NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.7% → 99.9%

Time: 4.8s

Alternatives: 14

Speedup: 2.1×

• 38.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -5.240252024512497e+52
  2. eps = 8.113916516740081e-231

Specification

Math

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

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: 72.7% accurate, 1.0× speedup

Math

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

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

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 9.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-\left|\varepsilon\right| \cdot x} + e^{\left(\left|\varepsilon\right| - 1\right) \cdot x}\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (fabs eps) 9.5e-20)
   (* (/ (+ (- x -1.0) (- x -1.0)) (exp x)) 0.5)
   (* (+ (exp (- (* (fabs eps) x))) (exp (* (- (fabs eps) 1.0) x))) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 9.5e-20) {
		tmp = (((x - -1.0) + (x - -1.0)) / exp(x)) * 0.5;
	} else {
		tmp = (exp(-(fabs(eps) * x)) + exp(((fabs(eps) - 1.0) * x))) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (abs(eps) <= 9.5d-20) then
        tmp = (((x - (-1.0d0)) + (x - (-1.0d0))) / exp(x)) * 0.5d0
    else
        tmp = (exp(-(abs(eps) * x)) + exp(((abs(eps) - 1.0d0) * x))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 9.5e-20) {
		tmp = (((x - -1.0) + (x - -1.0)) / Math.exp(x)) * 0.5;
	} else {
		tmp = (Math.exp(-(Math.abs(eps) * x)) + Math.exp(((Math.abs(eps) - 1.0) * x))) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 9.5e-20:
		tmp = (((x - -1.0) + (x - -1.0)) / math.exp(x)) * 0.5
	else:
		tmp = (math.exp(-(math.fabs(eps) * x)) + math.exp(((math.fabs(eps) - 1.0) * x))) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 9.5e-20)
		tmp = Float64(Float64(Float64(Float64(x - -1.0) + Float64(x - -1.0)) / exp(x)) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(-Float64(abs(eps) * x))) + exp(Float64(Float64(abs(eps) - 1.0) * x))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 9.5e-20)
		tmp = (((x - -1.0) + (x - -1.0)) / exp(x)) * 0.5;
	else
		tmp = (exp(-(abs(eps) * x)) + exp(((abs(eps) - 1.0) * x))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 9.5e-20], N[(N[(N[(N[(x - -1.0), $MachinePrecision] + N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[(-N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision] + N[Exp[N[(N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 9.5e-20

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.4

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

   6. Applied rewrites 57.5%

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

   if 9.5e-20 < eps

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.0

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in eps around inf

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

      1. lower-*.f64 88.7

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

   9. Applied rewrites 88.7%

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

Alternative 2: 99.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

2. 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. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 99.0

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

4. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.0

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

6. Applied rewrites 99.0%

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

Alternative 3: 84.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left|\varepsilon\right| - 1\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+89}:\\ \;\;\;\;\frac{1 \cdot 1 - \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x - -1}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-297}:\\ \;\;\;\;\left(e^{-\left|\varepsilon\right| \cdot x} + \left(1 + x \cdot t\_0\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{t\_0 \cdot x} - -1\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (fabs eps) 1.0)))
   (if (<= x -1.75e+89)
     (/ (- (* 1.0 1.0) (sqrt (* (* x x) (* x x)))) (- x -1.0))
     (if (<= x 5e-297)
       (* (+ (exp (- (* (fabs eps) x))) (+ 1.0 (* x t_0))) 0.5)
       (* (- (exp (* t_0 x)) -1.0) 0.5)))))

