NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 98.9%

Time: 5.6s

Alternatives: 13

Speedup: 2.5×

• 38.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -5.696572050608561e+136
  2. eps = -2.5439632492018115e-24

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2.75 \cdot 10^{-8}:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{eps\_m \cdot x} - \left(-e^{-x \cdot eps\_m}\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2.75e-8)
   (* (* (exp (- x)) 2.0) 0.5)
   (* (- (exp (* eps_m x)) (- (exp (- (* x eps_m))))) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.75e-8) {
		tmp = (exp(-x) * 2.0) * 0.5;
	} else {
		tmp = (exp((eps_m * x)) - -exp(-(x * eps_m))) * 0.5;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 2.75d-8) then
        tmp = (exp(-x) * 2.0d0) * 0.5d0
    else
        tmp = (exp((eps_m * x)) - -exp(-(x * eps_m))) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.75e-8) {
		tmp = (Math.exp(-x) * 2.0) * 0.5;
	} else {
		tmp = (Math.exp((eps_m * x)) - -Math.exp(-(x * eps_m))) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 2.75e-8:
		tmp = (math.exp(-x) * 2.0) * 0.5
	else:
		tmp = (math.exp((eps_m * x)) - -math.exp(-(x * eps_m))) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2.75e-8)
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(eps_m * x)) - Float64(-exp(Float64(-Float64(x * eps_m))))) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 2.75e-8)
		tmp = (exp(-x) * 2.0) * 0.5;
	else
		tmp = (exp((eps_m * x)) - -exp(-(x * eps_m))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 2.75e-8], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision] - (-N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.7500000000000001e-8

   1. Initial program 36.2%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 97.5

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

   7. Applied rewrites 97.5%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. count-2-rev N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-exp.f64 97.5

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

   9. Applied rewrites 97.5%

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

   if 2.7500000000000001e-8 < eps

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in eps around inf

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.3

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

   10. Applied rewrites 99.3%

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

Alternative 2: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

2. Taylor expanded in eps around inf

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

4. Applied rewrites 98.9%

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

Alternative 3: 85.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-293}:\\ \;\;\;\;\left(1 - \left(-e^{-x \cdot eps\_m}\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left(eps\_m - 1\right) \cdot x} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-293)
   (* (- 1.0 (- (exp (- (* x eps_m))))) 0.5)
   (* (- (exp (* (- eps_m 1.0) x)) -1.0) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-293) {
		tmp = (1.0 - -exp(-(x * eps_m))) * 0.5;
	} else {
		tmp = (exp(((eps_m - 1.0) * x)) - -1.0) * 0.5;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1d-293)) then
        tmp = (1.0d0 - -exp(-(x * eps_m))) * 0.5d0
    else
        tmp = (exp(((eps_m - 1.0d0) * x)) - (-1.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-293) {
		tmp = (1.0 - -Math.exp(-(x * eps_m))) * 0.5;
	} else {
		tmp = (Math.exp(((eps_m - 1.0) * x)) - -1.0) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-293:
		tmp = (1.0 - -math.exp(-(x * eps_m))) * 0.5
	else:
		tmp = (math.exp(((eps_m - 1.0) * x)) - -1.0) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-293)
		tmp = Float64(Float64(1.0 - Float64(-exp(Float64(-Float64(x * eps_m))))) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(Float64(eps_m - 1.0) * x)) - -1.0) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1e-293)
		tmp = (1.0 - -exp(-(x * eps_m))) * 0.5;
	else
		tmp = (exp(((eps_m - 1.0) * x)) - -1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1e-293], N[(N[(1.0 - (-N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0000000000000001e-293

   1. Initial program 69.9%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in eps around inf

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.1

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

   10. Applied rewrites 99.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.4%

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

   if -1.0000000000000001e-293 < x

   1. Initial program 76.1%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 75.4%

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

Alternative 4: 85.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-293}:\\ \;\;\;\;\left(1 - \left(-e^{-x \cdot eps\_m}\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{eps\_m \cdot x} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-293)
   (* (- 1.0 (- (exp (- (* x eps_m))))) 0.5)
   (* (- (exp (* eps_m x)) -1.0) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-293) {
		tmp = (1.0 - -exp(-(x * eps_m))) * 0.5;
	} else {
		tmp = (exp((eps_m * x)) - -1.0) * 0.5;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1d-293)) then
        tmp = (1.0d0 - -exp(-(x * eps_m))) * 0.5d0
    else
        tmp = (exp((eps_m * x)) - (-1.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-293) {
		tmp = (1.0 - -Math.exp(-(x * eps_m))) * 0.5;
	} else {
		tmp = (Math.exp((eps_m * x)) - -1.0) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-293:
		tmp = (1.0 - -math.exp(-(x * eps_m))) * 0.5
	else:
		tmp = (math.exp((eps_m * x)) - -1.0) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-293)
		tmp = Float64(Float64(1.0 - Float64(-exp(Float64(-Float64(x * eps_m))))) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(eps_m * x)) - -1.0) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1e-293)
		tmp = (1.0 - -exp(-(x * eps_m))) * 0.5;
	else
		tmp = (exp((eps_m * x)) - -1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1e-293], N[(N[(1.0 - (-N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0000000000000001e-293

