Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 13.7s

Alternatives: 14

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

4. Applied rewrites 99.8%

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

   1. lift-*.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. exp-neg N/A

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

   5. lift-exp.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-exp.f32 N/A

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

   4. exp-neg N/A

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

   5. lift-/.f32 N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lower-*.f32 99.8

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

   11. lift-/.f32 N/A

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

   12. lift-neg.f32 N/A

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

   13. distribute-frac-neg N/A

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

   14. lift-/.f32 N/A

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

   15. lift-neg.f32 99.8

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 200000000753664:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(x, \frac{x}{s \cdot s}, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x \cdot -0.0625}{s}, \frac{x}{s}, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 200000000753664.0)
     (/ 1.0 (* s (fma x (/ x (* s s)) 4.0)))
     (/ (fma (/ (* x -0.0625) s) (/ x s) 0.25) 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 / (t_1 * (s * t_1))) <= 200000000753664.0f) {
		tmp = 1.0f / (s * fmaf(x, (x / (s * s)), 4.0f));
	} else {
		tmp = fmaf(((x * -0.0625f) / s), (x / s), 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(200000000753664.0))
		tmp = Float32(Float32(1.0) / Float32(s * fma(x, Float32(x / Float32(s * s)), Float32(4.0))));
	else
		tmp = Float32(fma(Float32(Float32(x * Float32(-0.0625)) / s), Float32(x / s), Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 2.00000001e14

   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

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(s \cdot \left(-1 \cdot \frac{-1 \cdot \left(-4 \cdot \left|x\right| + 4 \cdot \left|x\right|\right) + -1 \cdot \frac{-1 \cdot \left(-4 \cdot {\left(\left|x\right|\right)}^{2} + \left(4 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right) + \left(-1 \cdot \frac{\left|x\right| \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{s} + \left(\frac{-2}{3} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \left(\frac{2}{3} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \frac{{\left(\left|x\right|\right)}^{3}}{s}\right)\right)\right)\right)}{s}}{s} - 4\right)\right)}} \]

   7. Applied rewrites 38.9%

      \[\leadsto \frac{1}{\color{blue}{\left(-4 + \frac{\frac{\mathsf{fma}\left(\left|x\right|, \frac{-3 \cdot \left(x \cdot x\right)}{s}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{\left|x\right|}{s}, 0\right)\right) - x \cdot x}{s}}{s}\right) \cdot \left(-s\right)}} \]

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 86.9%

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

   if 2.00000001e14 < (/.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

   4. Applied rewrites 99.6%

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

      1. lift-*.f32 N/A

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

      2. lift-exp.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f32 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f32 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in s around inf

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + -1 \cdot \frac{\frac{-1}{64} \cdot {\left(-4 \cdot \left|x\right| + 4 \cdot \left|x\right|\right)}^{2} + \frac{1}{16} \cdot \left(-4 \cdot {\left(\left|x\right|\right)}^{2} + \left(4 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{-4 \cdot \left|x\right| + 4 \cdot \left|x\right|}{s}}}{s} \]

   9. Applied rewrites 35.7%

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

   11. Applied rewrites 88.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\left|x\right|}{-s}}}{\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(s \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \leq 200000000753664:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(x, \frac{x}{s \cdot s}, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x \cdot -0.0625}{s}, \frac{x}{s}, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 9.999999778196308 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(x, \frac{x}{s \cdot s}, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 9.999999778196308e+21)
     (/ 1.0 (* s (fma x (/ x (* s s)) 4.0)))
     (/ 0.25 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 / (t_1 * (s * t_1))) <= 9.999999778196308e+21f) {
		tmp = 1.0f / (s * fmaf(x, (x / (s * s)), 4.0f));
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(9.999999778196308e+21))
		tmp = Float32(Float32(1.0) / Float32(s * fma(x, Float32(x / Float32(s * s)), Float32(4.0))));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 9.99999978e21

   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

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(s \cdot \left(-1 \cdot \frac{-1 \cdot \left(-4 \cdot \left|x\right| + 4 \cdot \left|x\right|\right) + -1 \cdot \frac{-1 \cdot \left(-4 \cdot {\left(\left|x\right|\right)}^{2} + \left(4 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right) + \left(-1 \cdot \frac{\left|x\right| \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{s} + \left(\frac{-2}{3} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \left(\frac{2}{3} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \frac{{\left(\left|x\right|\right)}^{3}}{s}\right)\right)\right)\right)}{s}}{s} - 4\right)\right)}} \]

