Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 9.5s

Alternatives: 12

Speedup: N/A×

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

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-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f32 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. pow2 N/A

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

   5. lift-+.f32 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift-pow.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 99.7

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

8. Applied rewrites 99.7%

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

Alternative 2: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0.10000000149011612:\\ \;\;\;\;\frac{e^{\frac{-s}{s \cdot \frac{s}{\left|x\right|}}}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{\frac{x}{s} \cdot x}{s}\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 0.10000000149011612)
     (/ (exp (/ (- s) (* s (/ s (fabs x))))) (* 4.0 s))
     (/ 1.0 (* (+ 4.0 (/ (* (/ x s) x) s)) s)))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 0.10000000149011612f) {
		tmp = expf((-s / (s * (s / fabsf(x))))) / (4.0f * s);
	} else {
		tmp = 1.0f / ((4.0f + (((x / s) * x) / s)) * 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 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 0.10000000149011612e0) then
        tmp = exp((-s / (s * (s / abs(x))))) / (4.0e0 * s)
    else
        tmp = 1.0e0 / ((4.0e0 + (((x / s) * x) / s)) * s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / 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(0.10000000149011612))
		tmp = Float32(exp(Float32(Float32(-s) / Float32(s * Float32(s / abs(x))))) / Float32(Float32(4.0) * s));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) + Float32(Float32(Float32(x / s) * x) / s)) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(0.10000000149011612))
		tmp = exp((-s / (s * (s / abs(x))))) / (single(4.0) * s);
	else
		tmp = single(1.0) / ((single(4.0) + (((x / s) * x) / s)) * s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.100000001

   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 99.6

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

   5. Applied rewrites 99.6%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s}} \]
   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-sub0 N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{0 - \left|x\right|}}{s}}}{4 \cdot s} \]

      4. div-sub N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{0}{s} - \frac{\left|x\right|}{s}}}}{4 \cdot s} \]

      5. clear-num N/A

         \[\leadsto \frac{e^{\frac{0}{s} - \color{blue}{\frac{1}{\frac{s}{\left|x\right|}}}}}{4 \cdot s} \]

      6. frac-sub N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{0 \cdot \frac{s}{\left|x\right|} - s \cdot 1}{s \cdot \frac{s}{\left|x\right|}}}}}{4 \cdot s} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{0 \cdot \frac{s}{\left|x\right|} - s \cdot 1}{s \cdot \frac{s}{\left|x\right|}}}}}{4 \cdot s} \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{e^{\frac{0 \cdot \frac{s}{\left|x\right|} - \color{blue}{s}}{s \cdot \frac{s}{\left|x\right|}}}}{4 \cdot s} \]

      9. lower--.f32 N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{0 \cdot \frac{s}{\left|x\right|} - s}}{s \cdot \frac{s}{\left|x\right|}}}}{4 \cdot s} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{e^{\frac{\color{blue}{0 \cdot \frac{s}{\left|x\right|}} - s}{s \cdot \frac{s}{\left|x\right|}}}}{4 \cdot s} \]

      11. lower-/.f32 N/A

          \[\leadsto \frac{e^{\frac{0 \cdot \color{blue}{\frac{s}{\left|x\right|}} - s}{s \cdot \frac{s}{\left|x\right|}}}}{4 \cdot s} \]

      12. lower-*.f32 N/A

          \[\leadsto \frac{e^{\frac{0 \cdot \frac{s}{\left|x\right|} - s}{\color{blue}{s \cdot \frac{s}{\left|x\right|}}}}}{4 \cdot s} \]

      13. lower-/.f32 99.6

          \[\leadsto \frac{e^{\frac{0 \cdot \frac{s}{\left|x\right|} - s}{s \cdot \color{blue}{\frac{s}{\left|x\right|}}}}}{4 \cdot s} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{e^{\color{blue}{\frac{0 \cdot \frac{s}{\left|x\right|} - s}{s \cdot \frac{s}{\left|x\right|}}}}}{4 \cdot s} \]

   if 0.100000001 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 98.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing
   3. 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. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

