NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 99.0%

Time: 5.7s

Alternatives: 12

Speedup: 1.6×

• 38.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.2893326905206917e+230
  2. eps = -5.517947112043823e-36

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: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

2. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 99.0

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

4. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.0

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

6. Applied rewrites 99.0%

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

Alternative 2: 88.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(e^{-\varepsilon \cdot x} + e^{\left(\varepsilon - 1\right) \cdot x}\right) \cdot 0.5 \end{array} \]

FPCore

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

C

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

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 = (exp(-(eps * x)) + exp(((eps - 1.0d0) * x))) * 0.5d0
end function

Java

public static double code(double x, double eps) {
	return (Math.exp(-(eps * x)) + Math.exp(((eps - 1.0) * x))) * 0.5;
}

Python

def code(x, eps):
	return (math.exp(-(eps * x)) + math.exp(((eps - 1.0) * x))) * 0.5

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (exp(-(eps * x)) + exp(((eps - 1.0) * x))) * 0.5;
end

Wolfram

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

Derivation

1. Initial program 73.8%

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

2. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 99.0

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

4. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.0

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in eps around inf

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

   1. lower-*.f64 88.8

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

9. Applied rewrites 88.8%

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

Alternative 3: 64.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\left(\frac{e^{-x}}{\varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-265}:\\ \;\;\;\;\left(e^{-\varepsilon \cdot x} + \left(1 + x \cdot \left(\varepsilon - 1\right)\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2e+17)
   (* (- (/ (exp (- x)) eps) (- (/ 1.0 eps) 1.0)) 0.5)
   (if (<= x 4e-265)
     (* (+ (exp (- (* eps x))) (+ 1.0 (* x (- eps 1.0)))) 0.5)
     (* 0.5 (- (exp (- (* x (- 1.0 eps)))) -1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2e+17) {
		tmp = ((exp(-x) / eps) - ((1.0 / eps) - 1.0)) * 0.5;
	} else if (x <= 4e-265) {
		tmp = (exp(-(eps * x)) + (1.0 + (x * (eps - 1.0)))) * 0.5;
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - eps))) - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2d+17)) then
        tmp = ((exp(-x) / eps) - ((1.0d0 / eps) - 1.0d0)) * 0.5d0
    else if (x <= 4d-265) then
        tmp = (exp(-(eps * x)) + (1.0d0 + (x * (eps - 1.0d0)))) * 0.5d0
    else
        tmp = 0.5d0 * (exp(-(x * (1.0d0 - eps))) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2e+17) {
		tmp = ((Math.exp(-x) / eps) - ((1.0 / eps) - 1.0)) * 0.5;
	} else if (x <= 4e-265) {
		tmp = (Math.exp(-(eps * x)) + (1.0 + (x * (eps - 1.0)))) * 0.5;
	} else {
		tmp = 0.5 * (Math.exp(-(x * (1.0 - eps))) - -1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2e+17:
		tmp = ((math.exp(-x) / eps) - ((1.0 / eps) - 1.0)) * 0.5
	elif x <= 4e-265:
		tmp = (math.exp(-(eps * x)) + (1.0 + (x * (eps - 1.0)))) * 0.5
	else:
		tmp = 0.5 * (math.exp(-(x * (1.0 - eps))) - -1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2e+17)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps) - Float64(Float64(1.0 / eps) - 1.0)) * 0.5);
	elseif (x <= 4e-265)
		tmp = Float64(Float64(exp(Float64(-Float64(eps * x))) + Float64(1.0 + Float64(x * Float64(eps - 1.0)))) * 0.5);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - eps)))) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e+17)
		tmp = ((exp(-x) / eps) - ((1.0 / eps) - 1.0)) * 0.5;
	elseif (x <= 4e-265)
		tmp = (exp(-(eps * x)) + (1.0 + (x * (eps - 1.0)))) * 0.5;
	else
		tmp = 0.5 * (exp(-(x * (1.0 - eps))) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2e+17], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4e-265], N[(N[(N[Exp[(-N[(eps * x), $MachinePrecision])], $MachinePrecision] + N[(1.0 + N[(x * N[(eps - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e17

