Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 4.1s

Alternatives: 13

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-x\_m}{s}}\\ \frac{e^{\frac{-\left|x\_m\right|}{s}}}{\mathsf{fma}\left(t\_0, s, s\right) \cdot \left(t\_0 - -1\right)} \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x_m) s))))
   (/ (exp (/ (- (fabs x_m)) s)) (* (fma t_0 s s) (- t_0 -1.0)))))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-fabs.f32 N/A

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

   6. flip3-+ N/A

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

   7. metadata-eval N/A

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

   8. div-add N/A

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 95.0%

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

8. Applied rewrites 94.8%

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

9. Final simplification 94.8%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\mathsf{fma}\left(e^{\frac{-x}{s}}, s, s\right) \cdot \left(e^{\frac{-x}{s}} - -1\right)} \]
10. Add Preprocessing

Alternative 2: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 4.099999904632568:\\ \;\;\;\;\frac{e^{\frac{-x\_m}{s}}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{x\_m}{s}}{s} \cdot x\_m, -0.0625, 0.25\right)}{s}\\ \end{array} \end{array} \]

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)) 4.099999904632568)
     (/ (exp (/ (- x_m) s)) (* 4.0 s))
     (/ (fma (* (/ (/ x_m s) s) x_m) -0.0625 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)) <= 4.099999904632568f) {
		tmp = expf((-x_m / s)) / (4.0f * s);
	} else {
		tmp = fmaf((((x_m / s) / s) * x_m), -0.0625f, 0.25f) / s;
	}
	return tmp;
}

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(4.099999904632568))
		tmp = Float32(exp(Float32(Float32(-x_m) / s)) / Float32(Float32(4.0) * s));
	else
		tmp = Float32(fma(Float32(Float32(Float32(x_m / s) / s) * x_m), Float32(-0.0625), Float32(0.25)) / s);
	end
	return 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))))) < 4.0999999

   1. Initial program 99.4%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-+.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. lift-/.f32 N/A

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

      11. lift-neg.f32 N/A

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

      12. lift-fabs.f32 N/A

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

      13. associate-*l* N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 98.3%

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

      1. lift-fabs.f32 N/A

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

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

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

      3. sqrt-unprod N/A

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

      4. rem-square-sqrt 46.0

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

   9. Applied rewrites 46.0%

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

   if 4.0999999 < (/.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 98.9%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

         \[\leadsto \frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{\color{blue}{s}} \]

   5. Applied rewrites 71.8%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\mathsf{fma}\left(x \cdot x, 0.125, -0.0625 \cdot \left(3 \cdot \left(x \cdot x\right)\right)\right)}{s \cdot s}}{s}} \]

   7. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot \frac{x}{s \cdot s}, -0.0625, 0.25\right)}{s}} \]
   8. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot \frac{x}{s \cdot s}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot \frac{x}{s \cdot s}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      3. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot \frac{x}{s \cdot s}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s \cdot s} \cdot x, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{{s}^{2}} \cdot x, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{{s}^{2}} \cdot x, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s \cdot s} \cdot x, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{x}{s}}{s} \cdot x, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      9. lower-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{x}{s}}{s} \cdot x, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      10. lift-/.f32 87.0

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{x}{s}}{s} \cdot x, -0.0625, 0.25\right)}{s} \]

   9. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{x}{s}}{s} \cdot x, -0.0625, 0.25\right)}{s}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 66.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 4.099999904632568:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{x\_m}{s} \cdot 3 - 4, x\_m, 4 \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{x\_m}{s}}{s} \cdot x\_m, -0.0625, 0.25\right)}{s}\\ \end{array} \end{array} \]

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)) 4.099999904632568)
     (/ 1.0 (fma (- (* (/ x_m s) 3.0) 4.0) x_m (* 4.0 s)))
     (/ (fma (* (/ (/ x_m s) s) x_m) -0.0625 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)) <= 4.099999904632568f) {
		tmp = 1.0f / fmaf((((x_m / s) * 3.0f) - 4.0f), x_m, (4.0f * s));
	} else {
		tmp = fmaf((((x_m / s) / s) * x_m), -0.0625f, 0.25f) / s;
	}
	return tmp;
}

