NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.5% → 99.9%

Time: 6.2s

Alternatives: 12

Speedup: 1.9×

• 38.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -5.54766714666258e+299
  2. eps = 2.5557725620741902e-300

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.5% 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: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot eps\_m}\\ t_1 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 0.005:\\ \;\;\;\;\left(\left(x + 1\right) \cdot t\_1 - \left(-t\_1 \cdot \left(x + 1\right)\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \left(-\frac{1}{t\_0}\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x eps_m))) (t_1 (exp (- x))))
   (if (<= eps_m 0.005)
     (* (- (* (+ x 1.0) t_1) (- (* t_1 (+ x 1.0)))) 0.5)
     (* (- t_0 (- (/ 1.0 t_0))) 0.5))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * eps_m));
	double t_1 = exp(-x);
	double tmp;
	if (eps_m <= 0.005) {
		tmp = (((x + 1.0) * t_1) - -(t_1 * (x + 1.0))) * 0.5;
	} else {
		tmp = (t_0 - -(1.0 / t_0)) * 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) :: t_1
    real(8) :: tmp
    t_0 = exp((x * eps_m))
    t_1 = exp(-x)
    if (eps_m <= 0.005d0) then
        tmp = (((x + 1.0d0) * t_1) - -(t_1 * (x + 1.0d0))) * 0.5d0
    else
        tmp = (t_0 - -(1.0d0 / t_0)) * 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 * eps_m));
	double t_1 = Math.exp(-x);
	double tmp;
	if (eps_m <= 0.005) {
		tmp = (((x + 1.0) * t_1) - -(t_1 * (x + 1.0))) * 0.5;
	} else {
		tmp = (t_0 - -(1.0 / t_0)) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * eps_m))
	t_1 = math.exp(-x)
	tmp = 0
	if eps_m <= 0.005:
		tmp = (((x + 1.0) * t_1) - -(t_1 * (x + 1.0))) * 0.5
	else:
		tmp = (t_0 - -(1.0 / t_0)) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * eps_m))
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 0.005)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) * t_1) - Float64(-Float64(t_1 * Float64(x + 1.0)))) * 0.5);
	else
		tmp = Float64(Float64(t_0 - Float64(-Float64(1.0 / t_0))) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * eps_m));
	t_1 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 0.005)
		tmp = (((x + 1.0) * t_1) - -(t_1 * (x + 1.0))) * 0.5;
	else
		tmp = (t_0 - -(1.0 / t_0)) * 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[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 0.005], N[(N[(N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] - (-N[(t$95$1 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t$95$0 - (-N[(1.0 / t$95$0), $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0050000000000000001

   1. Initial program 39.5%

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

   2. 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 100.0%

      \[\leadsto \color{blue}{\left(\left(x + 1\right) \cdot e^{-x} - -1 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if 0.0050000000000000001 < eps

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt1-in N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lift-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 99.8

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

   12. Applied rewrites 99.8%

       \[\leadsto \left(e^{x \cdot \varepsilon} - \left(-\frac{1}{e^{x \cdot \varepsilon}}\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot eps\_m}\\ \mathbf{if}\;eps\_m \leq 0.005:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \left(-\frac{1}{t\_0}\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x eps_m))))
   (if (<= eps_m 0.005)
     (* (* (exp (- x)) 2.0) 0.5)
     (* (- t_0 (- (/ 1.0 t_0))) 0.5))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * eps_m));
	double tmp;
	if (eps_m <= 0.005) {
		tmp = (exp(-x) * 2.0) * 0.5;
	} else {
		tmp = (t_0 - -(1.0 / t_0)) * 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 * eps_m))
    if (eps_m <= 0.005d0) then
        tmp = (exp(-x) * 2.0d0) * 0.5d0
    else
        tmp = (t_0 - -(1.0d0 / t_0)) * 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 * eps_m));
	double tmp;
	if (eps_m <= 0.005) {
		tmp = (Math.exp(-x) * 2.0) * 0.5;
	} else {
		tmp = (t_0 - -(1.0 / t_0)) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * eps_m))
	tmp = 0
	if eps_m <= 0.005:
		tmp = (math.exp(-x) * 2.0) * 0.5
	else:
		tmp = (t_0 - -(1.0 / t_0)) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * eps_m))
	tmp = 0.0
	if (eps_m <= 0.005)
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) * 0.5);
	else
		tmp = Float64(Float64(t_0 - Float64(-Float64(1.0 / t_0))) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * eps_m));
	tmp = 0.0;
	if (eps_m <= 0.005)
		tmp = (exp(-x) * 2.0) * 0.5;
	else
		tmp = (t_0 - -(1.0 / t_0)) * 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[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps$95$m, 0.005], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t$95$0 - (-N[(1.0 / t$95$0), $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0050000000000000001

