Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 10.9s

Alternatives: 13

Speedup: 1.1×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

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

FPCore

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

C

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

Fortran

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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f32 99.4

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.0625}{0.25 - {\left(\frac{\left|x\right|}{s}\right)}^{2} \cdot -0.0625}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (- t_0 -1.0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 9.999999747378752e-5)
     (/ t_0 (* 4.0 s))
     (/ (/ 0.0625 (- 0.25 (* (pow (/ (fabs x) s) 2.0) -0.0625))) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = t_0 - -1.0f;
	float tmp;
	if ((t_0 / ((t_1 * s) * t_1)) <= 9.999999747378752e-5f) {
		tmp = t_0 / (4.0f * s);
	} else {
		tmp = (0.0625f / (0.25f - (powf((fabsf(x) / s), 2.0f) * -0.0625f))) / s;
	}
	return tmp;
}

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
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 - (-1.0e0)
    if ((t_0 / ((t_1 * s) * t_1)) <= 9.999999747378752e-5) then
        tmp = t_0 / (4.0e0 * s)
    else
        tmp = (0.0625e0 / (0.25e0 - (((abs(x) / s) ** 2.0e0) * (-0.0625e0)))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(9.999999747378752e-5))
		tmp = Float32(t_0 / Float32(Float32(4.0) * s));
	else
		tmp = Float32(Float32(Float32(0.0625) / Float32(Float32(0.25) - Float32((Float32(abs(x) / s) ^ Float32(2.0)) * Float32(-0.0625)))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = t_0 - single(-1.0);
	tmp = single(0.0);
	if ((t_0 / ((t_1 * s) * t_1)) <= single(9.999999747378752e-5))
		tmp = t_0 / (single(4.0) * s);
	else
		tmp = (single(0.0625) / (single(0.25) - (((abs(x) / s) ^ single(2.0)) * single(-0.0625)))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 9.99999975e-5

   1. Initial program 99.4%

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

   3. Taylor expanded in s around inf

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

      1. lower-*.f32 99.1

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

   5. Applied rewrites 99.1%

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

   if 9.99999975e-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 99.5%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 91.3%

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

   9. Applied rewrites 93.5%

      \[\leadsto \frac{\frac{0.0625 - 0.00390625 \cdot {\left(\frac{\left|x\right|}{s}\right)}^{4}}{0.25 - -0.0625 \cdot {\left(\frac{\left|x\right|}{s}\right)}^{2}}}{s} \]

   10. Taylor expanded in s around inf

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

   12. Applied rewrites 94.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.0625}{0.25 - {\left(\frac{\left|x\right|}{s}\right)}^{2} \cdot -0.0625}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. lower-*.f32 N/A

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 4: 96.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   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

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. distribute-frac-neg N/A

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

   11. exp-neg N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-exp.f32 N/A

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

   15. lower-/.f32 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-fabs.f32 96.6

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

7. Applied rewrites 96.6%

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

8. Final simplification 96.6%

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

Alternative 5: 96.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. lower--.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-fabs.f32 96.0

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

5. Applied rewrites 96.0%

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

6. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 96.1

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

8. Applied rewrites 96.1%

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

9. Taylor expanded in s around 0

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

11. Applied rewrites 96.1%

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

12. Final simplification 96.1%

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

Alternative 6: 94.6% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ (/ 1.0 (* 4.0 s)) (exp (/ (fabs x) s))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 94.4

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

5. Applied rewrites 94.4%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. distribute-frac-neg N/A

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

   8. lift-/.f32 N/A

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

   9. rec-exp N/A

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

   10. lift-exp.f32 N/A

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

   11. un-div-inv N/A

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

7. Applied rewrites 94.4%

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

Alternative 7: 94.6% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ (exp (/ -1.0 (/ s (fabs x)))) (* 4.0 s)))

C

float code(float x, float s) {
	return expf((-1.0f / (s / fabsf(x)))) / (4.0f * s);
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 94.4