C

double code(double x, double eps) {
	double t_0 = fabs(eps) - 1.0;
	double tmp;
	if (x <= -1.75e+89) {
		tmp = ((1.0 * 1.0) - sqrt(((x * x) * (x * x)))) / (x - -1.0);
	} else if (x <= 5e-297) {
		tmp = (exp(-(fabs(eps) * x)) + (1.0 + (x * t_0))) * 0.5;
	} else {
		tmp = (exp((t_0 * x)) - -1.0) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(eps) - 1.0d0
    if (x <= (-1.75d+89)) then
        tmp = ((1.0d0 * 1.0d0) - sqrt(((x * x) * (x * x)))) / (x - (-1.0d0))
    else if (x <= 5d-297) then
        tmp = (exp(-(abs(eps) * x)) + (1.0d0 + (x * t_0))) * 0.5d0
    else
        tmp = (exp((t_0 * x)) - (-1.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.abs(eps) - 1.0;
	double tmp;
	if (x <= -1.75e+89) {
		tmp = ((1.0 * 1.0) - Math.sqrt(((x * x) * (x * x)))) / (x - -1.0);
	} else if (x <= 5e-297) {
		tmp = (Math.exp(-(Math.abs(eps) * x)) + (1.0 + (x * t_0))) * 0.5;
	} else {
		tmp = (Math.exp((t_0 * x)) - -1.0) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.fabs(eps) - 1.0
	tmp = 0
	if x <= -1.75e+89:
		tmp = ((1.0 * 1.0) - math.sqrt(((x * x) * (x * x)))) / (x - -1.0)
	elif x <= 5e-297:
		tmp = (math.exp(-(math.fabs(eps) * x)) + (1.0 + (x * t_0))) * 0.5
	else:
		tmp = (math.exp((t_0 * x)) - -1.0) * 0.5
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(abs(eps) - 1.0)
	tmp = 0.0
	if (x <= -1.75e+89)
		tmp = Float64(Float64(Float64(1.0 * 1.0) - sqrt(Float64(Float64(x * x) * Float64(x * x)))) / Float64(x - -1.0));
	elseif (x <= 5e-297)
		tmp = Float64(Float64(exp(Float64(-Float64(abs(eps) * x))) + Float64(1.0 + Float64(x * t_0))) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(t_0 * x)) - -1.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = abs(eps) - 1.0;
	tmp = 0.0;
	if (x <= -1.75e+89)
		tmp = ((1.0 * 1.0) - sqrt(((x * x) * (x * x)))) / (x - -1.0);
	elseif (x <= 5e-297)
		tmp = (exp(-(abs(eps) * x)) + (1.0 + (x * t_0))) * 0.5;
	else
		tmp = (exp((t_0 * x)) - -1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -1.75e+89], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-297], N[(N[(N[Exp[(-N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision] + N[(1.0 + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(t$95$0 * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e89

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.1

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot x \]

      3. fp-cancel-sign-sub-inv N/A

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

      4. flip-- N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-unsound-+.f32 N/A

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

      11. lower-+.f32 N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-flip N/A

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

      15. lift--.f64 N/A

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

      16. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 49.7%

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

       1. rem-square-sqrt N/A

          \[\leadsto \frac{1 \cdot 1 - \sqrt{x \cdot x} \cdot \sqrt{x \cdot x}}{x - -1} \]

       2. sqrt-unprod N/A

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

       3. lower-*.f32 N/A

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

       4. lower-unsound-*.f32 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-unsound-*.f64 53.1

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

   11. Applied rewrites 53.1%

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

   if -1.75e89 < x < 5e-297

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.0

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in eps around inf

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

      1. lower-*.f64 88.7

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

   9. Applied rewrites 88.7%

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

   10. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 64.5

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

   12. Applied rewrites 64.5%

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

   if 5e-297 < x

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.9%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.9

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 63.9

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

   9. Applied rewrites 63.9%

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

Alternative 4: 78.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{\frac{e^{-x}}{\left|\varepsilon\right|} - \left(\frac{1}{\left|\varepsilon\right|} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left(\left|\varepsilon\right| - 1\right) \cdot x} - -1\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -700.0)
   (/ (- (/ (exp (- x)) (fabs eps)) (- (/ 1.0 (fabs eps)) 1.0)) 2.0)
   (* (- (exp (* (- (fabs eps) 1.0) x)) -1.0) 0.5)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-700.0d0)) then
        tmp = ((exp(-x) / abs(eps)) - ((1.0d0 / abs(eps)) - 1.0d0)) / 2.0d0
    else
        tmp = (exp(((abs(eps) - 1.0d0) * x)) - (-1.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -700.0) {
		tmp = ((Math.exp(-x) / Math.abs(eps)) - ((1.0 / Math.abs(eps)) - 1.0)) / 2.0;
	} else {
		tmp = (Math.exp(((Math.abs(eps) - 1.0) * x)) - -1.0) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -700.0:
		tmp = ((math.exp(-x) / math.fabs(eps)) - ((1.0 / math.fabs(eps)) - 1.0)) / 2.0
	else:
		tmp = (math.exp(((math.fabs(eps) - 1.0) * x)) - -1.0) * 0.5
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -700.0)
		tmp = ((exp(-x) / abs(eps)) - ((1.0 / abs(eps)) - 1.0)) / 2.0;
	else
		tmp = (exp(((abs(eps) - 1.0) * x)) - -1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -700.0], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 37.3