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 38.1

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

   4. Applied rewrites 38.1%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 12.0

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

   7. Applied rewrites 12.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 12.0

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

   9. Applied rewrites 12.0%

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

   if -2e17 < x < 3.99999999999999994e-265

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.0

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in eps around inf

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

      1. lower-*.f64 88.8

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

   9. Applied rewrites 88.8%

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

   10. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 64.4

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

   12. Applied rewrites 64.4%

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

   if 3.99999999999999994e-265 < x

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.5%

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

Alternative 4: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -330:\\ \;\;\;\;\left(\frac{e^{-x}}{\varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -330.0)
   (* (- (/ (exp (- x)) eps) (- (/ 1.0 eps) 1.0)) 0.5)
   (* 0.5 (- (exp (- (* x (- 1.0 eps)))) -1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -330.0) {
		tmp = ((exp(-x) / eps) - ((1.0 / eps) - 1.0)) * 0.5;
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - eps))) - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-330.0d0)) then
        tmp = ((exp(-x) / eps) - ((1.0d0 / eps) - 1.0d0)) * 0.5d0
    else
        tmp = 0.5d0 * (exp(-(x * (1.0d0 - eps))) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -330.0) {
		tmp = ((Math.exp(-x) / eps) - ((1.0 / eps) - 1.0)) * 0.5;
	} else {
		tmp = 0.5 * (Math.exp(-(x * (1.0 - eps))) - -1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -330.0:
		tmp = ((math.exp(-x) / eps) - ((1.0 / eps) - 1.0)) * 0.5
	else:
		tmp = 0.5 * (math.exp(-(x * (1.0 - eps))) - -1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -330.0)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps) - Float64(Float64(1.0 / eps) - 1.0)) * 0.5);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - eps)))) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -330.0)
		tmp = ((exp(-x) / eps) - ((1.0 / eps) - 1.0)) * 0.5;
	else
		tmp = 0.5 * (exp(-(x * (1.0 - eps))) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -330.0], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -330

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 38.1

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

   4. Applied rewrites 38.1%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 12.0

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

   7. Applied rewrites 12.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 12.0

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

   9. Applied rewrites 12.0%

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

   if -330 < x

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.5%

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

Alternative 5: 63.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{1 - \left(\left(x \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(x, x, 1\right)}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e+154)
   (/ (fma x x -1.0) (- -1.0 x))
   (if (<= x 4e-265)
     (/ (/ (- 1.0 (* (* (* x x) x) x)) (fma x x 1.0)) (- x -1.0))
     (* 0.5 (- (exp (- (* x (- 1.0 eps)))) -1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else if (x <= 4e-265) {
		tmp = ((1.0 - (((x * x) * x) * x)) / fma(x, x, 1.0)) / (x - -1.0);
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - eps))) - -1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	elseif (x <= 4e-265)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(x * x) * x) * x)) / fma(x, x, 1.0)) / Float64(x - -1.0));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - eps)))) - -1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e+154], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e-265], N[(N[(N[(1.0 - N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.0

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

   7. Applied rewrites 43.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.0

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.0%

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

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. sub-negate-rev N/A

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

       3. metadata-eval N/A

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

       4. add-flip-rev N/A

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

       5. flip-+ N/A

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

       6. lift--.f64 N/A

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

       7. distribute-frac-neg2 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. mul-1-neg N/A

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

       11. add-flip-rev N/A

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

       12. lift-*.f64 N/A

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

       13. lower-fma.f64 N/A

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

       14. lift--.f64 N/A

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

       15. sub-negate-rev N/A

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

       16. lower--.f64 49.2

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

   11. Applied rewrites 49.2%

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

   if -1.35000000000000003e154 < x < 3.99999999999999994e-265

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.0

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

   7. Applied rewrites 43.0%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. metadata-eval N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sqr-neg-rev N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. add-flip-rev N/A

