Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 8.9s

Alternatives: 14

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

real(4) function code(x, s)
    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

real(4) function code(x, s)
    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

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))))
   (/ t_0 (* (+ (/ s (exp (/ x_m s))) 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 / (((s / expf((x_m / s))) + s) * (1.0f + t_0));
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f32 N/A

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

4. Applied rewrites 97.0%

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

Alternative 2: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\left(--0.125\right) \cdot \left(x\_m - \left|x\_m\right|\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{x\_m}{s} \cdot x\_m\right) \cdot 0.0625}{s} - 0.25}{-s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 4.999999858590343e-10)
     (/ (/ (* (- -0.125) (- x_m (fabs x_m))) s) s)
     (/ (- (/ (* (* (/ x_m s) x_m) 0.0625) s) 0.25) (- s)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-fabsf(x_m) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 4.999999858590343e-10f) {
		tmp = ((-(-0.125f) * (x_m - fabsf(x_m))) / s) / s;
	} else {
		tmp = (((((x_m / s) * x_m) * 0.0625f) / s) - 0.25f) / -s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x_m) / s))
    t_1 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 4.999999858590343e-10) then
        tmp = ((-(-0.125e0) * (x_m - abs(x_m))) / s) / s
    else
        tmp = (((((x_m / s) * x_m) * 0.0625e0) / s) - 0.25e0) / -s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(Float32(-abs(x_m)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_1) * t_1)) <= Float32(4.999999858590343e-10))
		tmp = Float32(Float32(Float32(Float32(-Float32(-0.125)) * Float32(x_m - abs(x_m))) / s) / s);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(x_m / s) * x_m) * Float32(0.0625)) / s) - Float32(0.25)) / Float32(-s));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	t_0 = exp((-abs(x_m) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(4.999999858590343e-10))
		tmp = ((-single(-0.125) * (x_m - abs(x_m))) / s) / s;
	else
		tmp = (((((x_m / s) * x_m) * single(0.0625)) / s) - single(0.25)) / -s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 4.99999986e-10

   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-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 4.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{4} - \frac{\mathsf{fma}\left(\frac{1}{4}, \left|x\right|, \frac{-1}{8} \cdot x + \frac{-1}{8} \cdot \left|x\right|\right)}{s}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 4.0%

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

   11. Taylor expanded in s around 0

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

   13. Applied rewrites 50.2%

       \[\leadsto \frac{\frac{--0.125 \cdot \left(x - \left|x\right|\right)}{s}}{s} \]

   if 4.99999986e-10 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 89.1%

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

   7. Applied rewrites 92.2%

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

   9. Applied rewrites 92.2%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\left(--0.125\right) \cdot \left(x - \left|x\right|\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{s} \cdot x\right) \cdot 0.0625}{s} - 0.25}{-s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\left(--0.125\right) \cdot \left(x\_m - \left|x\_m\right|\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{x\_m}{s} \cdot x\_m\right) \cdot \frac{-0.0625}{s} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 4.999999858590343e-10)
     (/ (/ (* (- -0.125) (- x_m (fabs x_m))) s) s)
     (/ (+ (* (* (/ x_m s) x_m) (/ -0.0625 s)) 0.25) s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-fabsf(x_m) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 4.999999858590343e-10f) {
		tmp = ((-(-0.125f) * (x_m - fabsf(x_m))) / s) / s;
	} else {
		tmp = ((((x_m / s) * x_m) * (-0.0625f / s)) + 0.25f) / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x_m) / s))
    t_1 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 4.999999858590343e-10) then
        tmp = ((-(-0.125e0) * (x_m - abs(x_m))) / s) / s
    else
        tmp = ((((x_m / s) * x_m) * ((-0.0625e0) / s)) + 0.25e0) / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(Float32(-abs(x_m)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_1) * t_1)) <= Float32(4.999999858590343e-10))
		tmp = Float32(Float32(Float32(Float32(-Float32(-0.125)) * Float32(x_m - abs(x_m))) / s) / s);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(x_m / s) * x_m) * Float32(Float32(-0.0625) / s)) + Float32(0.25)) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	t_0 = exp((-abs(x_m) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(4.999999858590343e-10))
		tmp = ((-single(-0.125) * (x_m - abs(x_m))) / s) / s;
	else
		tmp = ((((x_m / s) * x_m) * (single(-0.0625) / s)) + single(0.25)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 4.99999986e-10

   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-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 4.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{4} - \frac{\mathsf{fma}\left(\frac{1}{4}, \left|x\right|, \frac{-1}{8} \cdot x + \frac{-1}{8} \cdot \left|x\right|\right)}{s}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 4.0%