   1. Initial program 69.9%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in eps around inf

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.1

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

   10. Applied rewrites 99.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.4%

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

   if -1.0000000000000001e-293 < x

   1. Initial program 76.1%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in eps around inf

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 75.6%

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

Alternative 5: 78.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-293}:\\ \;\;\;\;\left(e^{-x} + 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{eps\_m \cdot x} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-293)
   (* (+ (exp (- x)) 1.0) 0.5)
   (* (- (exp (* eps_m x)) -1.0) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-293) {
		tmp = (exp(-x) + 1.0) * 0.5;
	} else {
		tmp = (exp((eps_m * x)) - -1.0) * 0.5;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1d-293)) then
        tmp = (exp(-x) + 1.0d0) * 0.5d0
    else
        tmp = (exp((eps_m * x)) - (-1.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-293) {
		tmp = (Math.exp(-x) + 1.0) * 0.5;
	} else {
		tmp = (Math.exp((eps_m * x)) - -1.0) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-293:
		tmp = (math.exp(-x) + 1.0) * 0.5
	else:
		tmp = (math.exp((eps_m * x)) - -1.0) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-293)
		tmp = Float64(Float64(exp(Float64(-x)) + 1.0) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(eps_m * x)) - -1.0) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1e-293)
		tmp = (exp(-x) + 1.0) * 0.5;
	else
		tmp = (exp((eps_m * x)) - -1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1e-293], N[(N[(N[Exp[(-x)], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0000000000000001e-293

   1. Initial program 69.9%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 82.4

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

   7. Applied rewrites 82.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 82.4%

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

   if -1.0000000000000001e-293 < x

   1. Initial program 76.1%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in eps around inf

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 75.6%

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

Alternative 6: 70.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\left(e^{-x} + 1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{eps\_m} - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 360.0)
   (* (+ (exp (- x)) 1.0) 0.5)
   (if (<= x 8.2e+102)
     (/ (- (/ 1.0 eps_m) (- (/ 1.0 eps_m) 1.0)) 2.0)
     (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 360.0) {
		tmp = (exp(-x) + 1.0) * 0.5;
	} else if (x <= 8.2e+102) {
		tmp = ((1.0 / eps_m) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 360.0)
		tmp = Float64(Float64(exp(Float64(-x)) + 1.0) * 0.5);
	elseif (x <= 8.2e+102)
		tmp = Float64(Float64(Float64(1.0 / eps_m) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 360.0], N[(N[(N[Exp[(-x)], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 8.2e+102], N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 360

   1. Initial program 63.2%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 79.0

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 79.0%

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

   if 360 < x < 8.1999999999999999e102

   1. Initial program 99.7%

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

   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. lift-/.f64 N/A

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

      2. lift--.f64 53.2

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

   4. Applied rewrites 53.2%

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

   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. +-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 46.3

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

   7. Applied rewrites 46.3%

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

   8. Taylor expanded in eps around 0

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

      1. lift-/.f64 46.3

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

   10. Applied rewrites 46.3%

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

   if 8.1999999999999999e102 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 51.2

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

   7. Applied rewrites 51.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 51.2

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

   9. Applied rewrites 51.2%

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

Alternative 7: 70.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 8.2e+102)
   (* (* (exp (- x)) 2.0) 0.5)
   (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 8.2e+102) {
		tmp = (exp(-x) * 2.0) * 0.5;
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 8.2e+102)
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) * 0.5);
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 8.2e+102], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $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 < 8.1999999999999999e102

   1. Initial program 67.2%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 75.6

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

   7. Applied rewrites 75.6%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. count-2-rev N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-exp.f64 75.6

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

   9. Applied rewrites 75.6%

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

   if 8.1999999999999999e102 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 51.2

