Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 2.5s

Alternatives: 7

Speedup: 1.1×

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, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\frac{-x\_m}{s} - \mathsf{log1p}\left(\frac{1}{e^{\frac{x\_m}{s}}}\right) \cdot 2}}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (exp (- (/ (- x_m) s) (* (log1p (/ 1.0 (exp (/ x_m s)))) 2.0))) s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf(((-x_m / s) - (log1pf((1.0f / expf((x_m / s)))) * 2.0f))) / s;
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(exp(Float32(Float32(Float32(-x_m) / s) - Float32(log1p(Float32(Float32(1.0) / exp(Float32(x_m / s)))) * Float32(2.0)))) / s)
end

Derivation

1. Initial program 99.6%

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

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 99.6

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. distribute-neg-frac2 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-neg.f32 99.6

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

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

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

3. Applied rewrites 99.6%

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

5. Applied rewrites 99.6%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. rec-exp N/A

      \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

   7. lift-exp.f32 N/A

      \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\frac{1}{\color{blue}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

   8. lower-/.f32 99.6

      \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

7. Applied rewrites 99.6%

   \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]
8. Add Preprocessing

Alternative 2: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 1.0000000200408773 \cdot 10^{+20}:\\ \;\;\;\;\frac{e^{\frac{-x\_m}{s} - \left(\log 4 + x\_m \cdot \left(0.25 \cdot \frac{x\_m}{s \cdot s} - \frac{1}{s}\right)\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 1.0000000200408773e+20)
     (/
      (exp
       (-
        (/ (- x_m) s)
        (+ (log 4.0) (* x_m (- (* 0.25 (/ x_m (* s s))) (/ 1.0 s))))))
      s)
     (/ 0.25 s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-fabsf(x_m) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 1.0000000200408773e+20f) {
		tmp = expf(((-x_m / s) - (logf(4.0f) + (x_m * ((0.25f * (x_m / (s * s))) - (1.0f / s)))))) / s;
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

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

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

real(4) function code(x_m, s)
use fmin_fmax_functions
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x_m) / s))
    t_1 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 1.0000000200408773e+20) then
        tmp = exp(((-x_m / s) - (log(4.0e0) + (x_m * ((0.25e0 * (x_m / (s * s))) - (1.0e0 / s)))))) / s
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(Float32(-abs(x_m)) / 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(1.0000000200408773e+20))
		tmp = Float32(exp(Float32(Float32(Float32(-x_m) / s) - Float32(log(Float32(4.0)) + Float32(x_m * Float32(Float32(Float32(0.25) * Float32(x_m / Float32(s * s))) - Float32(Float32(1.0) / s)))))) / s);
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	t_0 = exp((-abs(x_m) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(1.0000000200408773e+20))
		tmp = exp(((-x_m / s) - (log(single(4.0)) + (x_m * ((single(0.25) * (x_m / (s * s))) - (single(1.0) / s)))))) / s;
	else
		tmp = single(0.25) / 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))))) < 1.00000002e20

   1. Initial program 99.6%

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.7

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 99.7

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   3. Applied rewrites 99.7%

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

   5. Applied rewrites 99.7%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. lift-/.f32 N/A

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

      6. rec-exp N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

      7. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\frac{1}{\color{blue}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

      8. lower-/.f32 99.7

         \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

   7. Applied rewrites 99.7%

      \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

   8. Taylor expanded in x around 0

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

      1. lower-+.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \left(2 \cdot \log 2 + \color{blue}{x \cdot \left(\frac{1}{4} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}\right)}}{s} \]

      2. log-pow-rev N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log \left({2}^{2}\right) + \color{blue}{x} \cdot \left(\frac{1}{4} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)\right)}}{s} \]

      3. metadata-eval N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log 4 + x \cdot \left(\frac{1}{4} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)\right)}}{s} \]

      4. lower-log.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log 4 + \color{blue}{x} \cdot \left(\frac{1}{4} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)\right)}}{s} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log 4 + x \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}\right)}}{s} \]

      6. lower--.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log 4 + x \cdot \left(\frac{1}{4} \cdot \frac{x}{{s}^{2}} - \color{blue}{\frac{1}{s}}\right)\right)}}{s} \]

      7. lower-*.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log 4 + x \cdot \left(\frac{1}{4} \cdot \frac{x}{{s}^{2}} - \frac{\color{blue}{1}}{s}\right)\right)}}{s} \]