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(4.099999904632568))
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(Float32(x_m / s) * Float32(3.0)) - Float32(4.0)), x_m, Float32(Float32(4.0) * s)));
	else
		tmp = Float32(fma(Float32(Float32(Float32(x_m / s) / s) * x_m), Float32(-0.0625), Float32(0.25)) / s);
	end
	return 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))))) < 4.0999999

   1. Initial program 99.4%

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

      1. lift-fabs.f32 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sqrt.f32 42.3

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

   4. Applied rewrites 42.3%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sqrt-unprod N/A

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

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

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

      4. sqrt-unprod N/A

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

      5. rem-square-sqrt 42.3

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

   7. Applied rewrites 42.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s + x \cdot \left(3 \cdot \frac{x}{s} - 4\right)}} \]

   10. Applied rewrites 98.2%

       \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, x, 4 \cdot s\right)}} \]

   11. Taylor expanded in s around inf

       \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, x, 4 \cdot s\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 58.6%

       \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, x, 4 \cdot s\right)} \]

   if 4.0999999 < (/.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 98.9%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

         \[\leadsto \frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{\color{blue}{s}} \]

   5. Applied rewrites 71.8%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\mathsf{fma}\left(x \cdot x, 0.125, -0.0625 \cdot \left(3 \cdot \left(x \cdot x\right)\right)\right)}{s \cdot s}}{s}} \]

   7. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot \frac{x}{s \cdot s}, -0.0625, 0.25\right)}{s}} \]
   8. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot \frac{x}{s \cdot s}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot \frac{x}{s \cdot s}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      3. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot \frac{x}{s \cdot s}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s \cdot s} \cdot x, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{{s}^{2}} \cdot x, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{{s}^{2}} \cdot x, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s \cdot s} \cdot x, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{x}{s}}{s} \cdot x, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      9. lower-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{x}{s}}{s} \cdot x, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      10. lift-/.f32 87.0

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{x}{s}}{s} \cdot x, -0.0625, 0.25\right)}{s} \]

   9. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{x}{s}}{s} \cdot x, -0.0625, 0.25\right)}{s}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x_m) s)))) (/ (/ t_0 (pow (- t_0 -1.0) 2.0)) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-x_m / s));
	return (t_0 / powf((t_0 - -1.0f), 2.0f)) / 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
    real(4) :: t_0
    t_0 = exp((-x_m / s))
    code = (t_0 / ((t_0 - (-1.0e0)) ** 2.0e0)) / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-fabs.f32 N/A

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

   6. flip3-+ N/A

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

   7. metadata-eval N/A

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

   8. div-add N/A

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 95.0%

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

8. Applied rewrites 60.5%

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

9. Final simplification 60.5%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x_m) s)))) (/ (/ t_0 s) (pow (- t_0 -1.0) 2.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-x_m / s));
	return (t_0 / s) / powf((t_0 - -1.0f), 2.0f);
}

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
    t_0 = exp((-x_m / s))
    code = (t_0 / s) / ((t_0 - (-1.0e0)) ** 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-fabs.f32 N/A

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

   6. flip3-+ N/A

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

   7. metadata-eval N/A

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

   8. div-add N/A

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 95.0%

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

8. Applied rewrites 60.1%

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

9. Final simplification 60.1%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf((-x_m / s)) / (powf((expf((x_m * (-1.0f / s))) - -1.0f), 2.0f) * 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 = exp((-x_m / s)) / (((exp((x_m * ((-1.0e0) / s))) - (-1.0e0)) ** 2.0e0) * s)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-fabs.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-neg.f32 N/A

