Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 8.2s

Alternatives: 11

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

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))))
   (/ t_0 (* (* s (- 1.0 (/ -1.0 (exp (/ 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 * (1.0f - (-1.0f / expf((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 * (1.0e0 - ((-1.0e0) / exp((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(s * Float32(Float32(1.0) - Float32(Float32(-1.0) / exp(Float32(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 * (single(1.0) - (single(-1.0) / exp((x_m / s))))) * (single(1.0) + t_0));
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. 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. *-lft-identity N/A

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

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

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

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

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

   5. *-lft-identity N/A

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

   6. lower--.f32 N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

   9. lift-exp.f32 N/A

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

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

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

   11. flip-+ N/A

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

   12. sinh---cosh-rev N/A

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

4. Applied rewrites 97.7%

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

Alternative 2: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{-x\_m}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.0625 \cdot {\left(\frac{x\_m}{s}\right)}^{2} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 1.9999999494757503e-5)
     (/ (exp (/ (- x_m) s)) s)
     (/ (+ (* -0.0625 (pow (/ x_m s) 2.0)) 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)) <= 1.9999999494757503e-5f) {
		tmp = expf((-x_m / s)) / s;
	} else {
		tmp = ((-0.0625f * powf((x_m / s), 2.0f)) + 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)) <= 1.9999999494757503e-5) then
        tmp = exp((-x_m / s)) / s
    else
        tmp = (((-0.0625e0) * ((x_m / s) ** 2.0e0)) + 0.25e0) / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(Float32(-abs(x_m)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_1) * t_1)) <= Float32(1.9999999494757503e-5))
		tmp = Float32(exp(Float32(Float32(-x_m) / s)) / s);
	else
		tmp = Float32(Float32(Float32(Float32(-0.0625) * (Float32(x_m / s) ^ Float32(2.0))) + Float32(0.25)) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	t_0 = exp((-abs(x_m) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(1.9999999494757503e-5))
		tmp = exp((-x_m / s)) / s;
	else
		tmp = ((single(-0.0625) * ((x_m / s) ^ single(2.0))) + single(0.25)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 1.99999995e-5

   1. Initial program 99.9%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in s around -inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

   7. Applied rewrites 79.7%

      \[\leadsto \frac{e^{\frac{-x}{s} - \color{blue}{\left(\log 2 - \frac{-0.5 \cdot \left(\frac{\left(x \cdot x\right) \cdot 0.25}{s} - x\right)}{s}\right)} \cdot 2}}{s} \]
   8. Step-by-step derivation

   9. Applied rewrites 85.0%

      \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log 2 - \frac{-0.5 \cdot \left(x \cdot \frac{0.25 \cdot x}{s} - x\right)}{s}\right) \cdot 2}}{s} \]

   10. Taylor expanded in x around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f32 55.5

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

   12. Applied rewrites 55.5%

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

   if 1.99999995e-5 < (/.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 99.0%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 25.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   7. Applied rewrites 91.0%

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

   9. Applied rewrites 91.0%

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

   11. Applied rewrites 93.8%

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

Alternative 3: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{-x\_m}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.0625 \cdot \left(x\_m \cdot x\_m\right)}{s}}{s} + 0.25}{s}\\ \end{array} \end{array} \]

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)) 1.9999999494757503e-5)
     (/ (exp (/ (- x_m) s)) s)
     (/ (+ (/ (/ (* -0.0625 (* x_m 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)) <= 1.9999999494757503e-5f) {
		tmp = expf((-x_m / s)) / s;
	} else {
		tmp = ((((-0.0625f * (x_m * x_m)) / s) / 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)) <= 1.9999999494757503e-5) then
        tmp = exp((-x_m / s)) / s
    else
        tmp = (((((-0.0625e0) * (x_m * x_m)) / s) / 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(1.9999999494757503e-5))
		tmp = Float32(exp(Float32(Float32(-x_m) / s)) / s);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(-0.0625) * Float32(x_m * x_m)) / s) / 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(1.9999999494757503e-5))
		tmp = exp((-x_m / s)) / s;
	else
		tmp = ((((single(-0.0625) * (x_m * x_m)) / s) / 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))))) < 1.99999995e-5

   1. Initial program 99.9%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in s around -inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

   7. Applied rewrites 79.7%

      \[\leadsto \frac{e^{\frac{-x}{s} - \color{blue}{\left(\log 2 - \frac{-0.5 \cdot \left(\frac{\left(x \cdot x\right) \cdot 0.25}{s} - x\right)}{s}\right)} \cdot 2}}{s} \]
   8. Step-by-step derivation

   9. Applied rewrites 85.0%

      \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log 2 - \frac{-0.5 \cdot \left(x \cdot \frac{0.25 \cdot x}{s} - x\right)}{s}\right) \cdot 2}}{s} \]

   10. Taylor expanded in x around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f32 55.5

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

   12. Applied rewrites 55.5%

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

   if 1.99999995e-5 < (/.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 99.0%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 25.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   7. Applied rewrites 91.0%

