NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 99.0%

Time: 5.8s

Alternatives: 16

Speedup: 2.1×

• 61.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -5.931871856601423e+169
  2. eps = -5.235579845461429e-57

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

3. Applied rewrites 73.2%

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

4. Taylor expanded in eps around inf

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

   1. lower-+.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 99.0%

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

6. Applied rewrites 99.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.0%

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

8. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (exp(((-1.0 - eps) * x)) + exp((x * (eps - 1.0)))) * 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((((-1.0d0) - eps) * x)) + exp((x * (eps - 1.0d0)))) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

3. Applied rewrites 73.2%

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

4. Taylor expanded in eps around inf

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

   1. lower-+.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 99.0%

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

6. Applied rewrites 99.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. lift--.f64 N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. lower-+.f64 99.0%

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

8. Applied rewrites 99.0%

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

Alternative 3: 86.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left|\varepsilon\right| - 1\\ t_1 := x \cdot t\_0\\ t_2 := e^{-x}\\ \mathbf{if}\;\left|\varepsilon\right| \leq 6600:\\ \;\;\;\;0.5 \cdot \left(t\_2 - -1 \cdot t\_2\right)\\ \mathbf{elif}\;\left|\varepsilon\right| \leq 1.15 \cdot 10^{+134}:\\ \;\;\;\;\left(e^{t\_1} - \mathsf{fma}\left(\left|\varepsilon\right| - -1, x, -1\right)\right) \cdot 0.5\\ \mathbf{elif}\;\left|\varepsilon\right| \leq 8.2 \cdot 10^{+158}:\\ \;\;\;\;\left(\left(1 + t\_1\right) + \frac{1}{e^{x + \left|\varepsilon\right| \cdot x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left|\varepsilon\right| \cdot \left|\varepsilon\right| - 1 \cdot 1}{t\_0}}{\left|\varepsilon\right|} - \left(\frac{1}{\left|\varepsilon\right|} - 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (fabs eps) 1.0)) (t_1 (* x t_0)) (t_2 (exp (- x))))
   (if (<= (fabs eps) 6600.0)
     (* 0.5 (- t_2 (* -1.0 t_2)))
     (if (<= (fabs eps) 1.15e+134)
       (* (- (exp t_1) (fma (- (fabs eps) -1.0) x -1.0)) 0.5)
       (if (<= (fabs eps) 8.2e+158)
         (* (+ (+ 1.0 t_1) (/ 1.0 (exp (+ x (* (fabs eps) x))))) 0.5)
         (/
          (-
           (/ (/ (- (* (fabs eps) (fabs eps)) (* 1.0 1.0)) t_0) (fabs eps))
           (- (/ 1.0 (fabs eps)) 1.0))
          2.0))))))

C

double code(double x, double eps) {
	double t_0 = fabs(eps) - 1.0;
	double t_1 = x * t_0;
	double t_2 = exp(-x);
	double tmp;
	if (fabs(eps) <= 6600.0) {
		tmp = 0.5 * (t_2 - (-1.0 * t_2));
	} else if (fabs(eps) <= 1.15e+134) {
		tmp = (exp(t_1) - fma((fabs(eps) - -1.0), x, -1.0)) * 0.5;
	} else if (fabs(eps) <= 8.2e+158) {
		tmp = ((1.0 + t_1) + (1.0 / exp((x + (fabs(eps) * x))))) * 0.5;
	} else {
		tmp = (((((fabs(eps) * fabs(eps)) - (1.0 * 1.0)) / t_0) / fabs(eps)) - ((1.0 / fabs(eps)) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(abs(eps) - 1.0)
	t_1 = Float64(x * t_0)
	t_2 = exp(Float64(-x))
	tmp = 0.0
	if (abs(eps) <= 6600.0)
		tmp = Float64(0.5 * Float64(t_2 - Float64(-1.0 * t_2)));
	elseif (abs(eps) <= 1.15e+134)
		tmp = Float64(Float64(exp(t_1) - fma(Float64(abs(eps) - -1.0), x, -1.0)) * 0.5);
	elseif (abs(eps) <= 8.2e+158)
		tmp = Float64(Float64(Float64(1.0 + t_1) + Float64(1.0 / exp(Float64(x + Float64(abs(eps) * x))))) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(abs(eps) * abs(eps)) - Float64(1.0 * 1.0)) / t_0) / abs(eps)) - Float64(Float64(1.0 / abs(eps)) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[Abs[eps], $MachinePrecision], 6600.0], N[(0.5 * N[(t$95$2 - N[(-1.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[eps], $MachinePrecision], 1.15e+134], N[(N[(N[Exp[t$95$1], $MachinePrecision] - N[(N[(N[Abs[eps], $MachinePrecision] - -1.0), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[Abs[eps], $MachinePrecision], 8.2e+158], N[(N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(1.0 / N[Exp[N[(x + N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Abs[eps], $MachinePrecision] * N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 6600