   1. Initial program 39.5%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.1%

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

   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 98.1

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

   7. Applied rewrites 98.1%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. count-2-rev N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-exp.f64 98.1

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

   9. Applied rewrites 98.1%

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

   if 0.0050000000000000001 < eps

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt1-in N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lift-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 99.8

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

   12. Applied rewrites 99.8%

       \[\leadsto \left(e^{x \cdot \varepsilon} - \left(-\frac{1}{e^{x \cdot \varepsilon}}\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.005:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - \left(-e^{-x \cdot eps\_m}\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.005)
   (* (* (exp (- x)) 2.0) 0.5)
   (* (- (exp (* x eps_m)) (- (exp (- (* x eps_m))))) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.005) {
		tmp = (exp(-x) * 2.0) * 0.5;
	} else {
		tmp = (exp((x * eps_m)) - -exp(-(x * eps_m))) * 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) :: tmp
    if (eps_m <= 0.005d0) then
        tmp = (exp(-x) * 2.0d0) * 0.5d0
    else
        tmp = (exp((x * eps_m)) - -exp(-(x * eps_m))) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.005) {
		tmp = (Math.exp(-x) * 2.0) * 0.5;
	} else {
		tmp = (Math.exp((x * eps_m)) - -Math.exp(-(x * eps_m))) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.005:
		tmp = (math.exp(-x) * 2.0) * 0.5
	else:
		tmp = (math.exp((x * eps_m)) - -math.exp(-(x * eps_m))) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.005)
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - Float64(-exp(Float64(-Float64(x * eps_m))))) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.005)
		tmp = (exp(-x) * 2.0) * 0.5;
	else
		tmp = (exp((x * eps_m)) - -exp(-(x * eps_m))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.005], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - (-N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0050000000000000001

   1. Initial program 39.5%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.1%

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

   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 98.1

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

   7. Applied rewrites 98.1%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. count-2-rev N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-exp.f64 98.1

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

   9. Applied rewrites 98.1%

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

   if 0.0050000000000000001 < eps

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   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 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 99.8

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

   10. Applied rewrites 99.8%

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

Alternative 4: 99.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(exp(Float64(Float64(-x) * Float64(1.0 - eps_m))) - 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[((-x) * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - (-N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 74.5%

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

2. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.2%

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

Alternative 5: 84.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-284}:\\ \;\;\;\;\left(1 - \left(-e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+269}:\\ \;\;\;\;\left(e^{\left(-x\right) \cdot \left(1 - eps\_m\right)} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4e-284)
   (* (- 1.0 (- (exp (- (fma x eps_m x))))) 0.5)
   (if (<= x 3.8e+269)
     (* (- (exp (* (- x) (- 1.0 eps_m))) -1.0) 0.5)
     (* (* (exp (- x)) 2.0) 0.5))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4e-284) {
		tmp = (1.0 - -exp(-fma(x, eps_m, x))) * 0.5;
	} else if (x <= 3.8e+269) {
		tmp = (exp((-x * (1.0 - eps_m))) - -1.0) * 0.5;
	} else {
		tmp = (exp(-x) * 2.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4e-284)
		tmp = Float64(Float64(1.0 - Float64(-exp(Float64(-fma(x, eps_m, x))))) * 0.5);
	elseif (x <= 3.8e+269)
		tmp = Float64(Float64(exp(Float64(Float64(-x) * Float64(1.0 - eps_m))) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -4e-284], N[(N[(1.0 - (-N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 3.8e+269], N[(N[(N[Exp[N[((-x) * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.00000000000000015e-284