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

   4. Applied rewrites 37.3%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 12.0

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

   7. Applied rewrites 12.0%

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

   if -700 < x

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.9%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.9

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 63.9

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

   9. Applied rewrites 63.9%

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

Alternative 5: 78.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.4

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

   6. Applied rewrites 57.5%

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

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

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.9%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.9

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 63.9

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

   9. Applied rewrites 63.9%

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

Alternative 6: 74.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-297}:\\ \;\;\;\;\frac{1 \cdot 1 - \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left(\left|\varepsilon\right| - 1\right) \cdot x} - -1\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5e-297)
   (/ (- (* 1.0 1.0) (sqrt (* (* x x) (* x x)))) (- x -1.0))
   (* (- (exp (* (- (fabs eps) 1.0) x)) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5e-297) {
		tmp = ((1.0 * 1.0) - sqrt(((x * x) * (x * x)))) / (x - -1.0);
	} else {
		tmp = (exp(((fabs(eps) - 1.0) * x)) - -1.0) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 5d-297) then
        tmp = ((1.0d0 * 1.0d0) - sqrt(((x * x) * (x * x)))) / (x - (-1.0d0))
    else
        tmp = (exp(((abs(eps) - 1.0d0) * x)) - (-1.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 5e-297) {
		tmp = ((1.0 * 1.0) - Math.sqrt(((x * x) * (x * x)))) / (x - -1.0);
	} else {
		tmp = (Math.exp(((Math.abs(eps) - 1.0) * x)) - -1.0) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 5e-297:
		tmp = ((1.0 * 1.0) - math.sqrt(((x * x) * (x * x)))) / (x - -1.0)
	else:
		tmp = (math.exp(((math.fabs(eps) - 1.0) * x)) - -1.0) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5e-297)
		tmp = Float64(Float64(Float64(1.0 * 1.0) - sqrt(Float64(Float64(x * x) * Float64(x * x)))) / Float64(x - -1.0));
	else
		tmp = Float64(Float64(exp(Float64(Float64(abs(eps) - 1.0) * x)) - -1.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 5e-297)
		tmp = ((1.0 * 1.0) - sqrt(((x * x) * (x * x)))) / (x - -1.0);
	else
		tmp = (exp(((abs(eps) - 1.0) * x)) - -1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5e-297], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e-297

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.1

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot x \]

      3. fp-cancel-sign-sub-inv N/A

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

      4. flip-- N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-unsound-+.f32 N/A

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

      11. lower-+.f32 N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-flip N/A

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

      15. lift--.f64 N/A

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

      16. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 49.7%

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

       1. rem-square-sqrt N/A

          \[\leadsto \frac{1 \cdot 1 - \sqrt{x \cdot x} \cdot \sqrt{x \cdot x}}{x - -1} \]

       2. sqrt-unprod N/A

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

       3. lower-*.f32 N/A

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

       4. lower-unsound-*.f32 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-unsound-*.f64 53.1

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

   11. Applied rewrites 53.1%

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

   if 5e-297 < x

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.9%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.9

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 63.9

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

   9. Applied rewrites 63.9%

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

Alternative 7: 71.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-297}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -1\right) \cdot \frac{-1}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left(\left|\varepsilon\right| - 1\right) \cdot x} - -1\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5e-297)
   (* (fma x x -1.0) (/ -1.0 (- x -1.0)))
   (* (- (exp (* (- (fabs eps) 1.0) x)) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5e-297) {
		tmp = fma(x, x, -1.0) * (-1.0 / (x - -1.0));
	} else {
		tmp = (exp(((fabs(eps) - 1.0) * x)) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5e-297)
		tmp = Float64(fma(x, x, -1.0) * Float64(-1.0 / Float64(x - -1.0)));
	else
		tmp = Float64(Float64(exp(Float64(Float64(abs(eps) - 1.0) * x)) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5e-297], N[(N[(x * x + -1.0), $MachinePrecision] * N[(-1.0 / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e-297

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.1

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot x \]

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.1

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.1%

      \[\leadsto 1 - x \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. sub-negate-rev N/A

          \[\leadsto \mathsf{neg}\left(\left(x - 1\right)\right) \]

