Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 5.6s

Alternatives: 9

Speedup: 0.8×

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: 99.5% 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: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{\frac{t\_0}{s}}{e^{\mathsf{log1p}\left(t\_0\right) \cdot 2}} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))))
   (/ (/ t_0 s) (exp (* (log1p t_0) 2.0)))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return (t_0 / s) / expf((log1pf(t_0) * 2.0f));
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(Float32(t_0 / s) / exp(Float32(log1p(t_0) * Float32(2.0))))
end

Derivation

1. Initial program 99.4%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-pow.f32 N/A

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

   2. pow-to-exp N/A

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

   3. lower-exp.f32 N/A

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

   4. lower-*.f32 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{\frac{e^{\frac{\left|x\right|}{-s}}}{s}}{\color{blue}{e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right) \cdot 2}}} \]

7. Final simplification 99.5%

   \[\leadsto \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{e^{\mathsf{log1p}\left(e^{\frac{-\left|x\right|}{s}}\right) \cdot 2}} \]
8. Add Preprocessing

Alternative 2: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_0}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 - \frac{\left(x \cdot \frac{x}{s}\right) \cdot 0.0625}{s}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 9.999999747378752e-6)
     (/ t_0 s)
     (/ (- 0.25 (/ (* (* x (/ x s)) 0.0625) s)) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 9.999999747378752e-6f) {
		tmp = t_0 / s;
	} else {
		tmp = (0.25f - (((x * (x / s)) * 0.0625f) / s)) / s;
	}
	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(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
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 9.999999747378752e-6) then
        tmp = t_0 / s
    else
        tmp = (0.25e0 - (((x * (x / s)) * 0.0625e0) / s)) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_1) * t_1)) <= Float32(9.999999747378752e-6))
		tmp = Float32(t_0 / s);
	else
		tmp = Float32(Float32(Float32(0.25) - Float32(Float32(Float32(x * Float32(x / s)) * Float32(0.0625)) / s)) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(9.999999747378752e-6))
		tmp = t_0 / s;
	else
		tmp = (single(0.25) - (((x * (x / s)) * single(0.0625)) / s)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 9.99999975e-6

   1. Initial program 99.5%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f32 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f32 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f32 N/A

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

      4. lower-*.f32 N/A

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

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\frac{e^{\frac{\left|x\right|}{-s}}}{s}}{\color{blue}{e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right) \cdot 2}}} \]

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{e^{\frac{\left|x\right|}{-s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)}}{s}} \]

   9. Taylor expanded in s around 0

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

   11. Applied rewrites 99.5%

       \[\leadsto \frac{e^{\color{blue}{\frac{\left|x\right|}{-s}}}}{s} \]

   if 9.99999975e-6 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in s around inf

      \[\leadsto \frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s} \]
   7. Step-by-step derivation

   8. Applied rewrites 72.3%

      \[\leadsto \frac{0.25 - \frac{\mathsf{fma}\left(-0.125, x \cdot x, 0.1875 \cdot \left(x \cdot x\right)\right)}{s \cdot s}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.3%

       \[\leadsto \frac{0.25 - \frac{\left(x \cdot x\right) \cdot 0.0625}{s \cdot s}}{s} \]
   11. Step-by-step derivation

   12. Applied rewrites 88.9%

       \[\leadsto \frac{0.25 - \frac{\left(x \cdot \frac{x}{s}\right) \cdot 0.0625}{s}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 - \frac{\left(x \cdot \frac{x}{s}\right) \cdot 0.0625}{s}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{t\_0}\right), -2, t\_0\right)}}{s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s)))) (/ (exp (fma (log1p (exp t_0)) -2.0 t_0)) s)))

C

float code(float x, float s) {
	float t_0 = x / -s;
	return expf(fmaf(log1pf(expf(t_0)), -2.0f, t_0)) / s;
}

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	return Float32(exp(fma(log1p(exp(t_0)), Float32(-2.0), t_0)) / s)
end