   1. Initial program 70.5%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.4%

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

   if -4.00000000000000015e-284 < x < 3.79999999999999983e269

   1. Initial program 75.5%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 77.5%

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

   if 3.79999999999999983e269 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   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 54.8

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

   7. Applied rewrites 54.8%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. count-2-rev N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-exp.f64 54.8

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

   9. Applied rewrites 54.8%

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

Alternative 6: 78.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(e^{-x} \cdot 2\right) \cdot 0.5\\ \mathbf{if}\;x \leq 2 \cdot 10^{-286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+269}:\\ \;\;\;\;\left(e^{\left(-x\right) \cdot \left(1 - eps\_m\right)} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (* (exp (- x)) 2.0) 0.5)))
   (if (<= x 2e-286)
     t_0
     (if (<= x 3.8e+269) (* (- (exp (* (- x) (- 1.0 eps_m))) -1.0) 0.5) t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (exp(-x) * 2.0) * 0.5;
	double tmp;
	if (x <= 2e-286) {
		tmp = t_0;
	} else if (x <= 3.8e+269) {
		tmp = (exp((-x * (1.0 - eps_m))) - -1.0) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (exp(-x) * 2.0d0) * 0.5d0
    if (x <= 2d-286) then
        tmp = t_0
    else if (x <= 3.8d+269) then
        tmp = (exp((-x * (1.0d0 - eps_m))) - (-1.0d0)) * 0.5d0
    else
        tmp = t_0
    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) * 2.0) * 0.5;
	double tmp;
	if (x <= 2e-286) {
		tmp = t_0;
	} else if (x <= 3.8e+269) {
		tmp = (Math.exp((-x * (1.0 - eps_m))) - -1.0) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (math.exp(-x) * 2.0) * 0.5
	tmp = 0
	if x <= 2e-286:
		tmp = t_0
	elif x <= 3.8e+269:
		tmp = (math.exp((-x * (1.0 - eps_m))) - -1.0) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(exp(Float64(-x)) * 2.0) * 0.5)
	tmp = 0.0
	if (x <= 2e-286)
		tmp = t_0;
	elseif (x <= 3.8e+269)
		tmp = Float64(Float64(exp(Float64(Float64(-x) * Float64(1.0 - eps_m))) - -1.0) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (exp(-x) * 2.0) * 0.5;
	tmp = 0.0;
	if (x <= 2e-286)
		tmp = t_0;
	elseif (x <= 3.8e+269)
		tmp = (exp((-x * (1.0 - eps_m))) - -1.0) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, 2e-286], t$95$0, If[LessEqual[x, 3.8e+269], N[(N[(N[Exp[N[((-x) * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-286 or 3.79999999999999983e269 < x

   1. Initial program 71.6%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.1%

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

   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 81.0

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

   7. Applied rewrites 81.0%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. count-2-rev N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-exp.f64 81.0

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

   9. Applied rewrites 81.0%

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

   if 2.0000000000000001e-286 < x < 3.79999999999999983e269

   1. Initial program 77.1%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 75.7%

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

Alternative 7: 70.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \left(e^{-x} \cdot 2\right) \cdot 0.5 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (* (* (exp (- x)) 2.0) 0.5))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (exp(-x) * 2.0) * 0.5;
}

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 = (exp(-x) * 2.0d0) * 0.5d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return (Math.exp(-x) * 2.0) * 0.5;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (math.exp(-x) * 2.0) * 0.5

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(exp(Float64(-x)) * 2.0) * 0.5)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (exp(-x) * 2.0) * 0.5;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 74.5%