       3. flip-- N/A

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

       4. lower-unsound-+.f32 N/A

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

       5. lower-+.f32 N/A

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

       6. metadata-eval N/A

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

       7. sub-flip N/A

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

       8. lift--.f64 N/A

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

       9. lower-unsound-/.f64 N/A

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

       10. lower-unsound-*.f64 N/A

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

       11. lower-unsound--.f32 N/A

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

       12. lower-unsound-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. metadata-eval N/A

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

       15. lower--.f32 N/A

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

       16. sub-negate-rev N/A

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

       17. lift--.f64 N/A

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

       18. metadata-eval N/A

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

       19. lift-*.f64 N/A

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

       20. lower-/.f64 N/A

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

       21. distribute-frac-neg2 N/A

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

       22. mult-flip N/A

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

       23. lower-*.f64 N/A

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

   11. Applied rewrites 49.7%

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

   if 5e-297 < x

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.9%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.9

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 63.9

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

   9. Applied rewrites 63.9%

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

Alternative 8: 63.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -1\right) \cdot \frac{-1}{x - -1}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.3333333333333333 \cdot x\right) \cdot x, x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 7e-7)
   (* (fma x x -1.0) (/ -1.0 (- x -1.0)))
   (if (<= x 2.55e+212)
     (/ (- (+ 1.0 (/ 1.0 eps)) (- (/ 1.0 eps) 1.0)) 2.0)
     (fma (* (* 0.3333333333333333 x) x) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 7e-7) {
		tmp = fma(x, x, -1.0) * (-1.0 / (x - -1.0));
	} else if (x <= 2.55e+212) {
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0;
	} else {
		tmp = fma(((0.3333333333333333 * x) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 7e-7)
		tmp = Float64(fma(x, x, -1.0) * Float64(-1.0 / Float64(x - -1.0)));
	elseif (x <= 2.55e+212)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	else
		tmp = fma(Float64(Float64(0.3333333333333333 * x) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 7e-7], N[(N[(x * x + -1.0), $MachinePrecision] * N[(-1.0 / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e+212], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.99999999999999968e-7

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.1

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot x \]

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.1

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.1%

      \[\leadsto 1 - x \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. sub-negate-rev N/A

          \[\leadsto \mathsf{neg}\left(\left(x - 1\right)\right) \]

       3. flip-- N/A

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

       4. lower-unsound-+.f32 N/A

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

       5. lower-+.f32 N/A

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

       6. metadata-eval N/A

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

       7. sub-flip N/A

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

       8. lift--.f64 N/A

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

       9. lower-unsound-/.f64 N/A

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

       10. lower-unsound-*.f64 N/A

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

       11. lower-unsound--.f32 N/A

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

       12. lower-unsound-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. metadata-eval N/A

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

       15. lower--.f32 N/A

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

       16. sub-negate-rev N/A

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

       17. lift--.f64 N/A

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

       18. metadata-eval N/A

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

       19. lift-*.f64 N/A

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

       20. lower-/.f64 N/A

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

       21. distribute-frac-neg2 N/A

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

       22. mult-flip N/A

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

       23. lower-*.f64 N/A

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

   11. Applied rewrites 49.7%

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

   if 6.99999999999999968e-7 < x < 2.5500000000000001e212

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 37.3

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

   4. Applied rewrites 37.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 30.1

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

   7. Applied rewrites 30.1%

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

   if 2.5500000000000001e212 < x

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.4%

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

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 52.4

         \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   7. Applied rewrites 52.4%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 52.4

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 52.4

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 52.3

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

   12. Applied rewrites 52.3%

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

Alternative 9: 63.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.65:\\ \;\;\;\;\mathsf{fma}\left(x, x, -1\right) \cdot \frac{-1}{x - -1}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+203}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.3333333333333333 \cdot x\right) \cdot x, x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.65)
   (* (fma x x -1.0) (/ -1.0 (- x -1.0)))
   (if (<= x 1.1e+203)
     (/ x (exp x))
     (fma (* (* 0.3333333333333333 x) x) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.65) {
		tmp = fma(x, x, -1.0) * (-1.0 / (x - -1.0));
	} else if (x <= 1.1e+203) {
		tmp = x / exp(x);
	} else {
		tmp = fma(((0.3333333333333333 * x) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.65)
		tmp = Float64(fma(x, x, -1.0) * Float64(-1.0 / Float64(x - -1.0)));
	elseif (x <= 1.1e+203)
		tmp = Float64(x / exp(x));
	else
		tmp = fma(Float64(Float64(0.3333333333333333 * x) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.65], N[(N[(x * x + -1.0), $MachinePrecision] * N[(-1.0 / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e+203], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.650000000000000022