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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 eps around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

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

   7. Applied rewrites 70.6%

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

   if 6600 < eps < 1.1499999999999999e134

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 64.3%

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

   7. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.3%

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

   9. Applied rewrites 64.3%

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

   if 1.1499999999999999e134 < eps < 8.20000000000000008e158

   1. Initial program 73.4%

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in eps around inf

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

      1. lower-+.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 99.0%

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 64.7%

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

   9. Applied rewrites 64.7%

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

   if 8.20000000000000008e158 < eps

   1. Initial program 73.4%

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

   2. Taylor expanded in 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 38.6%

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

   4. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. add-to-fraction N/A

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

      10. *-lft-identity N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. lower-exp.f64 38.5%

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

   6. Applied rewrites 38.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 30.7%

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

   9. Applied rewrites 30.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. flip-+ N/A

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

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

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

       5. lower--.f32 N/A

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

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

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

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

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

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

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

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

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

       10. lift--.f64 51.2%

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

   11. Applied rewrites 51.2%

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

Alternative 4: 86.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 1 + \left|\varepsilon\right|\\ t_1 := \left|\varepsilon\right| - 1\\ t_2 := e^{-x}\\ \mathbf{if}\;\left|\varepsilon\right| \leq 6600:\\ \;\;\;\;0.5 \cdot \left(t\_2 - -1 \cdot t\_2\right)\\ \mathbf{elif}\;\left|\varepsilon\right| \leq 4 \cdot 10^{+124}:\\ \;\;\;\;\left(e^{x \cdot t\_1} - \mathsf{fma}\left(\left|\varepsilon\right| - -1, x, -1\right)\right) \cdot 0.5\\ \mathbf{elif}\;\left|\varepsilon\right| \leq 2.15 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{t\_0}{\left|\varepsilon\right|} - -1 \cdot e^{-x \cdot t\_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left|\varepsilon\right| \cdot \left|\varepsilon\right| - 1 \cdot 1}{t\_1}}{\left|\varepsilon\right|} - \left(\frac{1}{\left|\varepsilon\right|} - 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (fabs eps))) (t_1 (- (fabs eps) 1.0)) (t_2 (exp (- x))))
   (if (<= (fabs eps) 6600.0)
     (* 0.5 (- t_2 (* -1.0 t_2)))
     (if (<= (fabs eps) 4e+124)
       (* (- (exp (* x t_1)) (fma (- (fabs eps) -1.0) x -1.0)) 0.5)
       (if (<= (fabs eps) 2.15e+160)
         (/ (- (/ t_0 (fabs eps)) (* -1.0 (exp (- (* x t_0))))) 2.0)
         (/
          (-
           (/ (/ (- (* (fabs eps) (fabs eps)) (* 1.0 1.0)) t_1) (fabs eps))
           (- (/ 1.0 (fabs eps)) 1.0))
          2.0))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + fabs(eps);
	double t_1 = fabs(eps) - 1.0;
	double t_2 = exp(-x);
	double tmp;
	if (fabs(eps) <= 6600.0) {
		tmp = 0.5 * (t_2 - (-1.0 * t_2));
	} else if (fabs(eps) <= 4e+124) {
		tmp = (exp((x * t_1)) - fma((fabs(eps) - -1.0), x, -1.0)) * 0.5;
	} else if (fabs(eps) <= 2.15e+160) {
		tmp = ((t_0 / fabs(eps)) - (-1.0 * exp(-(x * t_0)))) / 2.0;
	} else {
		tmp = (((((fabs(eps) * fabs(eps)) - (1.0 * 1.0)) / t_1) / fabs(eps)) - ((1.0 / fabs(eps)) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + abs(eps))
	t_1 = Float64(abs(eps) - 1.0)
	t_2 = exp(Float64(-x))
	tmp = 0.0
	if (abs(eps) <= 6600.0)
		tmp = Float64(0.5 * Float64(t_2 - Float64(-1.0 * t_2)));
	elseif (abs(eps) <= 4e+124)
		tmp = Float64(Float64(exp(Float64(x * t_1)) - fma(Float64(abs(eps) - -1.0), x, -1.0)) * 0.5);
	elseif (abs(eps) <= 2.15e+160)
		tmp = Float64(Float64(Float64(t_0 / abs(eps)) - Float64(-1.0 * exp(Float64(-Float64(x * t_0))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(abs(eps) * abs(eps)) - Float64(1.0 * 1.0)) / t_1) / abs(eps)) - Float64(Float64(1.0 / abs(eps)) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[Abs[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[Abs[eps], $MachinePrecision], 6600.0], N[(0.5 * N[(t$95$2 - N[(-1.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[eps], $MachinePrecision], 4e+124], N[(N[(N[Exp[N[(x * t$95$1), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Abs[eps], $MachinePrecision] - -1.0), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[Abs[eps], $MachinePrecision], 2.15e+160], N[(N[(N[(t$95$0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[Exp[(-N[(x * t$95$0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Abs[eps], $MachinePrecision] * N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 6600