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

5. Applied rewrites 94.4%

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

   1. lift-/.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. neg-mul-1 N/A

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

   4. associate-/l* N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 94.4

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

7. Applied rewrites 94.4%

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

Alternative 8: 94.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 94.4

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

5. Applied rewrites 94.4%

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

Alternative 9: 74.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.600000031086329 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{s} \cdot x\right) \cdot -0.0625}{s} + 0.25}{s}\\ \mathbf{elif}\;\left|x\right| \leq 2.9999999880125916 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(x \cdot x, -0.0625, \left(s \cdot s\right) \cdot 0.25\right)}{s}}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 \cdot s}}{1 - \frac{-0.5 \cdot \frac{x \cdot x}{s} - \left|x\right|}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 2.600000031086329e-23)
   (/ (+ (/ (* (* (/ x s) x) -0.0625) s) 0.25) s)
   (if (<= (fabs x) 2.9999999880125916e-12)
     (/ (/ (/ (fma (* x x) -0.0625 (* (* s s) 0.25)) s) s) s)
     (/ (/ 1.0 (* 4.0 s)) (- 1.0 (/ (- (* -0.5 (/ (* x x) s)) (fabs x)) s))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 2.600000031086329e-23f) {
		tmp = (((((x / s) * x) * -0.0625f) / s) + 0.25f) / s;
	} else if (fabsf(x) <= 2.9999999880125916e-12f) {
		tmp = ((fmaf((x * x), -0.0625f, ((s * s) * 0.25f)) / s) / s) / s;
	} else {
		tmp = (1.0f / (4.0f * s)) / (1.0f - (((-0.5f * ((x * x) / s)) - fabsf(x)) / s));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(2.600000031086329e-23))
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(x / s) * x) * Float32(-0.0625)) / s) + Float32(0.25)) / s);
	elseif (abs(x) <= Float32(2.9999999880125916e-12))
		tmp = Float32(Float32(Float32(fma(Float32(x * x), Float32(-0.0625), Float32(Float32(s * s) * Float32(0.25))) / s) / s) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(4.0) * s)) / Float32(Float32(1.0) - Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(x * x) / s)) - abs(x)) / s)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (fabs.f32 x) < 2.60000003e-23

   1. Initial program 97.8%

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

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 86.9%

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

   9. Applied rewrites 90.0%

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

   if 2.60000003e-23 < (fabs.f32 x) < 2.99999999e-12

   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

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 53.8%

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

   9. Applied rewrites 53.7%

      \[\leadsto \frac{\frac{0.0625 - 0.00390625 \cdot {\left(\frac{\left|x\right|}{s}\right)}^{4}}{0.25 - -0.0625 \cdot {\left(\frac{\left|x\right|}{s}\right)}^{2}}}{s} \]

   10. Taylor expanded in s around 0

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

   12. Applied rewrites 92.3%

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

   if 2.99999999e-12 < (fabs.f32 x)

   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

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

      1. lower-*.f32 97.1

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

   5. Applied rewrites 97.1%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-frac-neg N/A

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

      8. lift-/.f32 N/A

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

      9. rec-exp N/A

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

      10. lift-exp.f32 N/A

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

      11. un-div-inv N/A

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. unpow2 N/A

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

      13. sqr-abs N/A

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

      14. lower-*.f32 N/A

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

      15. lower-fabs.f32 72.3

          \[\leadsto \frac{\frac{1}{4 \cdot s}}{1 - \frac{\frac{x \cdot x}{s} \cdot -0.5 - \color{blue}{\left|x\right|}}{s}} \]