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

2. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.2%

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

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 70.4

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

7. Applied rewrites 70.4%

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

   1. lift-+.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. count-2-rev N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. lift-exp.f64 70.4

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

9. Applied rewrites 70.4%

   \[\leadsto \left(e^{-x} \cdot 2\right) \cdot \color{blue}{0.5} \]
10. Add Preprocessing

Alternative 8: 59.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -700.0)
   (* (fma (- x 2.0) x 2.0) 0.5)
   (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -700.0) {
		tmp = fma((x - 2.0), x, 2.0) * 0.5;
	} else {
		tmp = fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(fma(Float64(x - 2.0), x, 2.0) * 0.5);
	else
		tmp = fma(Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -700.0], N[(N[(N[(x - 2.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   1. Initial program 99.6%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   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 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 54.3

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

   10. Applied rewrites 54.3%

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

   if -700 < x

   1. Initial program 70.4%

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

   2. 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 66.5%

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

   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. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.4

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

   7. Applied rewrites 60.4%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -700.0)
   (* (fma (- x 2.0) x 2.0) 0.5)
   (fma (* 0.3333333333333333 x) (* x x) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -700.0) {
		tmp = fma((x - 2.0), x, 2.0) * 0.5;
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(fma(Float64(x - 2.0), x, 2.0) * 0.5);
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -700.0], N[(N[(N[(x - 2.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   1. Initial program 99.6%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   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 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 54.3

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

   10. Applied rewrites 54.3%

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

   if -700 < x

   1. Initial program 70.4%

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

   2. 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 66.5%

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

   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. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 60.3

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

   10. Applied rewrites 60.3%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 59.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 9e+82)
   (* (fma (- x 2.0) x 2.0) 0.5)
   (* (* (* x x) x) 0.3333333333333333)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 9e+82) {
		tmp = fma((x - 2.0), x, 2.0) * 0.5;
	} else {
		tmp = ((x * x) * x) * 0.3333333333333333;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 9e+82)
		tmp = Float64(fma(Float64(x - 2.0), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(x * x) * x) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 9e+82], N[(N[(N[(x - 2.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.9999999999999993e82

   1. Initial program 67.7%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.0%

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

   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 75.5

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

   7. Applied rewrites 75.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 62.8

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

   10. Applied rewrites 62.8%

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

   if 8.9999999999999993e82 < x

   1. Initial program 100.0%

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

   2. 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 51.2%

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

   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. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.2

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

   7. Applied rewrites 45.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{3} \cdot {x}^{\color{blue}{3}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x}^{3} \cdot \frac{1}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \frac{1}{3} \]

      3. unpow3 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{3} \]

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{3} \]

      7. lift-*.f64 45.2

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.3333333333333333 \]

   10. Applied rewrites 45.2%

       \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.82:\\ \;\;\;\;\mathsf{fma}\left(-2, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.82)
   (* (fma -2.0 x 2.0) 0.5)
   (* (* (* x x) x) 0.3333333333333333)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.82) {
		tmp = fma(-2.0, x, 2.0) * 0.5;
	} else {
		tmp = ((x * x) * x) * 0.3333333333333333;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 0.82)
		tmp = Float64(fma(-2.0, x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(x * x) * x) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 0.82], N[(N[(-2.0 * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.819999999999999951

   1. Initial program 64.0%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.0%

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

   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 79.0

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 60.3

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

   10. Applied rewrites 60.3%

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

   if 0.819999999999999951 < x

   1. Initial program 99.6%

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

   2. 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 50.3%

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

   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. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 33.2

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

   7. Applied rewrites 33.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{3} \cdot {x}^{\color{blue}{3}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x}^{3} \cdot \frac{1}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \frac{1}{3} \]

      3. unpow3 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{3} \]

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{3} \]

      7. lift-*.f64 33.2

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.3333333333333333 \]

   10. Applied rewrites 33.2%

       \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 43.4% 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 74.5%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 43.4%

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

Reproduce

herbie shell --seed 2025117 
(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))