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.1

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot x \]

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.1

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.1%

      \[\leadsto 1 - x \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. sub-negate-rev N/A

          \[\leadsto \mathsf{neg}\left(\left(x - 1\right)\right) \]

       3. flip-- N/A

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

       4. lower-unsound-+.f32 N/A

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

       5. lower-+.f32 N/A

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

       6. metadata-eval N/A

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

       7. sub-flip N/A

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

       8. lift--.f64 N/A

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

       9. lower-unsound-/.f64 N/A

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

       10. lower-unsound-*.f64 N/A

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

       11. lower-unsound--.f32 N/A

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

       12. lower-unsound-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. metadata-eval N/A

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

       15. lower--.f32 N/A

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

       16. sub-negate-rev N/A

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

       17. lift--.f64 N/A

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

       18. metadata-eval N/A

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

       19. lift-*.f64 N/A

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

       20. lower-/.f64 N/A

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

       21. distribute-frac-neg2 N/A

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

       22. mult-flip N/A

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

       23. lower-*.f64 N/A

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

   11. Applied rewrites 49.7%

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

   if 0.650000000000000022 < x < 1.10000000000000002e203

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.4

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

   6. Applied rewrites 57.5%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{x}{\color{blue}{e^{x}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{e^{x}} \]

      2. lower-exp.f64 16.5

         \[\leadsto \frac{x}{e^{x}} \]

   9. Applied rewrites 16.5%

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

   if 1.10000000000000002e203 < x

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.4%

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

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 52.4

         \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   7. Applied rewrites 52.4%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 52.4

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 52.4

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 52.3

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

   12. Applied rewrites 52.3%

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

Alternative 10: 63.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.65:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+203}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.3333333333333333 \cdot x\right) \cdot x, x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.65)
   (/ (fma x x -1.0) (- -1.0 x))
   (if (<= x 1.1e+203)
     (/ x (exp x))
     (fma (* (* 0.3333333333333333 x) x) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.65) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else if (x <= 1.1e+203) {
		tmp = x / exp(x);
	} else {
		tmp = fma(((0.3333333333333333 * x) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.65)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	elseif (x <= 1.1e+203)
		tmp = Float64(x / exp(x));
	else
		tmp = fma(Float64(Float64(0.3333333333333333 * x) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.65], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e+203], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.650000000000000022

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.1

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot x \]

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.1

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.1%

      \[\leadsto 1 - x \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. sub-negate-rev N/A

          \[\leadsto \mathsf{neg}\left(\left(x - 1\right)\right) \]

       3. flip-- N/A

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

       4. lower-unsound-+.f32 N/A

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

       5. lower-+.f32 N/A

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

       6. metadata-eval N/A

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

       7. sub-flip N/A

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

       8. lift--.f64 N/A

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

       9. lower-unsound-/.f64 N/A

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

       10. lower-unsound-*.f64 N/A

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

       11. lower-unsound--.f32 N/A

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

       12. lower-unsound-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. metadata-eval N/A

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

       15. lower--.f32 N/A

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

       16. sub-negate-rev N/A

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

       17. lift--.f64 N/A

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

       18. metadata-eval N/A

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

       19. lift-*.f64 N/A

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

       20. lower-/.f64 N/A

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

       21. distribute-frac-neg2 N/A

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

       22. lower-/.f64 N/A

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

   11. Applied rewrites 49.7%

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

   if 0.650000000000000022 < x < 1.10000000000000002e203

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.4

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

   6. Applied rewrites 57.5%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{x}{\color{blue}{e^{x}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{e^{x}} \]

      2. lower-exp.f64 16.5

         \[\leadsto \frac{x}{e^{x}} \]

   9. Applied rewrites 16.5%

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

   if 1.10000000000000002e203 < x

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.4%

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

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 52.4

         \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   7. Applied rewrites 52.4%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 52.4

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 52.4

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 52.3

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

   12. Applied rewrites 52.3%

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

Alternative 11: 59.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.9e-6)
   (/ (fma x x -1.0) (- -1.0 x))
   (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.9e-6) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.9e-6)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.9e-6], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9e-6

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.1

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot x \]

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.1

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.1%

      \[\leadsto 1 - x \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. sub-negate-rev N/A

          \[\leadsto \mathsf{neg}\left(\left(x - 1\right)\right) \]