   10. Applied rewrites 72.3%

       \[\leadsto \frac{\frac{1}{4 \cdot s}}{\color{blue}{1 - \frac{\frac{x \cdot x}{s} \cdot -0.5 - \left|x\right|}{s}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.600000031086329 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{s} \cdot x\right) \cdot -0.0625}{s} + 0.25}{s}\\ \mathbf{elif}\;\left|x\right| \leq 2.9999999880125916 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(x \cdot x, -0.0625, \left(s \cdot s\right) \cdot 0.25\right)}{s}}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 \cdot s}}{1 - \frac{-0.5 \cdot \frac{x \cdot x}{s} - \left|x\right|}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \mathbf{if}\;\left|x\right| \leq 2.600000031086329 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{s} \cdot x\right) \cdot -0.0625}{s} + 0.25}{s}\\ \mathbf{elif}\;\left|x\right| \leq 230:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(x \cdot x, -0.0625, \left(s \cdot s\right) \cdot 0.25\right)}{s}}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(2 - t\_0\right) \cdot s\right) \cdot \left(1 + \left(1 - t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)))
   (if (<= (fabs x) 2.600000031086329e-23)
     (/ (+ (/ (* (* (/ x s) x) -0.0625) s) 0.25) s)
     (if (<= (fabs x) 230.0)
       (/ (/ (/ (fma (* x x) -0.0625 (* (* s s) 0.25)) s) s) s)
       (/ 1.0 (* (* (- 2.0 t_0) s) (+ 1.0 (- 1.0 t_0))))))))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	float tmp;
	if (fabsf(x) <= 2.600000031086329e-23f) {
		tmp = (((((x / s) * x) * -0.0625f) / s) + 0.25f) / s;
	} else if (fabsf(x) <= 230.0f) {
		tmp = ((fmaf((x * x), -0.0625f, ((s * s) * 0.25f)) / s) / s) / s;
	} else {
		tmp = 1.0f / (((2.0f - t_0) * s) * (1.0f + (1.0f - t_0)));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(2.600000031086329e-23))
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(x / s) * x) * Float32(-0.0625)) / s) + Float32(0.25)) / s);
	elseif (abs(x) <= Float32(230.0))
		tmp = Float32(Float32(Float32(fma(Float32(x * x), Float32(-0.0625), Float32(Float32(s * s) * Float32(0.25))) / s) / s) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(2.0) - t_0) * s) * Float32(Float32(1.0) + Float32(Float32(1.0) - t_0))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (fabs.f32 x) < 2.60000003e-23

   1. Initial program 97.8%

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

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 86.9%

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

   9. Applied rewrites 90.0%

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

   if 2.60000003e-23 < (fabs.f32 x) < 230

   1. Initial program 99.7%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 30.6%

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

   9. Applied rewrites 30.1%

      \[\leadsto \frac{\frac{0.0625 - 0.00390625 \cdot {\left(\frac{\left|x\right|}{s}\right)}^{4}}{0.25 - -0.0625 \cdot {\left(\frac{\left|x\right|}{s}\right)}^{2}}}{s} \]

   10. Taylor expanded in s around 0

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

   12. Applied rewrites 64.5%

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

   if 230 < (fabs.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-fabs.f32 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-fabs.f32 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in s around inf