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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 eps around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

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

   7. Applied rewrites 70.6%

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

   if 6600 < eps < 3.99999999999999979e124

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 64.3%

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

   7. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.3%

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

   9. Applied rewrites 64.3%

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

   if 3.99999999999999979e124 < eps < 2.14999999999999994e160

   1. Initial program 73.4%

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

   2. Taylor expanded in 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 38.6%

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

   4. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. add-to-fraction N/A

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

      10. *-lft-identity N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. lower-exp.f64 38.5%

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

   6. Applied rewrites 38.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 30.7%

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

   9. Applied rewrites 30.7%

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

   10. Taylor expanded in eps around inf

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

       1. lower-*.f64 N/A

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

       2. lower-exp.f64 N/A

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

       3. lower-neg.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 36.2%

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

   12. Applied rewrites 36.2%

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

   if 2.14999999999999994e160 < eps

   1. Initial program 73.4%

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

   2. Taylor expanded in 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 38.6%

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

   4. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. add-to-fraction N/A

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

      10. *-lft-identity N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. lower-exp.f64 38.5%

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

   6. Applied rewrites 38.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 30.7%

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

   9. Applied rewrites 30.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. flip-+ N/A

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

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

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

       5. lower--.f32 N/A

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

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

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

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

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

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

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

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

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

       10. lift--.f64 51.2%

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

   11. Applied rewrites 51.2%

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

Alternative 5: 86.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left|\varepsilon\right| - 1\\ t_1 := e^{-x}\\ \mathbf{if}\;\left|\varepsilon\right| \leq 6600:\\ \;\;\;\;0.5 \cdot \left(t\_1 - -1 \cdot t\_1\right)\\ \mathbf{elif}\;\left|\varepsilon\right| \leq 2.3 \cdot 10^{+157}:\\ \;\;\;\;\left(e^{x \cdot t\_0} - \mathsf{fma}\left(\left|\varepsilon\right| - -1, x, -1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left|\varepsilon\right| \cdot \left|\varepsilon\right| - 1 \cdot 1}{t\_0}}{\left|\varepsilon\right|} - \left(\frac{1}{\left|\varepsilon\right|} - 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (fabs eps) 1.0)) (t_1 (exp (- x))))
   (if (<= (fabs eps) 6600.0)
     (* 0.5 (- t_1 (* -1.0 t_1)))
     (if (<= (fabs eps) 2.3e+157)
       (* (- (exp (* x t_0)) (fma (- (fabs eps) -1.0) x -1.0)) 0.5)
       (/
        (-
         (/ (/ (- (* (fabs eps) (fabs eps)) (* 1.0 1.0)) t_0) (fabs eps))
         (- (/ 1.0 (fabs eps)) 1.0))
        2.0)))))

