul-1-neg N/A

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

      9. sub-negate-rev N/A

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

      10. sub-negate-rev N/A

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

      11. mul-1-neg N/A

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

      12. add-flip N/A

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

      13. lower--.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 43.9

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

   7. Applied rewrites 43.9%

      \[\leadsto \mathsf{fma}\left(\left(\varepsilon - \left(\varepsilon + 1\right)\right) - 1, x, 2\right) \cdot 0.5 \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 52.0

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

   9. Applied rewrites 52.0%

      \[\leadsto \mathsf{fma}\left(\left(\varepsilon - \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}\right) - 1, x, 2\right) \cdot 0.5 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. pow2 N/A

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

       3. lower--.f64 N/A

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

       4. sub-flip N/A

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

       5. metadata-eval N/A

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

       6. pow2 N/A

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

       7. lower-fma.f64 52.0

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

   11. Applied rewrites 52.0%

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

   if -4.5000000000000002e-150 < x < 2.2e104

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.0%

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

   if 2.2e104 < x

   1. Initial program 73.0%

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

   2. Taylor expanded in x around 0

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

      1. sub-to-mult N/A

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

      2. inv-pow N/A

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

      3. pow-neg N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. sub-flip N/A

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

      7. mul-1-neg N/A

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

      8. mult-flip N/A

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

      9. mul-1-neg N/A

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

      10. sub-flip N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]

      3. lift-exp.f64 19.2

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 18.9

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

   10. Applied rewrites 18.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{\varepsilon}} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+104}:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{eps\_m} - \frac{1 - eps\_m}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.2e+104)
   (* (- (exp (* x eps_m)) -1.0) 0.5)
   (/ (- (/ 1.0 eps_m) (/ (- 1.0 eps_m) eps_m)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.2e+104) {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = ((1.0 / eps_m) - ((1.0 - eps_m) / eps_m)) / 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) :: tmp
    if (x <= 2.2d+104) then
        tmp = (exp((x * eps_m)) - (-1.0d0)) * 0.5d0
    else
        tmp = ((1.0d0 / eps_m) - ((1.0d0 - eps_m) / eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.2e+104) {
		tmp = (Math.exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = ((1.0 / eps_m) - ((1.0 - eps_m) / eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.2e+104:
		tmp = (math.exp((x * eps_m)) - -1.0) * 0.5
	else:
		tmp = ((1.0 / eps_m) - ((1.0 - eps_m) / eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.2e+104)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 / eps_m) - Float64(Float64(1.0 - eps_m) / eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.2e+104)
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	else
		tmp = ((1.0 / eps_m) - ((1.0 - eps_m) / eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.2e+104], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - N[(N[(1.0 - eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2e104

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.0%

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

   if 2.2e104 < x

   1. Initial program 73.0%

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

   2. Taylor expanded in x around 0

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

      1. sub-to-mult N/A

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

      2. inv-pow N/A

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

      3. pow-neg N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. sub-flip N/A

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

      7. mul-1-neg N/A

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

      8. mult-flip N/A

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

      9. mul-1-neg N/A

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

      10. sub-flip N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]

      3. lift-exp.f64 19.2

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 18.9

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

   10. Applied rewrites 18.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{\varepsilon}} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 55.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{-196}:\\ \;\;\;\;\mathsf{fma}\left(-2, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\left(\left(\frac{1}{x \cdot x} - 0.5\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{eps\_m} - \frac{1 - eps\_m}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 4.3e-196)
   (* (fma -2.0 x 2.0) 0.5)
   (if (<= x 1.45)
     (* (* (- (/ 1.0 (* x x)) 0.5) x) x)
     (/ (- (/ 1.0 eps_m) (/ (- 1.0 eps_m) eps_m)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 4.3e-196) {
		tmp = fma(-2.0, x, 2.0) * 0.5;
	} else if (x <= 1.45) {
		tmp = (((1.0 / (x * x)) - 0.5) * x) * x;
	} else {
		tmp = ((1.0 / eps_m) - ((1.0 - eps_m) / eps_m)) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 4.3e-196)
		tmp = Float64(fma(-2.0, x, 2.0) * 0.5);
	elseif (x <= 1.45)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(x * x)) - 0.5) * x) * x);
	else
		tmp = Float64(Float64(Float64(1.0 / eps_m) - Float64(Float64(1.0 - eps_m) / eps_m)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 4.3e-196], N[(N[(-2.0 * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.45], N[(N[(N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - N[(N[(1.0 - eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.29999999999999979e-196

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. add-flip N/A

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

      6. mul-1-neg N/A

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

      7. add-flip N/A

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

      8. mul-1-neg N/A

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

      9. sub-negate-rev N/A

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

      10. sub-negate-rev N/A

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

      11. mul-1-neg N/A

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

      12. add-flip N/A

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

      13. lower--.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 43.9

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

   7. Applied rewrites 43.9%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 43.9%

       \[\leadsto \mathsf{fma}\left(-2, x, 2\right) \cdot 0.5 \]

   if 4.29999999999999979e-196 < x < 1.44999999999999996

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot {x}^{2} + 1 \]

      2. *-commutative N/A

         \[\leadsto {x}^{2} \cdot \frac{-1}{2} + 1 \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 18.5

         \[\leadsto \left(\frac{1}{x \cdot x} - 0.5\right) \cdot \left(x \cdot x\right) \]

   10. Applied rewrites 18.5%

       \[\leadsto \left(\frac{1}{x \cdot x} - 0.5\right) \cdot \left(x \cdot \color{blue}{x}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-/.f64 N/A

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

       11. lift--.f64 22.5

           \[\leadsto \left(\left(\frac{1}{x \cdot x} - 0.5\right) \cdot x\right) \cdot x \]

   12. Applied rewrites 22.5%

       \[\leadsto \left(\left(\frac{1}{x \cdot x} - 0.5\right) \cdot x\right) \cdot x \]

   if 1.44999999999999996 < x

   1. Initial program 73.0%

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

   2. Taylor expanded in x around 0

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

      1. sub-to-mult N/A

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

      2. inv-pow N/A

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

      3. pow-neg N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. sub-flip N/A

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

      7. mul-1-neg N/A

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

      8. mult-flip N/A

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

      9. mul-1-neg N/A

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

      10. sub-flip N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]

      3. lift-exp.f64 19.2

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 18.9

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

   10. Applied rewrites 18.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{\varepsilon}} - \frac{1 - \varepsilon}{\varepsilon}}{2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 52.6% accurate, 4.2× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0))

C

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

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 73.0%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 58.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

   3. lower-fma.f64 N/A

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

   4. sub-flip N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \frac{-1}{2}, {x}^{2}, 1\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), x \cdot x, 1\right) \]

   8. lower-*.f64 52.6

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

7. Applied rewrites 52.6%

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

Alternative 11: 44.3% accurate, 58.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

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

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
    code = 1.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 73.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 44.3%

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

Reproduce

herbie shell --seed 2025140 
(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))