       3. flip-- N/A

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

       4. lower-unsound-+.f32 N/A

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

       5. lower-+.f32 N/A

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

       6. metadata-eval N/A

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

       7. sub-flip N/A

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

       8. lift--.f64 N/A

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

       9. lower-unsound-/.f64 N/A

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

       10. lower-unsound-*.f64 N/A

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

       11. lower-unsound--.f32 N/A

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

       12. lower-unsound-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. metadata-eval N/A

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

       15. lower--.f32 N/A

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

       16. sub-negate-rev N/A

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

       17. lift--.f64 N/A

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

       18. metadata-eval N/A

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

       19. lift-*.f64 N/A

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

       20. lower-/.f64 N/A

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

       21. distribute-frac-neg2 N/A

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

       22. lower-/.f64 N/A

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

   11. Applied rewrites 49.7%

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

   if -1.9e-6 < x

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.4%

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

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 52.4

         \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   7. Applied rewrites 52.4%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 52.4

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 52.4

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

   9. Applied rewrites 52.4%

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

Alternative 12: 59.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.3333333333333333 \cdot x\right) \cdot x, x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.9e-6)
   (/ (fma x x -1.0) (- -1.0 x))
   (fma (* (* 0.3333333333333333 x) x) x 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.9e-6) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else {
		tmp = fma(((0.3333333333333333 * x) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.9e-6)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	else
		tmp = fma(Float64(Float64(0.3333333333333333 * x) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.9e-6], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9e-6

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.1

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot x \]

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.1

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.1%

      \[\leadsto 1 - x \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. sub-negate-rev N/A

          \[\leadsto \mathsf{neg}\left(\left(x - 1\right)\right) \]

       3. flip-- N/A

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

       4. lower-unsound-+.f32 N/A

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

       5. lower-+.f32 N/A

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

       6. metadata-eval N/A

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

       7. sub-flip N/A

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

       8. lift--.f64 N/A

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

       9. lower-unsound-/.f64 N/A

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

       10. lower-unsound-*.f64 N/A

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

       11. lower-unsound--.f32 N/A

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

       12. lower-unsound-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. metadata-eval N/A

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

       15. lower--.f32 N/A

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

       16. sub-negate-rev N/A

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

       17. lift--.f64 N/A

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

       18. metadata-eval N/A

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

       19. lift-*.f64 N/A

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

       20. lower-/.f64 N/A

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

       21. distribute-frac-neg2 N/A

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

       22. lower-/.f64 N/A

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

   11. Applied rewrites 49.7%

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

   if -1.9e-6 < x

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.4%

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

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 52.4

         \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   7. Applied rewrites 52.4%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 52.4

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 52.4

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 52.3

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

   12. Applied rewrites 52.3%

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

Alternative 13: 52.3% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (fma (* (* 0.3333333333333333 x) x) x 1.0))

C

double code(double x, double eps) {
	return fma(((0.3333333333333333 * x) * x), x, 1.0);
}

Julia

function code(x, eps)
	return fma(Float64(Float64(0.3333333333333333 * x) * x), x, 1.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]

Derivation

1. Initial program 72.7%

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

2. 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. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 57.4%

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

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. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 52.4

      \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

7. Applied rewrites 52.4%

   \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

      \[\leadsto \left(\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot x\right) \cdot x + 1 \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot x, x, 1\right) \]

   9. lower-*.f64 52.4

      \[\leadsto \mathsf{fma}\left(\left(0.3333333333333333 \cdot x - 0.5\right) \cdot x, x, 1\right) \]

   10. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot x, x, 1\right) \]

   11. sub-flip N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot x, x, 1\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot x, x, 1\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x + \frac{-1}{2}\right) \cdot x, x, 1\right) \]

   14. lower-fma.f64 52.4

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right) \]

9. Applied rewrites 52.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right) \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x\right) \cdot x, x, 1\right) \]
11. Step-by-step derivation

    1. lower-*.f64 52.3

       \[\leadsto \mathsf{fma}\left(\left(0.3333333333333333 \cdot x\right) \cdot x, x, 1\right) \]

12. Applied rewrites 52.3%

    \[\leadsto \mathsf{fma}\left(\left(0.3333333333333333 \cdot x\right) \cdot x, x, 1\right) \]
13. Add Preprocessing

Alternative 14: 43.7% accurate, 58.4× speedup

Math

\[1 \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

double code(double x, double eps) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 72.7%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 43.7%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025173 
(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))