C

double code(double x, double eps) {
	double t_0 = fabs(eps) - 1.0;
	double t_1 = exp(-x);
	double tmp;
	if (fabs(eps) <= 6600.0) {
		tmp = 0.5 * (t_1 - (-1.0 * t_1));
	} else if (fabs(eps) <= 2.3e+157) {
		tmp = (exp((x * t_0)) - fma((fabs(eps) - -1.0), x, -1.0)) * 0.5;
	} else {
		tmp = (((((fabs(eps) * fabs(eps)) - (1.0 * 1.0)) / t_0) / fabs(eps)) - ((1.0 / fabs(eps)) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(abs(eps) - 1.0)
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (abs(eps) <= 6600.0)
		tmp = Float64(0.5 * Float64(t_1 - Float64(-1.0 * t_1)));
	elseif (abs(eps) <= 2.3e+157)
		tmp = Float64(Float64(exp(Float64(x * t_0)) - fma(Float64(abs(eps) - -1.0), x, -1.0)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(abs(eps) * abs(eps)) - Float64(1.0 * 1.0)) / t_0) / abs(eps)) - Float64(Float64(1.0 / abs(eps)) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[Abs[eps], $MachinePrecision], 6600.0], N[(0.5 * N[(t$95$1 - N[(-1.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[eps], $MachinePrecision], 2.3e+157], N[(N[(N[Exp[N[(x * t$95$0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Abs[eps], $MachinePrecision] - -1.0), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Abs[eps], $MachinePrecision] * N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 6600

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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 eps around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

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

   7. Applied rewrites 70.6%

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

   if 6600 < eps < 2.30000000000000004e157

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 64.3%

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

   7. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.3%

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

   9. Applied rewrites 64.3%

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

   if 2.30000000000000004e157 < eps

   1. Initial program 73.4%

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

   2. Taylor expanded in 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 38.6%

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

   4. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. add-to-fraction N/A

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

      10. *-lft-identity N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. lower-exp.f64 38.5%

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

   6. Applied rewrites 38.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 30.7%

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

   9. Applied rewrites 30.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. flip-+ N/A

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

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

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

       5. lower--.f32 N/A

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

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

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

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

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

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

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

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

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

       10. lift--.f64 51.2%

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

   11. Applied rewrites 51.2%

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

Alternative 6: 83.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(e^{-x} - -1\right)\\ \mathbf{elif}\;x \leq 31000000000000:\\ \;\;\;\;\left(e^{x \cdot \left(\varepsilon - 1\right)} - \mathsf{fma}\left(\varepsilon - -1, x, -1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\varepsilon \cdot \varepsilon - 1 \cdot 1}{\varepsilon - 1}}{\varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8e-5)
   (* 0.5 (- (exp (- x)) -1.0))
   (if (<= x 31000000000000.0)
     (* (- (exp (* x (- eps 1.0))) (fma (- eps -1.0) x -1.0)) 0.5)
     (/
      (-
       (/ (/ (- (* eps eps) (* 1.0 1.0)) (- eps 1.0)) eps)
       (- (/ 1.0 eps) 1.0))
      2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8e-5) {
		tmp = 0.5 * (exp(-x) - -1.0);
	} else if (x <= 31000000000000.0) {
		tmp = (exp((x * (eps - 1.0))) - fma((eps - -1.0), x, -1.0)) * 0.5;
	} else {
		tmp = (((((eps * eps) - (1.0 * 1.0)) / (eps - 1.0)) / eps) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8e-5)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) - -1.0));
	elseif (x <= 31000000000000.0)
		tmp = Float64(Float64(exp(Float64(x * Float64(eps - 1.0))) - fma(Float64(eps - -1.0), x, -1.0)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(eps * eps) - Float64(1.0 * 1.0)) / Float64(eps - 1.0)) / eps) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8e-5], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 31000000000000.0], N[(N[(N[Exp[N[(x * N[(eps - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(N[(eps - -1.0), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.00000000000000065e-5