      8. lower-/.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log 4 + x \cdot \left(\frac{1}{4} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)\right)}}{s} \]

      9. unpow2 N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log 4 + x \cdot \left(\frac{1}{4} \cdot \frac{x}{s \cdot s} - \frac{1}{s}\right)\right)}}{s} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log 4 + x \cdot \left(\frac{1}{4} \cdot \frac{x}{s \cdot s} - \frac{1}{s}\right)\right)}}{s} \]

      11. lower-/.f32 98.0

          \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log 4 + x \cdot \left(0.25 \cdot \frac{x}{s \cdot s} - \frac{1}{\color{blue}{s}}\right)\right)}}{s} \]

   10. Applied rewrites 98.0%

       \[\leadsto \frac{e^{\frac{-x}{s} - \color{blue}{\left(\log 4 + x \cdot \left(0.25 \cdot \frac{x}{s \cdot s} - \frac{1}{s}\right)\right)}}}{s} \]

   if 1.00000002e20 < (/.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 98.5%

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 98.8

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 98.8

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   3. Applied rewrites 98.8%

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

   5. Applied rewrites 98.6%

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

      1. lift-exp.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-negate1 N/A

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

      4. exp-sum N/A

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

      5. lift-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f32 N/A

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

      12. metadata-eval 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 79.9%

       \[\leadsto \frac{\color{blue}{0.25}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{-1 \cdot \frac{x\_m}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-x\_m}{s}} \cdot \left(0.25 + 0.25 \cdot \frac{x\_m}{s}\right)}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 9.999999974752427e-7)
     (/ (exp (* -1.0 (/ x_m s))) s)
     (/ (* (exp (/ (- x_m) s)) (+ 0.25 (* 0.25 (/ x_m s)))) s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-fabsf(x_m) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 9.999999974752427e-7f) {
		tmp = expf((-1.0f * (x_m / s))) / s;
	} else {
		tmp = (expf((-x_m / s)) * (0.25f + (0.25f * (x_m / s)))) / s;
	}
	return tmp;
}

Fortran

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

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

real(4) function code(x_m, s)
use fmin_fmax_functions
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x_m) / s))
    t_1 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 9.999999974752427e-7) then
        tmp = exp(((-1.0e0) * (x_m / s))) / s
    else
        tmp = (exp((-x_m / s)) * (0.25e0 + (0.25e0 * (x_m / s)))) / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(Float32(-abs(x_m)) / 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(9.999999974752427e-7))
		tmp = Float32(exp(Float32(Float32(-1.0) * Float32(x_m / s))) / s);
	else
		tmp = Float32(Float32(exp(Float32(Float32(-x_m) / s)) * Float32(Float32(0.25) + Float32(Float32(0.25) * Float32(x_m / s)))) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	t_0 = exp((-abs(x_m) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(9.999999974752427e-7))
		tmp = exp((single(-1.0) * (x_m / s))) / s;
	else
		tmp = (exp((-x_m / s)) * (single(0.25) + (single(0.25) * (x_m / 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))))) < 9.99999997e-7

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.8

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 99.8

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. lift-/.f32 N/A

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

      6. rec-exp N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

      7. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\frac{1}{\color{blue}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

      8. lower-/.f32 99.8

         \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

   7. Applied rewrites 99.8%

      \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f32 N/A

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

      2. lift-/.f32 99.8

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

   10. Applied rewrites 99.8%

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

   if 9.99999997e-7 < (/.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. 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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.2

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 99.2

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   3. Applied rewrites 99.2%

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

   5. Applied rewrites 99.2%

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

      1. lift-exp.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-negate1 N/A

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

      4. exp-sum N/A

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

      5. lift-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f32 N/A

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

      12. metadata-eval 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s}} \cdot \left(\frac{1}{4} + \color{blue}{\frac{1}{4} \cdot \frac{x}{s}}\right)}{s} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s}} \cdot \left(\frac{1}{4} + \frac{1}{4} \cdot \color{blue}{\frac{x}{s}}\right)}{s} \]

      3. lift-/.f32 89.5

         \[\leadsto \frac{e^{\frac{-x}{s}} \cdot \left(0.25 + 0.25 \cdot \frac{x}{\color{blue}{s}}\right)}{s} \]