   7. Applied rewrites 91.3%

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

   9. Applied rewrites 92.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 0.10000000149011612:\\ \;\;\;\;\frac{e^{\frac{-s}{s \cdot \frac{s}{\left|x\right|}}}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{\frac{x}{s} \cdot x}{s}\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0.10000000149011612:\\ \;\;\;\;\frac{e^{\frac{-1}{\frac{s}{\left|x\right|}}}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{\frac{x}{s} \cdot x}{s}\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 0.10000000149011612)
     (/ (exp (/ -1.0 (/ s (fabs x)))) (* 4.0 s))
     (/ 1.0 (* (+ 4.0 (/ (* (/ x s) x) s)) s)))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 0.10000000149011612f) {
		tmp = expf((-1.0f / (s / fabsf(x)))) / (4.0f * s);
	} else {
		tmp = 1.0f / ((4.0f + (((x / s) * x) / s)) * 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 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 0.10000000149011612e0) then
        tmp = exp(((-1.0e0) / (s / abs(x)))) / (4.0e0 * s)
    else
        tmp = 1.0e0 / ((4.0e0 + (((x / s) * x) / s)) * s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / 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(0.10000000149011612))
		tmp = Float32(exp(Float32(Float32(-1.0) / Float32(s / abs(x)))) / Float32(Float32(4.0) * s));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) + Float32(Float32(Float32(x / s) * x) / s)) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(0.10000000149011612))
		tmp = exp((single(-1.0) / (s / abs(x)))) / (single(4.0) * s);
	else
		tmp = single(1.0) / ((single(4.0) + (((x / s) * x) / s)) * s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.100000001

   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 99.6

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

   5. Applied rewrites 99.6%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s}} \]
   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. distribute-frac-neg N/A

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

      4. clear-num N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f32 N/A

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

      8. lower-/.f32 99.6

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

   7. Applied rewrites 99.6%

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

   if 0.100000001 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 98.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing
   3. 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. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

   7. Applied rewrites 91.3%

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

   9. Applied rewrites 92.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 0.10000000149011612:\\ \;\;\;\;\frac{e^{\frac{-1}{\frac{s}{\left|x\right|}}}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{\frac{x}{s} \cdot x}{s}\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\left|x\right|}{s}\\ t_1 := e^{t\_0}\\ t_2 := 1 + t\_1\\ \mathbf{if}\;\frac{t\_1}{\left(s \cdot t\_2\right) \cdot t\_2} \leq 0.10000000149011612:\\ \;\;\;\;\frac{{\mathsf{E}\left(\right)}^{t\_0}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{\frac{x}{s} \cdot x}{s}\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- (fabs x)) s)) (t_1 (exp t_0)) (t_2 (+ 1.0 t_1)))
   (if (<= (/ t_1 (* (* s t_2) t_2)) 0.10000000149011612)
     (/ (pow (E) t_0) (* 4.0 s))
     (/ 1.0 (* (+ 4.0 (/ (* (/ x s) x) s)) s)))))

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))))) < 0.100000001

   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 99.6

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

   5. Applied rewrites 99.6%

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

      1. lift-exp.f32 N/A

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

      2. *-lft-identity N/A

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

      3. pow-exp N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-pow.f32 99.6

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

      6. lift-exp.f32 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f32 99.6

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

   7. Applied rewrites 99.6%

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

   if 0.100000001 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 98.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing
   3. 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. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