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

   9. Applied rewrites 93.8%

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

Alternative 4: 30.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{x\_m \cdot x\_m}{s}\\ t_1 := -\left|x\_m\right|\\ t_2 := e^{\frac{t\_1}{s}}\\ t_3 := 1 + t\_2\\ \mathbf{if}\;\frac{t\_2}{\left(s \cdot t\_3\right) \cdot t\_3} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 - \frac{\mathsf{fma}\left(t\_0, -0.5, \left|x\_m\right|\right)}{s}}{\left(s \cdot \left(\frac{\mathsf{fma}\left(t\_0, 0.5, t\_1\right)}{s} + 2\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.0625 \cdot \left(x\_m \cdot x\_m\right)}{s}}{s} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (/ (* x_m x_m) s))
        (t_1 (- (fabs x_m)))
        (t_2 (exp (/ t_1 s)))
        (t_3 (+ 1.0 t_2)))
   (if (<= (/ t_2 (* (* s t_3) t_3)) 1.9999999494757503e-5)
     (/
      (- 1.0 (/ (fma t_0 -0.5 (fabs x_m)) s))
      (* (* s (+ (/ (fma t_0 0.5 t_1) s) 2.0)) 2.0))
     (/ (+ (/ (/ (* -0.0625 (* x_m x_m)) s) s) 0.25) s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = (x_m * x_m) / s;
	float t_1 = -fabsf(x_m);
	float t_2 = expf((t_1 / s));
	float t_3 = 1.0f + t_2;
	float tmp;
	if ((t_2 / ((s * t_3) * t_3)) <= 1.9999999494757503e-5f) {
		tmp = (1.0f - (fmaf(t_0, -0.5f, fabsf(x_m)) / s)) / ((s * ((fmaf(t_0, 0.5f, t_1) / s) + 2.0f)) * 2.0f);
	} else {
		tmp = ((((-0.0625f * (x_m * x_m)) / s) / s) + 0.25f) / s;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = Float32(Float32(x_m * x_m) / s)
	t_1 = Float32(-abs(x_m))
	t_2 = exp(Float32(t_1 / s))
	t_3 = Float32(Float32(1.0) + t_2)
	tmp = Float32(0.0)
	if (Float32(t_2 / Float32(Float32(s * t_3) * t_3)) <= Float32(1.9999999494757503e-5))
		tmp = Float32(Float32(Float32(1.0) - Float32(fma(t_0, Float32(-0.5), abs(x_m)) / s)) / Float32(Float32(s * Float32(Float32(fma(t_0, Float32(0.5), t_1) / s) + Float32(2.0))) * Float32(2.0)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(-0.0625) * Float32(x_m * x_m)) / s) / s) + 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))))) < 1.99999995e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f32 N/A

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

      4. lower-neg.f32 N/A

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

      5. lower--.f32 N/A

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in s around -inf

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. lower--.f32 N/A

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

       5. lower-/.f32 N/A

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

   11. Applied rewrites 3.2%

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

   if 1.99999995e-5 < (/.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 99.0%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 25.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   7. Applied rewrites 91.0%

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

   9. Applied rewrites 93.8%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 - \frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, -0.5, \left|x\right|\right)}{s}}{\left(s \cdot \left(\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, 0.5, -\left|x\right|\right)}{s} + 2\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.0625 \cdot \left(x \cdot x\right)}{s}}{s} + 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0:\\ \;\;\;\;\frac{\frac{-0.0625}{s} \cdot \frac{x\_m \cdot x\_m}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.0625 \cdot \left(x\_m \cdot x\_m\right)}{s}}{s} + 0.25}{s}\\ \end{array} \end{array} \]

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)) 0.0)
     (/ (* (/ -0.0625 s) (/ (* x_m x_m) s)) s)
     (/ (+ (/ (/ (* -0.0625 (* x_m 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)) <= 0.0f) {
		tmp = ((-0.0625f / s) * ((x_m * x_m) / s)) / s;
	} else {
		tmp = ((((-0.0625f * (x_m * x_m)) / s) / 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)) <= 0.0e0) then
        tmp = (((-0.0625e0) / s) * ((x_m * x_m) / s)) / s
    else
        tmp = (((((-0.0625e0) * (x_m * x_m)) / s) / 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(0.0))
		tmp = Float32(Float32(Float32(Float32(-0.0625) / s) * Float32(Float32(x_m * x_m) / s)) / s);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(-0.0625) * Float32(x_m * x_m)) / s) / 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(0.0))
		tmp = ((single(-0.0625) / s) * ((x_m * x_m) / s)) / s;
	else
		tmp = ((((single(-0.0625) * (x_m * x_m)) / s) / 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))))) < 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