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

   11. Taylor expanded in s around 0

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

   13. Applied rewrites 50.2%

       \[\leadsto \frac{\frac{--0.125 \cdot \left(x - \left|x\right|\right)}{s}}{s} \]

   if 4.99999986e-10 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 89.1%

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

   7. Applied rewrites 92.2%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\left(--0.125\right) \cdot \left(x - \left|x\right|\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{x}{s} \cdot x\right) \cdot \frac{-0.0625}{s} + 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\left(--0.125\right) \cdot \left(x\_m - \left|x\_m\right|\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 4.999999858590343e-10)
     (/ (/ (* (- -0.125) (- x_m (fabs x_m))) s) s)
     (/ 0.25 s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-fabsf(x_m) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 4.999999858590343e-10f) {
		tmp = ((-(-0.125f) * (x_m - fabsf(x_m))) / s) / s;
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x_m) / s))
    t_1 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 4.999999858590343e-10) then
        tmp = ((-(-0.125e0) * (x_m - abs(x_m))) / s) / s
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(Float32(-abs(x_m)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_1) * t_1)) <= Float32(4.999999858590343e-10))
		tmp = Float32(Float32(Float32(Float32(-Float32(-0.125)) * Float32(x_m - abs(x_m))) / s) / s);
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	t_0 = exp((-abs(x_m) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(4.999999858590343e-10))
		tmp = ((-single(-0.125) * (x_m - abs(x_m))) / s) / s;
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 4.99999986e-10

   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-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 4.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{4} - \frac{\mathsf{fma}\left(\frac{1}{4}, \left|x\right|, \frac{-1}{8} \cdot x + \frac{-1}{8} \cdot \left|x\right|\right)}{s}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 4.0%

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

   11. Taylor expanded in s around 0

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

   13. Applied rewrites 50.2%

       \[\leadsto \frac{\frac{--0.125 \cdot \left(x - \left|x\right|\right)}{s}}{s} \]

   if 4.99999986e-10 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 89.8

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

   5. Applied rewrites 89.8%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\left(--0.125\right) \cdot \left(x - \left|x\right|\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 0.0) 0.0 (/ 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)) <= 0.0f) {
		tmp = 0.0f;
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x_m) / s))
    t_1 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 0.0e0) then
        tmp = 0.0e0
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0

   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-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 4.0%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 3.4%

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

   12. Applied rewrites 57.8%

       \[\leadsto \frac{0.125 \cdot \left(x - x\right)}{s \cdot s} \]

   if 0.0 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 = abs(x)
real(4) function code(x_m, s)
    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(x_m / Float32(-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 99.8%

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

3. Taylor expanded in x around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. mul-1-neg N/A

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

   6. distribute-neg-frac2 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-fabs.f32 N/A

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

   9. lower-neg.f32 N/A

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

   10. lower-pow.f32 N/A

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 96.7%

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

9. Applied rewrites 61.9%

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

Alternative 7: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-fabs.f32 N/A

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

   11. lower-neg.f32 99.7

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

5. Applied rewrites 99.7%

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

7. Applied rewrites 96.7%

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

Alternative 8: 96.7% accurate, 1.3× speedup

Math

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

FPCore

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

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((-abs(x_m) / s))
    code = t_0 / (((s * ((((((x_m * x_m) / s) * 0.5e0) - x_m) / s) - (-1.0e0))) + s) * (1.0e0 + t_0))
end function

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	t_0 = exp((-abs(x_m) / s));
	tmp = t_0 / (((s * ((((((x_m * x_m) / s) * single(0.5)) - x_m) / s) - single(-1.0))) + s) * (single(1.0) + t_0));
end

Derivation

1. Initial program 99.8%

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

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f32 N/A

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

4. Applied rewrites 97.0%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f32 N/A

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

   4. lower-neg.f32 N/A

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

   5. lower--.f32 N/A

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

7. Applied rewrites 96.1%

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

8. Final simplification 96.1%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - x}{s} - -1\right) + s\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
9. Add Preprocessing

Alternative 9: 96.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fabs.f32 96.6

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

5. Applied rewrites 96.6%

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

Alternative 10: 96.2% accurate, 1.5× speedup

Math

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

FPCore

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f32 N/A

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

4. Applied rewrites 97.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 95.8

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

7. Applied rewrites 95.8%

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

Alternative 11: 95.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 95.4%

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

Alternative 12: 95.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{1}{e^{\frac{x\_m}{s}}}}{4 \cdot s} \end{array} \]

FPCore

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

C

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

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = (1.0e0 / exp((x_m / s))) / (4.0e0 * s)
end function

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(1.0) / exp((x_m / s))) / (single(4.0) * s);
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 95.1

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

5. Applied rewrites 95.1%

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

   1. lift-exp.f32 N/A

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

   2. sinh-+-cosh-rev N/A

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

   3. flip-+ N/A

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

   4. sinh-cosh N/A

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

   5. flip-- N/A

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

   6. sinh-cosh N/A

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

   7. sinh-+-cosh-rev N/A

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

   8. lift-/.f32 N/A

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

   9. frac-2neg N/A

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

   10. lift-neg.f32 N/A

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

   11. remove-double-neg N/A

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

   12. lift-neg.f32 N/A

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

7. Applied rewrites 58.7%

   \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{x}{s}}}}}{4 \cdot s} \]
8. Add Preprocessing

Alternative 13: 95.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = exp((x_m / -s)) / (4.0e0 * s)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 95.1

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

5. Applied rewrites 95.1%

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

   1. lift-fabs.f32 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 58.7

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

7. Applied rewrites 58.7%

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

8. Final simplification 58.7%

   \[\leadsto \frac{e^{\frac{x}{-s}}}{4 \cdot s} \]
9. Add Preprocessing

Alternative 14: 27.3% 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 = abs(x)
real(4) function code(x_m, s)
    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 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. 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 27.1

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

5. Applied rewrites 27.1%

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

Reproduce

herbie shell --seed 2024339 
(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))))))