Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 14.1s

Alternatives: 3

Speedup: N/A×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

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(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

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(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Alternative 1: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{-1}\right)}^{\left(\frac{\frac{\left|x\right|}{s}}{2}\right)}\\ t_1 := e^{\frac{\left|x\right|}{-s}} + 1\\ \frac{\frac{\frac{t\_0}{s}}{t\_1} \cdot t\_0}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (pow (exp -1.0) (/ (/ (fabs x) s) 2.0)))
        (t_1 (+ (exp (/ (fabs x) (- s))) 1.0)))
   (/ (* (/ (/ t_0 s) t_1) t_0) t_1)))

C

float code(float x, float s) {
	float t_0 = powf(expf(-1.0f), ((fabsf(x) / s) / 2.0f));
	float t_1 = expf((fabsf(x) / -s)) + 1.0f;
	return (((t_0 / s) / t_1) * t_0) / t_1;
}

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(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-1.0e0)) ** ((abs(x) / s) / 2.0e0)
    t_1 = exp((abs(x) / -s)) + 1.0e0
    code = (((t_0 / s) / t_1) * t_0) / t_1
end function

Julia

function code(x, s)
	t_0 = exp(Float32(-1.0)) ^ Float32(Float32(abs(x) / s) / Float32(2.0))
	t_1 = Float32(exp(Float32(abs(x) / Float32(-s))) + Float32(1.0))
	return Float32(Float32(Float32(Float32(t_0 / s) / t_1) * t_0) / t_1)
end

MATLAB

function tmp = code(x, s)
	t_0 = exp(single(-1.0)) ^ ((abs(x) / s) / single(2.0));
	t_1 = exp((abs(x) / -s)) + single(1.0);
	tmp = (((t_0 / s) / t_1) * t_0) / t_1;
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-fabs.f32 N/A

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

   6. flip3-+ N/A

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

   7. metadata-eval N/A

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

   8. div-add N/A

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

4. Applied rewrites 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-fabs.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. mul-1-neg N/A

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

   7. pow-exp N/A

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

   8. lift-exp.f32 N/A

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

   9. sqr-pow N/A

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

   10. lower-*.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-fabs.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. lower-/.f32 N/A

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

   17. lift-fabs.f32 N/A

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

   18. lift-/.f32 99.0

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 99.5%

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

10. Applied rewrites 99.6%

    \[\leadsto \color{blue}{\frac{\frac{\frac{{\left(e^{-1}\right)}^{\left(\frac{\frac{\left|x\right|}{s}}{2}\right)}}{s}}{e^{\frac{\left|x\right|}{-s}} + 1} \cdot {\left(e^{-1}\right)}^{\left(\frac{\frac{\left|x\right|}{s}}{2}\right)}}{e^{\frac{\left|x\right|}{-s}} + 1}} \]
11. Add Preprocessing

Alternative 2: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{-1}\right)}^{\left(\frac{\frac{\left|x\right|}{s}}{2}\right)}\\ t_1 := 1 + e^{\frac{\left|x\right|}{-s}}\\ \frac{t\_0}{s \cdot t\_1} \cdot \frac{t\_0}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (pow (exp -1.0) (/ (/ (fabs x) s) 2.0)))
        (t_1 (+ 1.0 (exp (/ (fabs x) (- s))))))
   (* (/ t_0 (* s t_1)) (/ t_0 t_1))))

C

float code(float x, float s) {
	float t_0 = powf(expf(-1.0f), ((fabsf(x) / s) / 2.0f));
	float t_1 = 1.0f + expf((fabsf(x) / -s));
	return (t_0 / (s * t_1)) * (t_0 / t_1);
}