Derivation

1. Initial program 99.4%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-pow.f32 N/A

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

   2. pow-to-exp N/A

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

   3. lower-exp.f32 N/A

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

   4. lower-*.f32 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{\frac{e^{\frac{\left|x\right|}{-s}}}{s}}{\color{blue}{e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right) \cdot 2}}} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{\left|x\right|}{-s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)}}{s}} \]
9. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\left|x\right|}{-s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)}}}{s} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{e^{\frac{\left|x\right|}{-s} - \color{blue}{2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)}}}{s} \]

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

      \[\leadsto \frac{e^{\color{blue}{\frac{\left|x\right|}{-s} + \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)}}}{s} \]

   4. +-commutative N/A

      \[\leadsto \frac{e^{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right) + \frac{\left|x\right|}{-s}}}}{s} \]

   5. *-commutative N/A

      \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(\mathsf{neg}\left(2\right)\right)} + \frac{\left|x\right|}{-s}}}{s} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right), \mathsf{neg}\left(2\right), \frac{\left|x\right|}{-s}\right)}}}{s} \]

   7. lift-fabs.f32 N/A

      \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{\color{blue}{\left|x\right|}}{-s}}\right), \mathsf{neg}\left(2\right), \frac{\left|x\right|}{-s}\right)}}{s} \]

   8. rem-sqrt-square N/A

      \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{\color{blue}{\sqrt{x \cdot x}}}{-s}}\right), \mathsf{neg}\left(2\right), \frac{\left|x\right|}{-s}\right)}}{s} \]

   9. sqrt-prod N/A

      \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-s}}\right), \mathsf{neg}\left(2\right), \frac{\left|x\right|}{-s}\right)}}{s} \]

   10. rem-square-sqrt N/A

       \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{\color{blue}{x}}{-s}}\right), \mathsf{neg}\left(2\right), \frac{\left|x\right|}{-s}\right)}}{s} \]

   11. metadata-eval 97.6

       \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{x}{-s}}\right), \color{blue}{-2}, \frac{\left|x\right|}{-s}\right)}}{s} \]

   12. lift-fabs.f32 N/A

       \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{x}{-s}}\right), -2, \frac{\color{blue}{\left|x\right|}}{-s}\right)}}{s} \]

   13. rem-sqrt-square N/A

       \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{x}{-s}}\right), -2, \frac{\color{blue}{\sqrt{x \cdot x}}}{-s}\right)}}{s} \]

   14. sqrt-prod N/A

       \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{x}{-s}}\right), -2, \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-s}\right)}}{s} \]

   15. rem-square-sqrt 85.8

       \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{x}{-s}}\right), -2, \frac{\color{blue}{x}}{-s}\right)}}{s} \]

10. Applied rewrites 85.8%

    \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{x}{-s}}\right), -2, \frac{x}{-s}\right)}}}{s} \]
11. Add Preprocessing

Alternative 4: 96.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(2 - \frac{\left|x\right|}{s}\right)}^{2}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (/ (exp (/ (- (fabs x)) s)) s) (pow (- 2.0 (/ (fabs x) s)) 2.0)))

C

float code(float x, float s) {
	return (expf((-fabsf(x) / s)) / s) / powf((2.0f - (fabsf(x) / s)), 2.0f);
}

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
    code = (exp((-abs(x) / s)) / s) / ((2.0e0 - (abs(x) / s)) ** 2.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(exp(Float32(Float32(-abs(x)) / s)) / s) / (Float32(Float32(2.0) - Float32(abs(x) / s)) ^ Float32(2.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = (exp((-abs(x) / s)) / s) / ((single(2.0) - (abs(x) / s)) ^ single(2.0));
end

Derivation

1. Initial program 99.4%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 96.2%

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

8. Final simplification 96.2%

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

Alternative 5: 96.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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
    code = exp((-abs(x) / s)) / ((((1.0e0 - (abs(x) / s)) - (-1.0e0)) ** 2.0e0) * s)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 96.2%

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

Alternative 6: 94.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\mathsf{fma}\left(\frac{-0.25}{s}, \frac{x \cdot x}{s}, \log 2 \cdot -2\right)}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (exp (fma (/ -0.25 s) (/ (* x x) s) (* (log 2.0) -2.0))) s))