   7. Applied rewrites 42.2%

      \[\leadsto \frac{1}{\color{blue}{\left(-4 + \frac{\frac{\mathsf{fma}\left(\left|x\right|, \frac{-3 \cdot \left(x \cdot x\right)}{s}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{\left|x\right|}{s}, 0\right)\right) - x \cdot x}{s}}{s}\right) \cdot \left(-s\right)}} \]

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 87.5%

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

   if 9.99999978e21 < (/.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 98.6%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 72.9

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

   5. Applied rewrites 72.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\left|x\right|}{-s}}}{\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(s \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \leq 9.999999778196308 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(x, \frac{x}{s \cdot s}, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{-s}\\ \frac{e^{\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{t\_0}\right), t\_0\right)}}{s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) (- s))))
   (/ (exp (fma -2.0 (log1p (exp t_0)) t_0)) s)))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / -s;
	return expf(fmaf(-2.0f, log1pf(expf(t_0)), t_0)) / s;
}

Julia

function code(x, s)
	t_0 = Float32(abs(x) / Float32(-s))
	return Float32(exp(fma(Float32(-2.0), log1p(exp(t_0)), t_0)) / 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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

   \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right), \frac{\left|x\right|}{-s}\right)}}{s} \]
8. Add Preprocessing

Alternative 5: 97.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ \frac{t\_0}{\left(s \cdot \left(\left(--2\right) - \frac{\mathsf{fma}\left(x, x \cdot \frac{\mathsf{fma}\left(\frac{\left|x\right|}{s}, 0.16666666666666666, -0.5\right)}{s}, \left|x\right|\right)}{s}\right)\right) \cdot \left(1 + t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))))
   (/
    t_0
    (*
     (*
      s
      (-
       (- -2.0)
       (/
        (fma
         x
         (* x (/ (fma (/ (fabs x) s) 0.16666666666666666 -0.5) s))
         (fabs x))
        s)))
     (+ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	return t_0 / ((s * (-(-2.0f) - (fmaf(x, (x * (fmaf((fabsf(x) / s), 0.16666666666666666f, -0.5f) / s)), fabsf(x)) / s))) * (1.0f + t_0));
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	return Float32(t_0 / Float32(Float32(s * Float32(Float32(-Float32(-2.0)) - Float32(fma(x, Float32(x * Float32(fma(Float32(abs(x) / s), Float32(0.16666666666666666), Float32(-0.5)) / s)), abs(x)) / s))) * Float32(Float32(1.0) + t_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

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

   1. *-commutative N/A

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

   2. lower-*.f32 94.6

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

5. Applied rewrites 94.6%

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

6. Taylor expanded in s around -inf

   \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(-1 \cdot \left(s \cdot \left(-1 \cdot \frac{-1 \cdot \left|x\right| + -1 \cdot \frac{\frac{-1}{2} \cdot {\left(\left|x\right|\right)}^{2} + \frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s}}{s}}{s} - 2\right)\right)\right)} \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

8. Applied rewrites 96.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(\left(\frac{\mathsf{fma}\left(x, x \cdot \frac{\mathsf{fma}\left(\frac{\left|x\right|}{s}, 0.16666666666666666, -0.5\right)}{s}, \left|x\right|\right)}{s} + -2\right) \cdot \left(-s\right)\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

9. Final simplification 96.6%

   \[\leadsto \frac{e^{\frac{\left|x\right|}{-s}}}{\left(s \cdot \left(\left(--2\right) - \frac{\mathsf{fma}\left(x, x \cdot \frac{\mathsf{fma}\left(\frac{\left|x\right|}{s}, 0.16666666666666666, -0.5\right)}{s}, \left|x\right|\right)}{s}\right)\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)} \]
10. Add Preprocessing

Alternative 6: 96.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	return t_0 / ((1.0f + t_0) * (s * (2.0f - (fmaf(((x * x) / s), -0.5f, fabsf(x)) / s))));
}

Julia

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

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. associate-*r/ N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. div-sub N/A