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

   11. Applied rewrites 80.5%

       \[\leadsto \frac{\color{blue}{1}}{\left(\left(2 - \frac{\left|x\right|}{s}\right) \cdot s\right) \cdot \left(1 + \left(1 - \frac{\left|x\right|}{s}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.600000031086329 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{s} \cdot x\right) \cdot -0.0625}{s} + 0.25}{s}\\ \mathbf{elif}\;\left|x\right| \leq 230:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(x \cdot x, -0.0625, \left(s \cdot s\right) \cdot 0.25\right)}{s}}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(2 - \frac{\left|x\right|}{s}\right) \cdot s\right) \cdot \left(1 + \left(1 - \frac{\left|x\right|}{s}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.600000031086329 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{s} \cdot x\right) \cdot -0.0625}{s} + 0.25}{s}\\ \mathbf{elif}\;\left|x\right| \leq 1500000000:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(x \cdot x, -0.0625, \left(s \cdot s\right) \cdot 0.25\right)}{s}}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 \cdot s}}{1 + \frac{\left|x\right|}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 2.600000031086329e-23)
   (/ (+ (/ (* (* (/ x s) x) -0.0625) s) 0.25) s)
   (if (<= (fabs x) 1500000000.0)
     (/ (/ (/ (fma (* x x) -0.0625 (* (* s s) 0.25)) s) s) s)
     (/ (/ 1.0 (* 4.0 s)) (+ 1.0 (/ (fabs x) s))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 2.600000031086329e-23f) {
		tmp = (((((x / s) * x) * -0.0625f) / s) + 0.25f) / s;
	} else if (fabsf(x) <= 1500000000.0f) {
		tmp = ((fmaf((x * x), -0.0625f, ((s * s) * 0.25f)) / s) / s) / s;
	} else {
		tmp = (1.0f / (4.0f * s)) / (1.0f + (fabsf(x) / s));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(2.600000031086329e-23))
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(x / s) * x) * Float32(-0.0625)) / s) + Float32(0.25)) / s);
	elseif (abs(x) <= Float32(1500000000.0))
		tmp = Float32(Float32(Float32(fma(Float32(x * x), Float32(-0.0625), Float32(Float32(s * s) * Float32(0.25))) / s) / s) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(4.0) * s)) / Float32(Float32(1.0) + Float32(abs(x) / s)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (fabs.f32 x) < 2.60000003e-23

   1. Initial program 97.8%

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

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 86.9%

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

   9. Applied rewrites 90.0%

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

   if 2.60000003e-23 < (fabs.f32 x) < 1.5e9

   1. Initial program 99.7%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 26.2%

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

   9. Applied rewrites 25.5%

      \[\leadsto \frac{\frac{0.0625 - 0.00390625 \cdot {\left(\frac{\left|x\right|}{s}\right)}^{4}}{0.25 - -0.0625 \cdot {\left(\frac{\left|x\right|}{s}\right)}^{2}}}{s} \]

   10. Taylor expanded in s around 0

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

   12. Applied rewrites 59.2%

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

   if 1.5e9 < (fabs.f32 x)

   1. Initial program 100.0%

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

   3. 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 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-frac-neg N/A

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

      8. lift-/.f32 N/A

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

      9. rec-exp N/A

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

      10. lift-exp.f32 N/A

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

      11. un-div-inv N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-fabs.f32 70.0

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

   10. Applied rewrites 70.0%

       \[\leadsto \frac{\frac{1}{4 \cdot s}}{\color{blue}{\frac{\left|x\right|}{s} + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.600000031086329 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{s} \cdot x\right) \cdot -0.0625}{s} + 0.25}{s}\\ \mathbf{elif}\;\left|x\right| \leq 1500000000:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(x \cdot x, -0.0625, \left(s \cdot s\right) \cdot 0.25\right)}{s}}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 \cdot s}}{1 + \frac{\left|x\right|}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{4 \cdot s}}{1 + \frac{\left|x\right|}{s}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 1.0 (* 4.0 s)) (+ 1.0 (/ (fabs x) s))))

C

float code(float x, float s) {
	return (1.0f / (4.0f * s)) / (1.0f + (fabsf(x) / s));
}

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / Float32(Float32(4.0) * s)) / Float32(Float32(1.0) + Float32(abs(x) / s)))
end

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 94.4

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

5. Applied rewrites 94.4%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. distribute-frac-neg N/A

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

   8. lift-/.f32 N/A

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

   9. rec-exp N/A

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

   10. lift-exp.f32 N/A

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

   11. un-div-inv N/A

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

7. Applied rewrites 94.4%

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

8. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-fabs.f32 54.7

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

10. Applied rewrites 54.7%

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

11. Final simplification 54.7%

    \[\leadsto \frac{\frac{1}{4 \cdot s}}{1 + \frac{\left|x\right|}{s}} \]
12. Add Preprocessing

Alternative 13: 27.4% accurate, 31.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 31.7

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

5. Applied rewrites 31.7%

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

Reproduce

herbie shell --seed 2024285 
(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))))))