   10. Applied rewrites 89.5%

       \[\leadsto \frac{e^{\frac{-x}{s}} \cdot \color{blue}{\left(0.25 + 0.25 \cdot \frac{x}{s}\right)}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{-1 \cdot \frac{x\_m}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 9.999999974752427e-7)
     (/ (exp (* -1.0 (/ x_m s))) s)
     (/ 0.25 s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-fabsf(x_m) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 9.999999974752427e-7f) {
		tmp = expf((-1.0f * (x_m / s))) / s;
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

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

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

real(4) function code(x_m, s)
use fmin_fmax_functions
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x_m) / s))
    t_1 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 9.999999974752427e-7) then
        tmp = exp(((-1.0e0) * (x_m / s))) / s
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(Float32(-abs(x_m)) / 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(9.999999974752427e-7))
		tmp = Float32(exp(Float32(Float32(-1.0) * Float32(x_m / s))) / s);
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	t_0 = exp((-abs(x_m) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(9.999999974752427e-7))
		tmp = exp((single(-1.0) * (x_m / s))) / s;
	else
		tmp = single(0.25) / 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))))) < 9.99999997e-7

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.8

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 99.8

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. lift-/.f32 N/A

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

      6. rec-exp N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

      7. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\frac{1}{\color{blue}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

      8. lower-/.f32 99.8

         \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

   7. Applied rewrites 99.8%

      \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f32 N/A

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

      2. lift-/.f32 99.8

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

   10. Applied rewrites 99.8%

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

   if 9.99999997e-7 < (/.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. 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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.2

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 99.2

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   3. Applied rewrites 99.2%

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

   5. Applied rewrites 99.2%

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

      1. lift-exp.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-negate1 N/A

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

      4. exp-sum N/A

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

      5. lift-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f32 N/A

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

      12. metadata-eval 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 88.7%

       \[\leadsto \frac{\color{blue}{0.25}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (/ (- x_m) s))) (/ (exp (fma (log1p (exp t_0)) -2.0 t_0)) s)))

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 99.6

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. distribute-neg-frac2 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-neg.f32 99.6

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

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

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

3. Applied rewrites 99.6%

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

5. Applied rewrites 99.6%

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

   1. lift--.f32 N/A

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

   2. sub-negate1 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

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

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

   6. lower-fma.f32 N/A

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

   7. metadata-eval 99.6

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

7. Applied rewrites 99.6%

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

Alternative 6: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{e^{\frac{-x\_m}{s}} \cdot \left(0.25 + 0.25 \cdot \frac{x\_m}{s}\right)}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-0.25 \cdot \frac{x\_m \cdot x\_m}{s \cdot s} - \log 4}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 3.999999999279835e-23)
   (/ (* (exp (/ (- x_m) s)) (+ 0.25 (* 0.25 (/ x_m s)))) s)
   (/ (exp (- (* -0.25 (/ (* x_m x_m) (* s s))) (log 4.0))) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 3.999999999279835e-23f) {
		tmp = (expf((-x_m / s)) * (0.25f + (0.25f * (x_m / s)))) / s;
	} else {
		tmp = expf(((-0.25f * ((x_m * x_m) / (s * s))) - logf(4.0f))) / s;
	}
	return tmp;
}

Fortran

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

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

real(4) function code(x_m, s)
use fmin_fmax_functions
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 3.999999999279835e-23) then
        tmp = (exp((-x_m / s)) * (0.25e0 + (0.25e0 * (x_m / s)))) / s
    else
        tmp = exp((((-0.25e0) * ((x_m * x_m) / (s * s))) - log(4.0e0))) / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(3.999999999279835e-23))
		tmp = Float32(Float32(exp(Float32(Float32(-x_m) / s)) * Float32(Float32(0.25) + Float32(Float32(0.25) * Float32(x_m / s)))) / s);
	else
		tmp = Float32(exp(Float32(Float32(Float32(-0.25) * Float32(Float32(x_m * x_m) / Float32(s * s))) - log(Float32(4.0)))) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(3.999999999279835e-23))
		tmp = (exp((-x_m / s)) * (single(0.25) + (single(0.25) * (x_m / s)))) / s;
	else
		tmp = exp(((single(-0.25) * ((x_m * x_m) / (s * s))) - log(single(4.0)))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4e-23

   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. 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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.1

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 99.1

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   3. Applied rewrites 99.1%