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

   11. unsub-neg N/A

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

   12. mul-1-neg N/A

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

   13. +-commutative N/A

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

5. Applied rewrites 96.0%

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

6. Final simplification 96.0%

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

Alternative 7: 96.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 95.4

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

5. Applied rewrites 95.4%

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

6. Final simplification 95.4%

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

Alternative 8: 95.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 94.6

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

5. Applied rewrites 94.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-exp.f32 N/A

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

   6. rec-exp N/A

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

   7. lift-/.f32 N/A

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

   8. distribute-frac-neg2 N/A

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

   9. lift-neg.f32 N/A

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

   10. frac-2neg N/A

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

   11. lift-/.f32 N/A

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

7. Applied rewrites 94.6%

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

8. Final simplification 94.6%

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

Alternative 9: 95.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	t_0 = Float32(abs(x) / Float32(-s))
	return Float32((Float32(exp(1)) ^ t_0) / Float32(Float32(Float32(1.0) + exp(t_0)) * Float32(s * Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	t_0 = abs(x) / -s;
	tmp = (single(2.71828182845904523536) ^ t_0) / ((single(1.0) + exp(t_0)) * (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

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

   1. *-commutative N/A

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

   2. lower-*.f32 94.6

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

5. Applied rewrites 94.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 94.6

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

7. Applied rewrites 94.6%

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

   1. lift-exp.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 94.6

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

   11. lift-/.f32 N/A

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

   12. lift-neg.f32 N/A

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

   13. distribute-frac-neg N/A

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

   14. lift-/.f32 N/A

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

   15. lift-neg.f32 94.6

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

9. Applied rewrites 94.6%

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

10. Final simplification 94.6%

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

Alternative 10: 95.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 94.6

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

5. Applied rewrites 94.6%

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

6. Final simplification 94.6%

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

Alternative 11: 94.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 94.6

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

5. Applied rewrites 94.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 94.6

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

7. Applied rewrites 94.6%

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

   1. lift-*.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. distribute-rgt-neg-out N/A

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

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

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

   5. lift-/.f32 N/A

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

   6. distribute-frac-neg2 N/A

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

   7. lift-neg.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. neg-mul-1 N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 94.6

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

9. Applied rewrites 94.6%

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

10. Taylor expanded in s around inf

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

    1. *-commutative N/A

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

    2. lower-*.f32 94.2

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

12. Applied rewrites 94.2%

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

13. Final simplification 94.2%

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

Alternative 12: 94.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{{e}^{\left(\frac{\left|x\right|}{-s}\right)} \cdot 0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (* (pow E (/ (fabs x) (- s))) 0.25) s))

C

float code(float x, float s) {
	return (powf(((float) M_E), (fabsf(x) / -s)) * 0.25f) / s;
}

Julia

function code(x, s)
	return Float32(Float32((Float32(exp(1)) ^ Float32(abs(x) / Float32(-s))) * Float32(0.25)) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = ((single(2.71828182845904523536) ^ (abs(x) / -s)) * 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

4. Applied rewrites 99.8%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 94.2%

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

   1. lift-exp.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-neg.f32 N/A

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

   6. *-lft-identity N/A

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

   7. associate-/l* N/A

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

   8. lift-/.f32 N/A

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

   9. exp-prod N/A

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

   10. lower-pow.f32 N/A

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

   11. exp-1-e N/A

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

   12. lower-E.f32 94.2

       \[\leadsto \frac{0.25 \cdot {\color{blue}{e}}^{\left(\frac{-\left|x\right|}{s}\right)}}{s} \]

   13. lift-/.f32 N/A

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

   14. lift-neg.f32 N/A

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

   15. distribute-frac-neg N/A

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

   16. lift-/.f32 N/A

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

   17. lift-neg.f32 94.2

       \[\leadsto \frac{0.25 \cdot {e}^{\color{blue}{\left(-\frac{\left|x\right|}{s}\right)}}}{s} \]

9. Applied rewrites 94.2%

   \[\leadsto \frac{0.25 \cdot \color{blue}{{e}^{\left(-\frac{\left|x\right|}{s}\right)}}}{s} \]

10. Final simplification 94.2%

    \[\leadsto \frac{{e}^{\left(\frac{\left|x\right|}{-s}\right)} \cdot 0.25}{s} \]
11. Add Preprocessing

Alternative 13: 94.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 94.2

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

5. Applied rewrites 94.2%

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

6. Final simplification 94.2%

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

Alternative 14: 27.7% 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 30.4

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

5. Applied rewrites 30.4%

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

Reproduce

herbie shell --seed 2024233 
(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))))))