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 57.4%

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

   10. Applied rewrites 57.4%

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

   if -8.00000000000000065e-5 < x < 3.1e13

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 64.3%

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

   7. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.3%

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

   9. Applied rewrites 64.3%

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

   if 3.1e13 < x

   1. Initial program 73.4%

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

   2. Taylor expanded in 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 38.6%

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

   4. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. add-to-fraction N/A

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

      10. *-lft-identity N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. lower-exp.f64 38.5%

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

   6. Applied rewrites 38.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 30.7%

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

   9. Applied rewrites 30.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. flip-+ N/A

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

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

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

       5. lower--.f32 N/A

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

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

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

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

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

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

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

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

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

       10. lift--.f64 51.2%

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

   11. Applied rewrites 51.2%

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

Alternative 7: 78.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \left(e^{-x} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left(\left|\varepsilon\right| - 1\right)} + \frac{1}{1 + x}\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.5e-267)
   (* 0.5 (- (exp (- x)) -1.0))
   (* (+ (exp (* x (- (fabs eps) 1.0))) (/ 1.0 (+ 1.0 x))) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.5e-267) {
		tmp = 0.5 * (exp(-x) - -1.0);
	} else {
		tmp = (exp((x * (fabs(eps) - 1.0))) + (1.0 / (1.0 + x))) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.5d-267) then
        tmp = 0.5d0 * (exp(-x) - (-1.0d0))
    else
        tmp = (exp((x * (abs(eps) - 1.0d0))) + (1.0d0 / (1.0d0 + x))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.5e-267) {
		tmp = 0.5 * (Math.exp(-x) - -1.0);
	} else {
		tmp = (Math.exp((x * (Math.abs(eps) - 1.0))) + (1.0 / (1.0 + x))) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.5e-267:
		tmp = 0.5 * (math.exp(-x) - -1.0)
	else:
		tmp = (math.exp((x * (math.fabs(eps) - 1.0))) + (1.0 / (1.0 + x))) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.5e-267)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) - -1.0));
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(abs(eps) - 1.0))) + Float64(1.0 / Float64(1.0 + x))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.5e-267)
		tmp = 0.5 * (exp(-x) - -1.0);
	else
		tmp = (exp((x * (abs(eps) - 1.0))) + (1.0 / (1.0 + x))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.5e-267], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5e-267

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 57.4%

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

   10. Applied rewrites 57.4%

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

   if 2.5e-267 < x

   1. Initial program 73.4%

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in eps around inf

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

      1. lower-+.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 99.0%

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 64.6%

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

   9. Applied rewrites 64.6%

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

   10. Taylor expanded in eps around 0

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

       1. lower-+.f64 64.6%

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

   12. Applied rewrites 64.6%

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

Alternative 8: 78.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \left(e^{-x} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left(\left|\varepsilon\right| - 1\right)} - -1\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.5e-267)
   (* 0.5 (- (exp (- x)) -1.0))
   (* (- (exp (* x (- (fabs eps) 1.0))) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.5e-267) {
		tmp = 0.5 * (exp(-x) - -1.0);
	} else {
		tmp = (exp((x * (fabs(eps) - 1.0))) - -1.0) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.5d-267) then
        tmp = 0.5d0 * (exp(-x) - (-1.0d0))
    else
        tmp = (exp((x * (abs(eps) - 1.0d0))) - (-1.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.5e-267) {
		tmp = 0.5 * (Math.exp(-x) - -1.0);
	} else {
		tmp = (Math.exp((x * (Math.abs(eps) - 1.0))) - -1.0) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.5e-267:
		tmp = 0.5 * (math.exp(-x) - -1.0)
	else:
		tmp = (math.exp((x * (math.fabs(eps) - 1.0))) - -1.0) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.5e-267)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) - -1.0));
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(abs(eps) - 1.0))) - -1.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.5e-267)
		tmp = 0.5 * (exp(-x) - -1.0);
	else
		tmp = (exp((x * (abs(eps) - 1.0))) - -1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.5e-267], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5e-267

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 57.4%

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

   10. Applied rewrites 57.4%

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

   if 2.5e-267 < x

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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