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

      14. +-commutative N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. add-flip N/A

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

      21. metadata-eval N/A

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

      22. lower--.f64 49.2

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

   9. Applied rewrites 49.2%

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

       1. lift--.f64 N/A

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

       2. flip-- N/A

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

       3. lower-/.f64 N/A

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

       4. metadata-eval N/A

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

       5. lower--.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. lower-fma.f64 45.6

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

   11. Applied rewrites 45.6%

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

   if 3.99999999999999994e-265 < x

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.5%

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

Alternative 6: 63.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x - -1, x, 1\right) \cdot \left(1 - x\right)}{\mathsf{fma}\left(x, x, x\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e+154)
   (/ (fma x x -1.0) (- -1.0 x))
   (if (<= x 4e-265)
     (/ (* (fma (- x -1.0) x 1.0) (- 1.0 x)) (+ (fma x x x) 1.0))
     (* 0.5 (- (exp (- (* x (- 1.0 eps)))) -1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else if (x <= 4e-265) {
		tmp = (fma((x - -1.0), x, 1.0) * (1.0 - x)) / (fma(x, x, x) + 1.0);
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - eps))) - -1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	elseif (x <= 4e-265)
		tmp = Float64(Float64(fma(Float64(x - -1.0), x, 1.0) * Float64(1.0 - x)) / Float64(fma(x, x, x) + 1.0));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - eps)))) - -1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e+154], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e-265], N[(N[(N[(N[(x - -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.0

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

   7. Applied rewrites 43.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.0

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.0%

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

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. sub-negate-rev N/A

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

       3. metadata-eval N/A

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

       4. add-flip-rev N/A

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

       5. flip-+ N/A

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

       6. lift--.f64 N/A

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

       7. distribute-frac-neg2 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. mul-1-neg N/A

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

       11. add-flip-rev N/A

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

       12. lift-*.f64 N/A

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

       13. lower-fma.f64 N/A

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

       14. lift--.f64 N/A

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

       15. sub-negate-rev N/A

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

       16. lower--.f64 49.2

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

   11. Applied rewrites 49.2%

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

   if -1.35000000000000003e154 < x < 3.99999999999999994e-265

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.0

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

   7. Applied rewrites 43.0%

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

   9. Applied rewrites 44.5%

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

       1. lift--.f64 N/A

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

       2. metadata-eval N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. pow3 N/A

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

       6. difference-cubes N/A

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

       7. metadata-eval N/A

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

       8. *-lft-identity N/A

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

       9. lift-fma.f64 N/A

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

       10. +-commutative N/A

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

       11. lift-+.f64 N/A

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

       12. lift--.f64 N/A

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

       13. lower-*.f64 44.5

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

       14. lift-+.f64 N/A

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

       15. lift-fma.f64 N/A

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

       16. distribute-lft1-in N/A

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

       17. metadata-eval N/A

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

       18. sub-flip N/A

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

       19. lift--.f64 N/A

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

       20. lower-fma.f64 44.5

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

   11. Applied rewrites 44.5%

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

   if 3.99999999999999994e-265 < x

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.5%

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

Alternative 7: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x, -1\right)}{-1 - \mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e+154)
   (/ (fma x x -1.0) (- -1.0 x))
   (if (<= x 4e-265)
     (/ (fma (* x x) x -1.0) (- -1.0 (fma x x x)))
     (* 0.5 (- (exp (- (* x (- 1.0 eps)))) -1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else if (x <= 4e-265) {
		tmp = fma((x * x), x, -1.0) / (-1.0 - fma(x, x, x));
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - eps))) - -1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	elseif (x <= 4e-265)
		tmp = Float64(fma(Float64(x * x), x, -1.0) / Float64(-1.0 - fma(x, x, x)));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - eps)))) - -1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e+154], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e-265], N[(N[(N[(x * x), $MachinePrecision] * x + -1.0), $MachinePrecision] / N[(-1.0 - N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.0