   7. Applied rewrites 91.3%

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

   9. Applied rewrites 92.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 0.10000000149011612:\\ \;\;\;\;\frac{{\mathsf{E}\left(\right)}^{\left(\frac{-\left|x\right|}{s}\right)}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{\frac{x}{s} \cdot x}{s}\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0.10000000149011612:\\ \;\;\;\;\frac{t\_0}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{\frac{x}{s} \cdot x}{s}\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 0.10000000149011612)
     (/ t_0 (* 4.0 s))
     (/ 1.0 (* (+ 4.0 (/ (* (/ x s) x) s)) s)))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 0.10000000149011612f) {
		tmp = t_0 / (4.0f * s);
	} else {
		tmp = 1.0f / ((4.0f + (((x / s) * x) / s)) * 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 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 0.10000000149011612e0) then
        tmp = t_0 / (4.0e0 * s)
    else
        tmp = 1.0e0 / ((4.0e0 + (((x / s) * x) / s)) * s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / 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(0.10000000149011612))
		tmp = Float32(t_0 / Float32(Float32(4.0) * s));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) + Float32(Float32(Float32(x / s) * x) / s)) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(0.10000000149011612))
		tmp = t_0 / (single(4.0) * s);
	else
		tmp = single(1.0) / ((single(4.0) + (((x / s) * x) / s)) * s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.100000001

   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 99.6

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

   5. Applied rewrites 99.6%

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

   if 0.100000001 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 98.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing
   3. 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. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

   7. Applied rewrites 91.3%

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

   9. Applied rewrites 92.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 0.10000000149011612:\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{\frac{x}{s} \cdot x}{s}\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

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

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lift-pow.f32 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lower-*.f32 99.5

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. inv-pow N/A

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

   7. lower-/.f32 N/A

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

8. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. log-E N/A

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

   6. lift-E.f32 N/A

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

   7. clear-num N/A

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

   8. lift-/.f32 N/A

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

   9. pow-to-exp N/A

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

   10. lower-pow.f32 99.7

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

10. Applied rewrites 99.7%

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

Alternative 7: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lift-pow.f32 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lower-*.f32 99.5

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. inv-pow N/A

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

   7. lower-/.f32 N/A

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

8. Applied rewrites 99.7%

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

Alternative 8: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t\_0}{\left(s \cdot \left(1 + t\_0\right)\right) \cdot \left(\frac{\left(\frac{x}{s} \cdot x\right) \cdot 0.5 - \left|x\right|}{s} + 2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))))
   (/
    t_0
    (* (* s (+ 1.0 t_0)) (+ (/ (- (* (* (/ x s) x) 0.5) (fabs x)) s) 2.0)))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / ((s * (1.0f + t_0)) * ((((((x / s) * x) * 0.5f) - fabsf(x)) / s) + 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 * (1.0e0 + t_0)) * ((((((x / s) * x) * 0.5e0) - abs(x)) / s) + 2.0e0))
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(t_0 / Float32(Float32(s * Float32(Float32(1.0) + t_0)) * Float32(Float32(Float32(Float32(Float32(Float32(x / s) * x) * Float32(0.5)) - abs(x)) / s) + Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = t_0 / ((s * (single(1.0) + t_0)) * ((((((x / s) * x) * single(0.5)) - abs(x)) / s) + single(2.0)));
end

Derivation

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. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

5. Applied rewrites 97.1%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 97.4%

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

Alternative 9: 95.5% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

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-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f32 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in s around inf

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

   1. lower-*.f32 94.9

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

9. Applied rewrites 94.9%

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

Alternative 10: 95.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / ((s * (1.0f + t_0)) * 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 * (1.0e0 + t_0)) * 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

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. 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 94.9%

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

Alternative 11: 81.8% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

7. Applied rewrites 77.7%

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

9. Applied rewrites 82.4%

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

10. Final simplification 82.4%

    \[\leadsto \frac{1}{\left(4 + x \cdot \frac{x}{s \cdot s}\right) \cdot s} \]
11. Add Preprocessing

Alternative 12: 26.9% 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 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 26.8

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

5. Applied rewrites 26.8%

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

Reproduce

herbie shell --seed 2024313 
(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))))))