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

   7. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.5%

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 64.5%

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

   9. Applied rewrites 64.5%

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

Alternative 9: 70.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \left(e^{-x} - -1\right)\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{1 + \varepsilon}{\varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.2e-17)
   (* 0.5 (- (exp (- x)) -1.0))
   (if (<= x 2.55e+138)
     (/ (- (/ (+ 1.0 eps) eps) (- (/ 1.0 eps) 1.0)) 2.0)
     (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.2e-17) {
		tmp = 0.5 * (exp(-x) - -1.0);
	} else if (x <= 2.55e+138) {
		tmp = (((1.0 + eps) / eps) - ((1.0 / eps) - 1.0)) / 2.0;
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.2e-17)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) - -1.0));
	elseif (x <= 2.55e+138)
		tmp = Float64(Float64(Float64(Float64(1.0 + eps) / eps) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.2e-17], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e+138], N[(N[(N[(N[(1.0 + eps), $MachinePrecision] / eps), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.2e-17

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 57.4%

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

   10. Applied rewrites 57.4%

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

   if 2.2e-17 < x < 2.5499999999999999e138

   1. Initial program 73.4%

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

   2. Taylor expanded in 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 38.6%

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

   4. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. add-to-fraction N/A

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

      10. *-lft-identity N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. lower-exp.f64 38.5%

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

   6. Applied rewrites 38.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 30.7%

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

   9. Applied rewrites 30.7%

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

   if 2.5499999999999999e138 < x

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 53.3%

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

   7. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 53.3%

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 53.3%

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

   9. Applied rewrites 53.3%

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

Alternative 10: 70.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \left(e^{-x} - -1\right)\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+138}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.2e-17)
   (* 0.5 (- (exp (- x)) -1.0))
   (if (<= x 2.55e+138)
     (/ (- (+ 1.0 (/ 1.0 eps)) (- (/ 1.0 eps) 1.0)) 2.0)
     (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.2e-17) {
		tmp = 0.5 * (exp(-x) - -1.0);
	} else if (x <= 2.55e+138) {
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0;
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.2e-17)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) - -1.0));
	elseif (x <= 2.55e+138)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.2e-17], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e+138], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.2e-17

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 57.4%

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

   10. Applied rewrites 57.4%

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

   if 2.2e-17 < x < 2.5499999999999999e138

   1. Initial program 73.4%

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

   2. Taylor expanded in 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 38.6%

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 30.9%

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

   7. Applied rewrites 30.9%

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

   if 2.5499999999999999e138 < x

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 53.3%

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

   7. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 53.3%

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 53.3%

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

   9. Applied rewrites 53.3%

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

Alternative 11: 70.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \left(e^{-x} - -1\right)\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{1 + \varepsilon}{\varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.2e-17)
   (* 0.5 (- (exp (- x)) -1.0))
   (if (<= x 2.55e+138)
     (/ (- (/ (+ 1.0 eps) eps) (- (/ 1.0 eps) 1.0)) 2.0)
     (* 0.3333333333333333 (pow x 3.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.2e-17) {
		tmp = 0.5 * (exp(-x) - -1.0);
	} else if (x <= 2.55e+138) {
		tmp = (((1.0 + eps) / eps) - ((1.0 / eps) - 1.0)) / 2.0;
	} else {
		tmp = 0.3333333333333333 * pow(x, 3.0);
	}
	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 <= 2.2d-17) then
        tmp = 0.5d0 * (exp(-x) - (-1.0d0))
    else if (x <= 2.55d+138) then
        tmp = (((1.0d0 + eps) / eps) - ((1.0d0 / eps) - 1.0d0)) / 2.0d0
    else
        tmp = 0.3333333333333333d0 * (x ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.2e-17) {
		tmp = 0.5 * (Math.exp(-x) - -1.0);
	} else if (x <= 2.55e+138) {
		tmp = (((1.0 + eps) / eps) - ((1.0 / eps) - 1.0)) / 2.0;
	} else {
		tmp = 0.3333333333333333 * Math.pow(x, 3.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.2e-17:
		tmp = 0.5 * (math.exp(-x) - -1.0)
	elif x <= 2.55e+138:
		tmp = (((1.0 + eps) / eps) - ((1.0 / eps) - 1.0)) / 2.0
	else:
		tmp = 0.3333333333333333 * math.pow(x, 3.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.2e-17)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) - -1.0));
	elseif (x <= 2.55e+138)
		tmp = Float64(Float64(Float64(Float64(1.0 + eps) / eps) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	else
		tmp = Float64(0.3333333333333333 * (x ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.2e-17)
		tmp = 0.5 * (exp(-x) - -1.0);
	elseif (x <= 2.55e+138)
		tmp = (((1.0 + eps) / eps) - ((1.0 / eps) - 1.0)) / 2.0;
	else
		tmp = 0.3333333333333333 * (x ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.2e-17], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e+138], N[(N[(N[(N[(1.0 + eps), $MachinePrecision] / eps), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.2e-17