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

   7. Applied rewrites 43.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.0

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.0%

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

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. sub-negate-rev N/A

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

       3. metadata-eval N/A

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

       4. add-flip-rev N/A

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

       5. flip-+ N/A

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

       6. lift--.f64 N/A

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

       7. distribute-frac-neg2 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. mul-1-neg N/A

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

       11. add-flip-rev N/A

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

       12. lift-*.f64 N/A

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

       13. lower-fma.f64 N/A

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

       14. lift--.f64 N/A

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

       15. sub-negate-rev N/A

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

       16. lower--.f64 49.2

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

   11. Applied rewrites 49.2%

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

   if -1.35000000000000003e154 < x < 3.99999999999999994e-265

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.0

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

   7. Applied rewrites 43.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.0

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.0%

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

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. flip3-- N/A

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

       3. metadata-eval N/A

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

       4. pow3 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift--.f64 N/A

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

       8. metadata-eval N/A

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

       9. *-lft-identity N/A

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

       10. lift-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lift-+.f64 N/A

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

       13. frac-2neg N/A

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

       14. lower-/.f64 N/A

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

       15. lift--.f64 N/A

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

       16. sub-negate-rev N/A

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

       17. metadata-eval N/A

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

       18. add-flip-rev N/A

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

       19. lift-*.f64 N/A

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

       20. lower-fma.f64 N/A

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

       21. lift-+.f64 N/A

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

       22. add-flip N/A

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

       23. metadata-eval N/A

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

   11. Applied rewrites 44.5%

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

   if 3.99999999999999994e-265 < x

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.5%

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

Alternative 8: 62.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x, -1\right)}{-1 - \mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e+154)
   (/ (fma x x -1.0) (- -1.0 x))
   (if (<= x 2.2e-8)
     (/ (fma (* x x) x -1.0) (- -1.0 (fma x x x)))
     (/ (- (+ 1.0 (/ 1.0 eps)) (- (/ 1.0 eps) 1.0)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else if (x <= 2.2e-8) {
		tmp = fma((x * x), x, -1.0) / (-1.0 - fma(x, x, x));
	} else {
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	elseif (x <= 2.2e-8)
		tmp = Float64(fma(Float64(x * x), x, -1.0) / Float64(-1.0 - fma(x, x, x)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e+154], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-8], N[(N[(N[(x * x), $MachinePrecision] * x + -1.0), $MachinePrecision] / N[(-1.0 - N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.0

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

   7. Applied rewrites 43.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.0

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.0%

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

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. sub-negate-rev N/A

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

       3. metadata-eval N/A

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

       4. add-flip-rev N/A

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

       5. flip-+ N/A

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

       6. lift--.f64 N/A

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

       7. distribute-frac-neg2 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. mul-1-neg N/A