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

   7. Applied rewrites 51.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 51.2

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

   9. Applied rewrites 51.2%

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

Alternative 8: 63.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{eps\_m} - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 360.0)
   (* (fma (- x 2.0) x 2.0) 0.5)
   (if (<= x 8.2e+102)
     (/ (- (/ 1.0 eps_m) (- (/ 1.0 eps_m) 1.0)) 2.0)
     (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 360.0) {
		tmp = fma((x - 2.0), x, 2.0) * 0.5;
	} else if (x <= 8.2e+102) {
		tmp = ((1.0 / eps_m) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 360.0)
		tmp = Float64(fma(Float64(x - 2.0), x, 2.0) * 0.5);
	elseif (x <= 8.2e+102)
		tmp = Float64(Float64(Float64(1.0 / eps_m) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 360.0], N[(N[(N[(x - 2.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 8.2e+102], N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 360

   1. Initial program 63.2%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 79.0

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 69.5

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

   10. Applied rewrites 69.5%

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

   if 360 < x < 8.1999999999999999e102

   1. Initial program 99.7%

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

   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. lift-/.f64 N/A

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

      2. lift--.f64 53.2

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

   4. Applied rewrites 53.2%

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

   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. +-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 46.3

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

   7. Applied rewrites 46.3%

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

   8. Taylor expanded in eps around 0

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

      1. lift-/.f64 46.3

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

   10. Applied rewrites 46.3%

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

   if 8.1999999999999999e102 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 51.2

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

   7. Applied rewrites 51.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 51.2

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

   9. Applied rewrites 51.2%

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

Alternative 9: 60.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -108:\\ \;\;\;\;\mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -108.0)
   (* (fma (- x 2.0) x 2.0) 0.5)
   (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -108.0) {
		tmp = fma((x - 2.0), x, 2.0) * 0.5;
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -108.0)
		tmp = Float64(fma(Float64(x - 2.0), x, 2.0) * 0.5);
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -108.0], N[(N[(N[(x - 2.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $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 < -108

   1. Initial program 99.5%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 52.2

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

   10. Applied rewrites 52.2%

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

   if -108 < x

   1. Initial program 69.1%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 62.1

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

   7. Applied rewrites 62.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 62.1

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

   9. Applied rewrites 62.1%

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

Alternative 10: 60.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -108:\\ \;\;\;\;\mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -108.0)
   (* (fma (- x 2.0) x 2.0) 0.5)
   (fma (* 0.3333333333333333 x) (* x x) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -108.0) {
		tmp = fma((x - 2.0), x, 2.0) * 0.5;
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -108.0)
		tmp = Float64(fma(Float64(x - 2.0), x, 2.0) * 0.5);
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -108.0], N[(N[(N[(x - 2.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -108

   1. Initial program 99.5%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 52.2

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

   10. Applied rewrites 52.2%

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

   if -108 < x

   1. Initial program 69.1%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 62.1

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

   7. Applied rewrites 62.1%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 61.9

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

   10. Applied rewrites 61.9%

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

Alternative 11: 60.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 53000000000000:\\ \;\;\;\;\mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 53000000000000.0)
   (* (fma (- x 2.0) x 2.0) 0.5)
   (* (* (* x x) x) 0.3333333333333333)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 53000000000000.0) {
		tmp = fma((x - 2.0), x, 2.0) * 0.5;
	} else {
		tmp = ((x * x) * x) * 0.3333333333333333;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 53000000000000.0)
		tmp = Float64(fma(Float64(x - 2.0), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(x * x) * x) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 53000000000000.0], N[(N[(N[(x - 2.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.3e13

   1. Initial program 63.7%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 78.5

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

   7. Applied rewrites 78.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 68.6

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

   10. Applied rewrites 68.6%

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

   if 5.3e13 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 37.4

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

   7. Applied rewrites 37.4%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 37.4

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

   10. Applied rewrites 37.4%

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

Alternative 12: 53.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 120000:\\ \;\;\;\;\mathsf{fma}\left(-2, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 120000.0)
   (* (fma -2.0 x 2.0) 0.5)
   (* (* (* x x) x) 0.3333333333333333)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 120000.0) {
		tmp = fma(-2.0, x, 2.0) * 0.5;
	} else {
		tmp = ((x * x) * x) * 0.3333333333333333;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 120000.0)
		tmp = Float64(fma(-2.0, x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(x * x) * x) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 120000.0], N[(N[(-2.0 * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e5

   1. Initial program 63.3%

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

   2. Taylor expanded in eps around inf

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 78.9

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

   7. Applied rewrites 78.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 59.9

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

   10. Applied rewrites 59.9%

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

   if 1.2e5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 36.4

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 36.4

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

   10. Applied rewrites 36.4%

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

Alternative 13: 44.2% accurate, 58.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 44.2%

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

Reproduce

herbie shell --seed 2025110 
(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))