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 57.4%

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

   10. Applied rewrites 57.4%

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

   if 2.2e-17 < x < 2.5499999999999999e138

   1. Initial program 73.4%

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

   2. Taylor expanded in 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 38.6%

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

   4. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. add-to-fraction N/A

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

      10. *-lft-identity N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. lower-exp.f64 38.5%

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

   6. Applied rewrites 38.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 30.7%

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

   9. Applied rewrites 30.7%

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

   if 2.5499999999999999e138 < x

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 53.3%

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

   7. Applied rewrites 53.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 11.9%

         \[\leadsto 0.3333333333333333 \cdot {x}^{3} \]

   10. Applied rewrites 11.9%

       \[\leadsto 0.3333333333333333 \cdot {x}^{\color{blue}{3}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \left(e^{-x} - -1\right)\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5e-11)
   (* 0.5 (- (exp (- x)) -1.0))
   (if (<= x 2.55e+138)
     (/ (- (/ 1.0 eps) (- (/ 1.0 eps) 1.0)) 2.0)
     (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5e-11) {
		tmp = 0.5 * (exp(-x) - -1.0);
	} else if (x <= 2.55e+138) {
		tmp = ((1.0 / eps) - ((1.0 / eps) - 1.0)) / 2.0;
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5e-11)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) - -1.0));
	elseif (x <= 2.55e+138)
		tmp = Float64(Float64(Float64(1.0 / eps) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5e-11], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e+138], N[(N[(N[(1.0 / eps), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.00000000000000018e-11

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 57.4%

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

   10. Applied rewrites 57.4%

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

   if 5.00000000000000018e-11 < x < 2.5499999999999999e138

   1. Initial program 73.4%

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

   2. Taylor expanded in 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 38.6%

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

   4. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. add-to-fraction N/A

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

      10. *-lft-identity N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. lower-exp.f64 38.5%

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

   6. Applied rewrites 38.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 30.7%

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

   9. Applied rewrites 30.7%

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

   10. Taylor expanded in eps around 0

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

       1. lower-/.f64 18.3%

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

   12. Applied rewrites 18.3%

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

   if 2.5499999999999999e138 < x

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 53.3%

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

   7. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 53.3%

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 53.3%

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

   9. Applied rewrites 53.3%

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

Alternative 13: 64.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{1 \cdot 1 - x \cdot x}{x - -1}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5e-11)
   (/ (- (* 1.0 1.0) (* x x)) (- x -1.0))
   (if (<= x 2.55e+138)
     (/ (- (/ 1.0 eps) (- (/ 1.0 eps) 1.0)) 2.0)
     (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5e-11) {
		tmp = ((1.0 * 1.0) - (x * x)) / (x - -1.0);
	} else if (x <= 2.55e+138) {
		tmp = ((1.0 / eps) - ((1.0 / eps) - 1.0)) / 2.0;
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5e-11)
		tmp = Float64(Float64(Float64(1.0 * 1.0) - Float64(x * x)) / Float64(x - -1.0));
	elseif (x <= 2.55e+138)
		tmp = Float64(Float64(Float64(1.0 / eps) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5e-11], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e+138], N[(N[(N[(1.0 / eps), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.00000000000000018e-11

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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 43.6%

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

   7. Applied rewrites 43.6%

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   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 43.6%