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(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-1.0e0)) ** ((abs(x) / s) / 2.0e0)
    t_1 = 1.0e0 + exp((abs(x) / -s))
    code = (t_0 / (s * t_1)) * (t_0 / t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(-1.0)) ^ Float32(Float32(abs(x) / s) / Float32(2.0))
	t_1 = Float32(Float32(1.0) + exp(Float32(abs(x) / Float32(-s))))
	return Float32(Float32(t_0 / Float32(s * t_1)) * Float32(t_0 / t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp(single(-1.0)) ^ ((abs(x) / s) / single(2.0));
	t_1 = single(1.0) + exp((abs(x) / -s));
	tmp = (t_0 / (s * t_1)) * (t_0 / t_1);
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-fabs.f32 N/A

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

   6. flip3-+ N/A

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

   7. metadata-eval N/A

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

   8. div-add N/A

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

4. Applied rewrites 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-fabs.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. mul-1-neg N/A

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

   7. pow-exp N/A

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

   8. lift-exp.f32 N/A

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

   9. sqr-pow N/A

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

   10. lower-*.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-fabs.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. lower-/.f32 N/A

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

   17. lift-fabs.f32 N/A

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

   18. lift-/.f32 99.0

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{{\left(e^{-1}\right)}^{\left(\frac{\frac{\left|x\right|}{s}}{2}\right)}}{s \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)} \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{\frac{\left|x\right|}{s}}{2}\right)}}{1 + e^{\frac{\left|x\right|}{-s}}}} \]
9. Add Preprocessing

Alternative 3: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\left|x\right|}{s}}{2}\\ t_1 := 1 + e^{\frac{\left|x\right|}{-s}}\\ \frac{{\left(e^{-1}\right)}^{t\_0}}{s \cdot t\_1} \cdot \frac{e^{\log \left(e^{-1}\right) \cdot t\_0}}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (/ (fabs x) s) 2.0)) (t_1 (+ 1.0 (exp (/ (fabs x) (- s))))))
   (*
    (/ (pow (exp -1.0) t_0) (* s t_1))
    (/ (exp (* (log (exp -1.0)) t_0)) t_1))))

C

float code(float x, float s) {
	float t_0 = (fabsf(x) / s) / 2.0f;
	float t_1 = 1.0f + expf((fabsf(x) / -s));
	return (powf(expf(-1.0f), t_0) / (s * t_1)) * (expf((logf(expf(-1.0f)) * t_0)) / t_1);
}

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(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = (abs(x) / s) / 2.0e0
    t_1 = 1.0e0 + exp((abs(x) / -s))
    code = ((exp((-1.0e0)) ** t_0) / (s * t_1)) * (exp((log(exp((-1.0e0))) * t_0)) / t_1)
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(abs(x) / s) / Float32(2.0))
	t_1 = Float32(Float32(1.0) + exp(Float32(abs(x) / Float32(-s))))
	return Float32(Float32((exp(Float32(-1.0)) ^ t_0) / Float32(s * t_1)) * Float32(exp(Float32(log(exp(Float32(-1.0))) * t_0)) / t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = (abs(x) / s) / single(2.0);
	t_1 = single(1.0) + exp((abs(x) / -s));
	tmp = ((exp(single(-1.0)) ^ t_0) / (s * t_1)) * (exp((log(exp(single(-1.0))) * t_0)) / t_1);
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-fabs.f32 N/A

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

   6. flip3-+ N/A

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

   7. metadata-eval N/A

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

   8. div-add N/A

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

4. Applied rewrites 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-fabs.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. mul-1-neg N/A

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

   7. pow-exp N/A

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

   8. lift-exp.f32 N/A

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

   9. sqr-pow N/A

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

   10. lower-*.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-fabs.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. lower-/.f32 N/A

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

   17. lift-fabs.f32 N/A

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

   18. lift-/.f32 99.0

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-log.f32 N/A

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

   7. lift-exp.f32 99.5

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

10. Applied rewrites 99.5%

    \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\frac{\frac{\left|x\right|}{s}}{2}\right)}}{s \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)} \cdot \frac{\color{blue}{e^{\log \left(e^{-1}\right) \cdot \frac{\frac{\left|x\right|}{s}}{2}}}}{1 + e^{\frac{\left|x\right|}{-s}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))