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

   12. lift-fabs.f32 N/A

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

   13. associate-*l* N/A

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

4. Applied rewrites 99.2%

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

   1. lift-fabs.f32 N/A

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

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

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

   3. sqrt-unprod N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. lift-*.f32 41.9

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

   7. lift-*.f32 N/A

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

   8. lift-sqrt.f32 N/A

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

   9. lift-sqrt.f32 N/A

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

   10. rem-square-sqrt 57.2

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

6. Applied rewrites 60.5%

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

   1. lift-neg.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. mul-1-neg N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 60.4

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

8. Applied rewrites 60.4%

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

9. Final simplification 60.4%

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

Alternative 7: 99.5% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x_m) s)))) (/ t_0 (* (pow (- t_0 -1.0) 2.0) s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-x_m / s));
	return t_0 / (powf((t_0 - -1.0f), 2.0f) * 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
    real(4) :: t_0
    t_0 = exp((-x_m / s))
    code = t_0 / (((t_0 - (-1.0e0)) ** 2.0e0) * s)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-fabs.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-neg.f32 N/A

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

   12. lift-fabs.f32 N/A

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

   13. associate-*l* N/A

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

4. Applied rewrites 99.2%

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

   1. lift-fabs.f32 N/A

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

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

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

   3. sqrt-unprod N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. lift-*.f32 41.9

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

   7. lift-*.f32 N/A

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

   8. lift-sqrt.f32 N/A

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

   9. lift-sqrt.f32 N/A

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

   10. rem-square-sqrt 57.2

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

6. Applied rewrites 60.5%

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

7. Final simplification 60.5%

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

Alternative 8: 97.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\frac{-\left|x\_m\right|}{s}}}{\mathsf{fma}\left(0.5 \cdot \frac{x\_m}{s} - 1, x\_m, 2 \cdot s\right) \cdot \left(1 + e^{\frac{\sqrt{x\_m} \cdot \sqrt{x\_m}}{-s}}\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/
  (exp (/ (- (fabs x_m)) s))
  (*
   (fma (- (* 0.5 (/ x_m s)) 1.0) x_m (* 2.0 s))
   (+ 1.0 (exp (/ (* (sqrt x_m) (sqrt x_m)) (- s)))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf((-fabsf(x_m) / s)) / (fmaf(((0.5f * (x_m / s)) - 1.0f), x_m, (2.0f * s)) * (1.0f + expf(((sqrtf(x_m) * sqrtf(x_m)) / -s))));
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(exp(Float32(Float32(-abs(x_m)) / s)) / Float32(fma(Float32(Float32(Float32(0.5) * Float32(x_m / s)) - Float32(1.0)), x_m, Float32(Float32(2.0) * s)) * Float32(Float32(1.0) + exp(Float32(Float32(sqrt(x_m) * sqrt(x_m)) / Float32(-s))))))
end

Derivation

1. Initial program 99.3%

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

   1. lift-fabs.f32 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-sqrt.f32 42.1

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

4. Applied rewrites 42.1%

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

5. Taylor expanded in x around 0

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

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

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

   2. sqrt-unprod N/A

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

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

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

   4. sqrt-unprod N/A

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

   5. rem-square-sqrt 42.1

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

7. Applied rewrites 42.1%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 40.1%

    \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\mathsf{fma}\left(0.5 \cdot \frac{x}{s} - 1, x, 2 \cdot s\right)} \cdot \left(1 + e^{\frac{-\sqrt{x} \cdot \sqrt{x}}{s}}\right)} \]

11. Final simplification 40.1%

    \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\mathsf{fma}\left(0.5 \cdot \frac{x}{s} - 1, x, 2 \cdot s\right) \cdot \left(1 + e^{\frac{\sqrt{x} \cdot \sqrt{x}}{-s}}\right)} \]
12. Add Preprocessing