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.6%

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

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. flip-- N/A

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

       3. lower-unsound-+.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

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

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

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

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

       8. lower-*.f32 N/A

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

       9. unpow2 N/A

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

       10. lift-pow.f64 N/A

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

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

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

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

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

       13. lift-pow.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. add-flip N/A

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

       17. metadata-eval N/A

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

       18. lower--.f64 50.4%

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

   11. Applied rewrites 50.4%

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

   if 5.00000000000000018e-11 < x < 2.5499999999999999e138

   1. Initial program 73.4%

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

   2. Taylor expanded in 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 38.6%

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

   4. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. add-to-fraction N/A

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

      10. *-lft-identity N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. lower-exp.f64 38.5%

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

   6. Applied rewrites 38.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 30.7%

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

   9. Applied rewrites 30.7%

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

   10. Taylor expanded in eps around 0

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

       1. lower-/.f64 18.3%

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

   12. Applied rewrites 18.3%

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

   if 2.5499999999999999e138 < x

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 53.3%

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

   7. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 53.3%

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 53.3%

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

   9. Applied rewrites 53.3%

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

Alternative 14: 60.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;\frac{1 \cdot 1 - x \cdot x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.68)
   (/ (- (* 1.0 1.0) (* x x)) (- x -1.0))
   (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.68) {
		tmp = ((1.0 * 1.0) - (x * x)) / (x - -1.0);
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.68)
		tmp = Float64(Float64(Float64(1.0 * 1.0) - Float64(x * x)) / Float64(x - -1.0));
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.68], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 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 43.6%

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

   7. Applied rewrites 43.6%

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   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 43.6%

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.6%

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

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. flip-- N/A

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

       3. lower-unsound-+.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

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

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

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

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

       8. lower-*.f32 N/A

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

       9. unpow2 N/A

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

       10. lift-pow.f64 N/A

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

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

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

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

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

       13. lift-pow.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. add-flip N/A

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

       17. metadata-eval N/A

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

       18. lower--.f64 50.4%

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

   11. Applied rewrites 50.4%

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

   if -0.680000000000000049 < x

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 53.3%

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

   7. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1 \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot x\right) \cdot x + 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot x, x, 1\right) \]

      9. lower-*.f64 53.3%

         \[\leadsto \mathsf{fma}\left(\left(0.3333333333333333 \cdot x - 0.5\right) \cdot x, x, 1\right) \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot x, x, 1\right) \]

      11. sub-flip N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot x, x, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot x, x, 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x, x, 1\right) \]

      14. metadata-eval 53.3%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right) \]

   9. Applied rewrites 53.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 53.3% accurate, 4.2× speedup

Math

\[\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right) \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0))

C

double code(double x, double eps) {
	return fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
}

Julia

function code(x, eps)
	return fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]

Derivation

1. Initial program 73.4%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \color{blue}{\left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(\color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   4. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(\color{blue}{-1} \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   7. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   8. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \mathsf{fma}\left(-1, \color{blue}{e^{\mathsf{neg}\left(x\right)}}, -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

4. Applied rewrites 57.5%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \mathsf{fma}\left(-1, e^{-x}, -1 \cdot \left(x \cdot e^{-x}\right)\right)\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \color{blue}{\frac{1}{2}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

   4. lower--.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

   5. lower-*.f64 53.3%

      \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

7. Applied rewrites 53.3%

   \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

   3. lift-*.f64 N/A

      \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

   5. lift-pow.f64 N/A

      \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1 \]

   7. associate-*r* N/A

      \[\leadsto \left(\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot x\right) \cdot x + 1 \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot x, x, 1\right) \]

   9. lower-*.f64 53.3%

      \[\leadsto \mathsf{fma}\left(\left(0.3333333333333333 \cdot x - 0.5\right) \cdot x, x, 1\right) \]

   10. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot x, x, 1\right) \]

   11. sub-flip N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot x, x, 1\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot x, x, 1\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x, x, 1\right) \]

   14. metadata-eval 53.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right) \]

9. Applied rewrites 53.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right) \]
10. Add Preprocessing

Alternative 16: 44.1% accurate, 58.4× speedup

Math

\[1 \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

double code(double x, double eps) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 73.4%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 44.1%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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))