C

float code(float x, float s) {
	return expf(fmaf((-0.25f / s), ((x * x) / s), (logf(2.0f) * -2.0f))) / s;
}

Julia

function code(x, s)
	return Float32(exp(fma(Float32(Float32(-0.25) / s), Float32(Float32(x * x) / s), Float32(log(Float32(2.0)) * Float32(-2.0)))) / s)
end

Derivation

1. Initial program 99.4%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-pow.f32 N/A

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

   2. pow-to-exp N/A

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

   3. lower-exp.f32 N/A

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

   4. lower-*.f32 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{\frac{e^{\frac{\left|x\right|}{-s}}}{s}}{\color{blue}{e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right) \cdot 2}}} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{\left|x\right|}{-s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)}}{s}} \]

9. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot {\left(\left|x\right|\right)}^{2} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}}{{s}^{2}} - 2 \cdot \log 2}}}{s} \]
10. Step-by-step derivation

11. Applied rewrites 95.2%

    \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(\frac{-0.25}{s}, \frac{x \cdot x}{s}, \log 2 \cdot -2\right)}}}{s} \]
12. Add Preprocessing

Alternative 7: 94.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (exp (/ (- (fabs x)) s)) (* 4.0 s)))

C

float code(float x, float s) {
	return expf((-fabsf(x) / s)) / (4.0f * s);
}

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
    code = exp((-abs(x) / s)) / (4.0e0 * s)
end function

Julia

function code(x, s)
	return Float32(exp(Float32(Float32(-abs(x)) / s)) / Float32(Float32(4.0) * s))
end

MATLAB

function tmp = code(x, s)
	tmp = exp((-abs(x) / s)) / (single(4.0) * s);
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s}} \]
4. Step-by-step derivation

5. Applied rewrites 95.2%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s}} \]
6. Add Preprocessing

Alternative 8: 44.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.25, \frac{x}{s}, 0.25\right) - -0.25 \cdot \frac{\left|x\right|}{s}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (- (fma -0.25 (/ x s) 0.25) (* -0.25 (/ (fabs x) s))) s))

C

float code(float x, float s) {
	return (fmaf(-0.25f, (x / s), 0.25f) - (-0.25f * (fabsf(x) / s))) / s;
}

Julia

function code(x, s)
	return Float32(Float32(fma(Float32(-0.25), Float32(x / s), Float32(0.25)) - Float32(Float32(-0.25) * Float32(abs(x) / s))) / s)
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s}} \]
4. Step-by-step derivation

5. Applied rewrites 95.2%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s}} \]
6. Step-by-step derivation

   1. lift-fabs.f32 N/A

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left|x\right|}}{s}}}{4 \cdot s} \]

   2. rem-sqrt-square-rev N/A

      \[\leadsto \frac{e^{\frac{-\color{blue}{\sqrt{x \cdot x}}}{s}}}{4 \cdot s} \]

   3. lower-sqrt.f32 N/A

      \[\leadsto \frac{e^{\frac{-\color{blue}{\sqrt{x \cdot x}}}{s}}}{4 \cdot s} \]

   4. lower-*.f32 94.6

      \[\leadsto \frac{e^{\frac{-\sqrt{\color{blue}{x \cdot x}}}{s}}}{4 \cdot s} \]

7. Applied rewrites 94.6%

   \[\leadsto \frac{e^{\frac{-\color{blue}{\sqrt{x \cdot x}}}{s}}}{4 \cdot s} \]

8. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{-1}{4} \cdot \frac{x}{s}\right) - \frac{-1}{4} \cdot \frac{\left|x\right|}{s}}{s}} \]
9. Step-by-step derivation

10. Applied rewrites 43.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.25, \frac{x}{s}, 0.25\right) - -0.25 \cdot \frac{\left|x\right|}{s}}{s}} \]
11. Add Preprocessing

Alternative 9: 27.5% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 s))

C

float code(float x, float s) {
	return 0.25f / s;
}

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
    code = 0.25e0 / s
end function

Julia

function code(x, s)
	return Float32(Float32(0.25) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.25) / s;
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{s}} \]
4. Step-by-step derivation

5. Applied rewrites 28.2%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025026 
(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))))))