Alternative 9: 97.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ \frac{t\_0}{\mathsf{fma}\left(s, 1, \mathsf{fma}\left(0.5 \cdot \frac{x\_m}{s} - 1, x\_m, s\right)\right) \cdot \left(1 + t\_0\right)} \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))))
   (/ t_0 (* (fma s 1.0 (fma (- (* 0.5 (/ x_m s)) 1.0) x_m s)) (+ 1.0 t_0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-fabsf(x_m) / s));
	return t_0 / (fmaf(s, 1.0f, fmaf(((0.5f * (x_m / s)) - 1.0f), x_m, s)) * (1.0f + t_0));
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(Float32(-abs(x_m)) / s))
	return Float32(t_0 / Float32(fma(s, Float32(1.0), fma(Float32(Float32(Float32(0.5) * Float32(x_m / s)) - Float32(1.0)), x_m, s)) * Float32(Float32(1.0) + t_0)))
end

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-fabs.f32 N/A

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

   6. flip3-+ N/A

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

   7. metadata-eval N/A

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

   8. div-add N/A

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 95.0%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 93.2

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

9. Applied rewrites 93.2%

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

Alternative 10: 96.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\frac{-x\_m}{s}}}{\mathsf{fma}\left(\frac{x\_m}{s} \cdot 3 - 4, x\_m, 4 \cdot s\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (exp (/ (- x_m) s)) (fma (- (* (/ x_m s) 3.0) 4.0) x_m (* 4.0 s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf((-x_m / s)) / fmaf((((x_m / s) * 3.0f) - 4.0f), x_m, (4.0f * s));
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(exp(Float32(Float32(-x_m) / s)) / fma(Float32(Float32(Float32(x_m / s) * Float32(3.0)) - Float32(4.0)), x_m, Float32(Float32(4.0) * s)))
end

Derivation

1. Initial program 99.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-fabs.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-neg.f32 N/A

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

   12. lift-fabs.f32 N/A

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

   13. associate-*l* N/A

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

4. Applied rewrites 99.2%

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

   1. lift-fabs.f32 N/A

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

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

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

   3. sqrt-unprod N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. lift-*.f32 41.9

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

   7. lift-*.f32 N/A

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

   8. lift-sqrt.f32 N/A

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

   9. lift-sqrt.f32 N/A

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

   10. rem-square-sqrt 57.2

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

6. Applied rewrites 60.5%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{e^{\frac{-x}{s}}}{\color{blue}{4 \cdot s + x \cdot \left(3 \cdot \frac{x}{s} - 4\right)}} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e^{\frac{-x}{s}}}{x \cdot \left(3 \cdot \frac{x}{s} - 4\right) + \color{blue}{4 \cdot s}} \]

   2. *-commutative N/A

      \[\leadsto \frac{e^{\frac{-x}{s}}}{\left(3 \cdot \frac{x}{s} - 4\right) \cdot x + \color{blue}{4} \cdot s} \]

   3. *-commutative N/A

      \[\leadsto \frac{e^{\frac{-x}{s}}}{\left(\frac{x}{s} \cdot 3 - 4\right) \cdot x + 4 \cdot s} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{e^{\frac{-x}{s}}}{\left(\frac{x}{s} \cdot 3 - 4\right) \cdot x + 4 \cdot s} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{e^{\frac{-x}{s}}}{\left(\frac{x}{s} \cdot 3 - 4\right) \cdot x + 4 \cdot s} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{e^{\frac{-x}{s}}}{\left(\frac{x}{s} \cdot 3 - 4\right) \cdot x + 4 \cdot s} \]

   7. lift-fma.f32 N/A

      \[\leadsto \frac{e^{\frac{-x}{s}}}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, \color{blue}{x}, 4 \cdot s\right)} \]

   8. lift-*.f32 56.5

      \[\leadsto \frac{e^{\frac{-x}{s}}}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, x, 4 \cdot s\right)} \]

9. Applied rewrites 56.5%

   \[\leadsto \frac{e^{\frac{-x}{s}}}{\color{blue}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, x, 4 \cdot s\right)}} \]