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 85.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   7. Applied rewrites 4.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\frac{{x}^{2}}{{s}^{2}}}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 8.5%

       \[\leadsto \frac{\frac{-0.0625}{s} \cdot \color{blue}{\frac{x \cdot x}{s}}}{s} \]

   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 98.7%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 26.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   7. Applied rewrites 88.7%

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

   9. Applied rewrites 91.3%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (exp (/ (- x_m) s)) (* (pow (+ (exp (/ (- (fabs x_m)) 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((-fabsf(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((-x_m / s)) / (((exp((-abs(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(-x_m) / s)) / Float32((Float32(exp(Float32(Float32(-abs(x_m)) / 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((-abs(x_m) / s)) + single(1.0)) ^ single(2.0)) * s);
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. 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 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lift-fabs.f32 N/A

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

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

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

   7. sqrt-prod N/A

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

   8. rem-square-sqrt N/A

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

   9. distribute-neg-frac2 N/A

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

   10. distribute-frac-neg N/A

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

   11. lift-neg.f32 N/A

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

   12. lift-/.f32 65.7

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

7. Applied rewrites 65.7%

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

8. Final simplification 65.7%

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

Alternative 7: 96.9% accurate, 1.3× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))))
   (/ t_0 (* (* s (- 1.0 (/ -1.0 (+ (/ x_m s) 1.0)))) (+ 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 * (1.0f - (-1.0f / ((x_m / s) + 1.0f)))) * (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 * (1.0e0 - ((-1.0e0) / ((x_m / s) + 1.0e0)))) * (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(s * Float32(Float32(1.0) - Float32(Float32(-1.0) / Float32(Float32(x_m / s) + Float32(1.0))))) * 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 * (single(1.0) - (single(-1.0) / ((x_m / s) + single(1.0))))) * (single(1.0) + t_0));
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. 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. *-lft-identity N/A

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

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

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

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

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

   5. *-lft-identity N/A

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

   6. lower--.f32 N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

   9. lift-exp.f32 N/A

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

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

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

   11. flip-+ N/A

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

   12. sinh---cosh-rev N/A

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

4. Applied rewrites 97.7%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 96.9

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

7. Applied rewrites 96.9%

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

Alternative 8: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\frac{-x\_m}{s} - \left(\log 2 - \frac{-0.5 \cdot \left(x\_m \cdot \frac{0.25 \cdot x\_m}{s} - x\_m\right)}{s}\right) \cdot 2}}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/
  (exp
   (-
    (/ (- x_m) s)
    (* (- (log 2.0) (/ (* -0.5 (- (* x_m (/ (* 0.25 x_m) s)) x_m)) s)) 2.0)))
  s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf(((-x_m / s) - ((logf(2.0f) - ((-0.5f * ((x_m * ((0.25f * x_m) / s)) - x_m)) / s)) * 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(((-x_m / s) - ((log(2.0e0) - (((-0.5e0) * ((x_m * ((0.25e0 * x_m) / s)) - x_m)) / s)) * 2.0e0))) / s
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(exp(Float32(Float32(Float32(-x_m) / s) - Float32(Float32(log(Float32(2.0)) - Float32(Float32(Float32(-0.5) * Float32(Float32(x_m * Float32(Float32(Float32(0.25) * x_m) / s)) - x_m)) / s)) * Float32(2.0)))) / s)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = exp(((-x_m / s) - ((log(single(2.0)) - ((single(-0.5) * ((x_m * ((single(0.25) * x_m) / s)) - x_m)) / s)) * single(2.0)))) / s;
end

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 68.8%

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

5. Taylor expanded in s around -inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. lower-log.f32 N/A

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

   6. lower-/.f32 N/A

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

7. Applied rewrites 83.8%

   \[\leadsto \frac{e^{\frac{-x}{s} - \color{blue}{\left(\log 2 - \frac{-0.5 \cdot \left(\frac{\left(x \cdot x\right) \cdot 0.25}{s} - x\right)}{s}\right)} \cdot 2}}{s} \]
8. Step-by-step derivation

9. Applied rewrites 87.5%

   \[\leadsto \frac{e^{\frac{-x}{s} - \left(\log 2 - \frac{-0.5 \cdot \left(x \cdot \frac{0.25 \cdot x}{s} - x\right)}{s}\right) \cdot 2}}{s} \]
10. Add Preprocessing

Alternative 9: 96.1% accurate, 1.4× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))))
   (/ t_0 (* (* s (- 2.0 (/ (fabs 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 * (2.0f - (fabsf(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 * (2.0e0 - (abs(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(s * Float32(Float32(2.0) - Float32(abs(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 * (single(2.0) - (abs(x_m) / s))) * (single(1.0) + t_0));
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fabs.f32 96.9

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

5. Applied rewrites 96.9%

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

Alternative 10: 95.0% 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 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. 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 11: 27.2% accurate, 31.1× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = 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 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. 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 28.8

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

5. Applied rewrites 28.8%

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

Reproduce

herbie shell --seed 2024326 
(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))))))