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

       11. add-flip-rev N/A

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

       12. lift-*.f64 N/A

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

       13. lower-fma.f64 N/A

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

       14. lift--.f64 N/A

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

       15. sub-negate-rev N/A

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

       16. lower--.f64 49.2

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

   11. Applied rewrites 49.2%

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

   if -1.35000000000000003e154 < x < 2.1999999999999998e-8

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.0

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

   7. Applied rewrites 43.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.0

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.0%

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

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. flip3-- N/A

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

       3. metadata-eval N/A

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

       4. pow3 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift--.f64 N/A

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

       8. metadata-eval N/A

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

       9. *-lft-identity N/A

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

       10. lift-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lift-+.f64 N/A

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

       13. frac-2neg N/A

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

       14. lower-/.f64 N/A

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

       15. lift--.f64 N/A

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

       16. sub-negate-rev N/A

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

       17. metadata-eval N/A

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

       18. add-flip-rev N/A

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

       19. lift-*.f64 N/A

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

       20. lower-fma.f64 N/A

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

       21. lift-+.f64 N/A

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

       22. add-flip N/A

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

       23. metadata-eval N/A

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

   11. Applied rewrites 44.5%

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

   if 2.1999999999999998e-8 < x

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 38.1

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

   4. Applied rewrites 38.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 31.0

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

   7. Applied rewrites 31.0%

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

Alternative 9: 62.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.2e-8)
   (/ (fma x x -1.0) (- -1.0 x))
   (/ (- (+ 1.0 (/ 1.0 eps)) (- (/ 1.0 eps) 1.0)) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.2e-8) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else {
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.2e-8)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.2e-8], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1999999999999998e-8

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.0

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

   7. Applied rewrites 43.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.0

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.0%

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

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. sub-negate-rev N/A

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

       3. metadata-eval N/A

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

       4. add-flip-rev N/A

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

       5. flip-+ N/A

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

       6. lift--.f64 N/A

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

       7. distribute-frac-neg2 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. mul-1-neg N/A

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

       11. add-flip-rev N/A

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

       12. lift-*.f64 N/A

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

       13. lower-fma.f64 N/A

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

       14. lift--.f64 N/A

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

       15. sub-negate-rev N/A

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

       16. lower--.f64 49.2

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

   11. Applied rewrites 49.2%

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

   if 2.1999999999999998e-8 < x

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 38.1

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

   4. Applied rewrites 38.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 31.0

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

   7. Applied rewrites 31.0%

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

Alternative 10: 62.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.0)
   (/ (fma x x -1.0) (- -1.0 x))
   (/ (- (/ 1.0 eps) (- (/ 1.0 eps) 1.0)) 2.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 73.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 43.0

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

   7. Applied rewrites 43.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.0

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.0%

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

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. sub-negate-rev N/A

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

       3. metadata-eval N/A

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

       4. add-flip-rev N/A

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

       5. flip-+ N/A

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

       6. lift--.f64 N/A

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

       7. distribute-frac-neg2 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. mul-1-neg N/A

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

       11. add-flip-rev N/A

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

       12. lift-*.f64 N/A

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

       13. lower-fma.f64 N/A

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

       14. lift--.f64 N/A

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

       15. sub-negate-rev N/A

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

       16. lower--.f64 49.2

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

   11. Applied rewrites 49.2%

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

   if 1 < x

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 38.1

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

   4. Applied rewrites 38.1%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 18.6

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

   10. Applied rewrites 18.6%

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

Alternative 11: 49.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (fma x x -1.0) (- -1.0 x)))

C

double code(double x, double eps) {
	return fma(x, x, -1.0) / (-1.0 - x);
}

Julia

function code(x, eps)
	return Float64(fma(x, x, -1.0) / Float64(-1.0 - x))
end

Wolfram

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

Derivation

1. Initial program 73.8%

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

2. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 43.0

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

7. Applied rewrites 43.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. sub-flip-reverse N/A

      \[\leadsto 1 - x \]

   5. lower--.f64 43.0

      \[\leadsto 1 - x \]

9. Applied rewrites 43.0%

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

    1. lift--.f64 N/A

       \[\leadsto 1 - x \]

    2. sub-negate-rev N/A

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

    3. metadata-eval N/A

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

    4. add-flip-rev N/A

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

    5. flip-+ N/A

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

    6. lift--.f64 N/A

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

    7. distribute-frac-neg2 N/A

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

    8. lower-/.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. mul-1-neg N/A

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

    11. add-flip-rev N/A

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

    12. lift-*.f64 N/A

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

    13. lower-fma.f64 N/A

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

    14. lift--.f64 N/A

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

    15. sub-negate-rev N/A

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

    16. lower--.f64 49.2

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

11. Applied rewrites 49.2%

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

Alternative 12: 43.6% accurate, 58.4× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 73.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 43.6%

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

Reproduce

herbie shell --seed 2025156 
(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))