10. Final simplification 56.5%

    \[\leadsto \frac{e^{\frac{-x}{s}}}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, x, 4 \cdot s\right)} \]
11. Add Preprocessing

Alternative 11: 86.5% accurate, 6.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4999999913984:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x\_m}{s}, -0.25, 0.25\right) - \frac{x\_m}{s} \cdot -0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{x\_m}{s} \cdot 3 - 4, x\_m, 4 \cdot s\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 4999999913984.0)
   (/ (- (fma (/ x_m s) -0.25 0.25) (* (/ x_m s) -0.25)) s)
   (/ 1.0 (fma (- (* (/ x_m s) 3.0) 4.0) x_m (* 4.0 s)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 4999999913984.0f) {
		tmp = (fmaf((x_m / s), -0.25f, 0.25f) - ((x_m / s) * -0.25f)) / s;
	} else {
		tmp = 1.0f / fmaf((((x_m / s) * 3.0f) - 4.0f), x_m, (4.0f * s));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(4999999913984.0))
		tmp = Float32(Float32(fma(Float32(x_m / s), Float32(-0.25), Float32(0.25)) - Float32(Float32(x_m / s) * Float32(-0.25))) / s);
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(Float32(x_m / s) * Float32(3.0)) - Float32(4.0)), x_m, Float32(Float32(4.0) * s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 4999999910000

   1. Initial program 99.1%

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

      1. lift-fabs.f32 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sqrt.f32 32.3

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

   4. Applied rewrites 32.3%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sqrt-unprod N/A

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

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

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

      4. sqrt-unprod N/A

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

      5. rem-square-sqrt 32.3

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

   7. Applied rewrites 32.3%

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

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 65.5%

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

   if 4999999910000 < x

   1. Initial program 100.0%

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

      1. lift-fabs.f32 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sqrt.f32 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sqrt-unprod N/A

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

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

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

      4. sqrt-unprod N/A

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

      5. rem-square-sqrt 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s + x \cdot \left(3 \cdot \frac{x}{s} - 4\right)}} \]

   10. Applied rewrites 100.0%

       \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, x, 4 \cdot s\right)}} \]

   11. Taylor expanded in s around inf

       \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, x, 4 \cdot s\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 97.7%

       \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, x, 4 \cdot s\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 64.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\mathsf{fma}\left(\frac{x\_m}{s} \cdot 3 - 4, x\_m, 4 \cdot s\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ 1.0 (fma (- (* (/ x_m s) 3.0) 4.0) x_m (* 4.0 s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 1.0f / fmaf((((x_m / s) * 3.0f) - 4.0f), x_m, (4.0f * s));
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(1.0) / fma(Float32(Float32(Float32(x_m / s) * Float32(3.0)) - Float32(4.0)), x_m, Float32(Float32(4.0) * s)))
end

Derivation

1. Initial program 99.3%

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

   1. lift-fabs.f32 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-sqrt.f32 42.1

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

4. Applied rewrites 42.1%

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

5. Taylor expanded in x around 0

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

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

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

   2. sqrt-unprod N/A

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

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

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

   4. sqrt-unprod N/A

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

   5. rem-square-sqrt 42.1

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

7. Applied rewrites 42.1%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s + x \cdot \left(3 \cdot \frac{x}{s} - 4\right)}} \]

10. Applied rewrites 92.7%

    \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, x, 4 \cdot s\right)}} \]

11. Taylor expanded in s around inf

    \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, x, 4 \cdot s\right)} \]
12. Step-by-step derivation

13. Applied rewrites 63.2%

    \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\frac{x}{s} \cdot 3 - 4, x, 4 \cdot s\right)} \]
14. Add Preprocessing

Alternative 13: 27.4% 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 99.3%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 28.3

      \[\leadsto \frac{0.25}{\color{blue}{s}} \]

5. Applied rewrites 28.3%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]

6. Final simplification 28.3%

   \[\leadsto \frac{0.25}{s} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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))))))