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

   5. Applied rewrites 99.1%

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

      1. lift-exp.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-negate1 N/A

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

      4. exp-sum N/A

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

      5. lift-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f32 N/A

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

      12. metadata-eval 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s}} \cdot \left(\frac{1}{4} + \color{blue}{\frac{1}{4} \cdot \frac{x}{s}}\right)}{s} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s}} \cdot \left(\frac{1}{4} + \frac{1}{4} \cdot \color{blue}{\frac{x}{s}}\right)}{s} \]

      3. lift-/.f32 93.5

         \[\leadsto \frac{e^{\frac{-x}{s}} \cdot \left(0.25 + 0.25 \cdot \frac{x}{\color{blue}{s}}\right)}{s} \]

   10. Applied rewrites 93.5%

       \[\leadsto \frac{e^{\frac{-x}{s}} \cdot \color{blue}{\left(0.25 + 0.25 \cdot \frac{x}{s}\right)}}{s} \]

   if 4e-23 < x

   1. Initial program 99.7%

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.7

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 99.7

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   3. Applied rewrites 99.7%

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

   5. Applied rewrites 99.7%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. lift-/.f32 N/A

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

      6. rec-exp N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

      7. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\frac{1}{\color{blue}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

      8. lower-/.f32 99.8

         \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

   7. Applied rewrites 99.8%

      \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot 2}}{s} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{e^{\color{blue}{\frac{-1}{4} \cdot \frac{{x}^{2}}{{s}^{2}} - 2 \cdot \log 2}}}{s} \]
   9. Step-by-step derivation

      1. lower--.f32 N/A

         \[\leadsto \frac{e^{\frac{-1}{4} \cdot \frac{{x}^{2}}{{s}^{2}} - \color{blue}{2 \cdot \log 2}}}{s} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{e^{\frac{-1}{4} \cdot \frac{{x}^{2}}{{s}^{2}} - \color{blue}{2} \cdot \log 2}}{s} \]

      3. lower-/.f32 N/A

         \[\leadsto \frac{e^{\frac{-1}{4} \cdot \frac{{x}^{2}}{{s}^{2}} - 2 \cdot \log 2}}{s} \]

      4. unpow2 N/A

         \[\leadsto \frac{e^{\frac{-1}{4} \cdot \frac{x \cdot x}{{s}^{2}} - 2 \cdot \log 2}}{s} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{e^{\frac{-1}{4} \cdot \frac{x \cdot x}{{s}^{2}} - 2 \cdot \log 2}}{s} \]

      6. unpow2 N/A

         \[\leadsto \frac{e^{\frac{-1}{4} \cdot \frac{x \cdot x}{s \cdot s} - 2 \cdot \log 2}}{s} \]

      7. lower-*.f32 N/A

         \[\leadsto \frac{e^{\frac{-1}{4} \cdot \frac{x \cdot x}{s \cdot s} - 2 \cdot \log 2}}{s} \]

      8. log-pow-rev N/A

         \[\leadsto \frac{e^{\frac{-1}{4} \cdot \frac{x \cdot x}{s \cdot s} - \log \left({2}^{2}\right)}}{s} \]

      9. metadata-eval N/A

         \[\leadsto \frac{e^{\frac{-1}{4} \cdot \frac{x \cdot x}{s \cdot s} - \log 4}}{s} \]

      10. lower-log.f32 98.2

          \[\leadsto \frac{e^{-0.25 \cdot \frac{x \cdot x}{s \cdot s} - \log 4}}{s} \]

   10. Applied rewrites 98.2%

       \[\leadsto \frac{e^{\color{blue}{-0.25 \cdot \frac{x \cdot x}{s \cdot s} - \log 4}}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 27.2% accurate, 31.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.25}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.25 s))

C

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

Fortran

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

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

real(4) function code(x_m, s)
use fmin_fmax_functions
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 99.6

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. distribute-neg-frac2 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-neg.f32 99.6

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

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

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

3. Applied rewrites 99.6%

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

5. Applied rewrites 99.6%

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

   1. lift-exp.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-negate1 N/A

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

   4. exp-sum N/A

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

   5. lift-exp.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-neg-in N/A

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

   11. lower-*.f32 N/A

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

   12. metadata-eval 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 27.2%

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

Reproduce

herbie shell --seed 2025107 
(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))))))