Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 11.5s

Alternatives: 11

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (exp((-2.0e0)) ** (0.5e0 * (abs(x) / s))) / (((exp((-abs(x) / s)) + 1.0e0) ** 2.0e0) * s)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-exp.f32 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-pow.f32 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f32 N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. prod-exp N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. div-inv N/A

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

   11. lower-*.f32 N/A

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

   12. metadata-eval 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 27.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(x \cdot x, -0.0625, \left(s \cdot s\right) \cdot 0.25\right)}{s}}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 5.000000018137469e-16)
     (/ (/ (/ (fma (* x x) -0.0625 (* (* s s) 0.25)) s) s) s)
     (/ (+ (/ (* (/ x s) (* -0.0625 x)) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / ((t_1 * s) * t_1)) <= 5.000000018137469e-16f) {
		tmp = ((fmaf((x * x), -0.0625f, ((s * s) * 0.25f)) / s) / s) / s;
	} else {
		tmp = ((((x / s) * (-0.0625f * x)) / s) + 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(5.000000018137469e-16))
		tmp = Float32(Float32(Float32(fma(Float32(x * x), Float32(-0.0625), Float32(Float32(s * s) * Float32(0.25))) / s) / s) / s);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(x / s) * Float32(Float32(-0.0625) * 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))))) < 5.00000002e-16

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   5. Applied rewrites 3.3%

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in s around 0

      \[\leadsto \frac{\frac{\frac{-1}{16} \cdot {x}^{2} + \frac{1}{4} \cdot {s}^{2}}{{s}^{2}}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.6%

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

   if 5.00000002e-16 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 98.7%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 85.7%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(x \cdot x, -0.0625, \left(s \cdot s\right) \cdot 0.25\right)}{s}}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 29.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 0.0)
     (/ (* (/ (/ (* x x) s) s) -0.0625) s)
     (/ (+ (/ (* (/ x s) (* -0.0625 x)) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / ((t_1 * s) * t_1)) <= 0.0f) {
		tmp = ((((x * x) / s) / s) * -0.0625f) / s;
	} else {
		tmp = ((((x / s) * (-0.0625f * x)) / s) + 0.25f) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 + 1.0e0
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.0e0) then
        tmp = ((((x * x) / s) / s) * (-0.0625e0)) / s
    else
        tmp = ((((x / s) * ((-0.0625e0) * x)) / s) + 0.25e0) / s
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   5. Applied rewrites 3.3%

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 8.6%

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

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

   1. Initial program 98.6%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 84.6%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \leq 0:\\ \;\;\;\;\frac{\frac{\frac{x \cdot x}{s}}{s} \cdot -0.0625}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f32 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 96.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp((-abs(x) / s)) / (((((((((x * x) / s) * 0.5e0) - abs(x)) / s) + 1.0e0) + 1.0e0) ** 2.0e0) * s)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f32 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

7. Applied rewrites 95.4%

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

8. Final simplification 95.4%

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

Alternative 6: 95.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in 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 93.8%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lift-neg.f32 N/A

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

   5. remove-double-neg N/A

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

   6. lower-*.f32 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. remove-double-neg N/A

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

   10. lower-/.f32 93.8

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

7. Applied rewrites 93.8%

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

8. Final simplification 93.8%

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

Alternative 7: 95.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in 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 93.8%

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

6. Final simplification 93.8%

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

Alternative 8: 94.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{s}{\left|x\right|}\\ \frac{e^{\frac{t\_0 \cdot 0 - s}{t\_0 \cdot s}}}{\frac{1}{\frac{\frac{1}{2}}{2 \cdot s}}} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ s (fabs x))))
   (/
    (exp (/ (- (* t_0 0.0) s) (* t_0 s)))
    (/ 1.0 (/ (/ 1.0 2.0) (* 2.0 s))))))

C

float code(float x, float s) {
	float t_0 = s / fabsf(x);
	return expf((((t_0 * 0.0f) - s) / (t_0 * s))) / (1.0f / ((1.0f / 2.0f) / (2.0f * s)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = s / abs(x)
    code = exp((((t_0 * 0.0e0) - s) / (t_0 * s))) / (1.0e0 / ((1.0e0 / 2.0e0) / (2.0e0 * s)))
end function

Julia

function code(x, s)
	t_0 = Float32(s / abs(x))
	return Float32(exp(Float32(Float32(Float32(t_0 * Float32(0.0)) - s) / Float32(t_0 * s))) / Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(2.0)) / Float32(Float32(2.0) * s))))
end

MATLAB

function tmp = code(x, s)
	t_0 = s / abs(x);
	tmp = exp((((t_0 * single(0.0)) - s) / (t_0 * s))) / (single(1.0) / ((single(1.0) / single(2.0)) / (single(2.0) * s)));
end

Derivation

1. Initial program 99.6%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 93.5%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. lift-pow.f32 N/A

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

   5. sqr-pow N/A

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

   6. associate-/r* N/A

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

   7. clear-num N/A

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

   8. lower-/.f32 N/A

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

   9. div-inv N/A

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

9. Applied rewrites 93.5%

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

    1. lift-/.f32 N/A

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

    2. lift-neg.f32 N/A

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

    3. neg-sub0 N/A

       \[\leadsto \frac{e^{\frac{\color{blue}{0 - \left|x\right|}}{s}}}{\frac{1}{\frac{\frac{1}{2}}{s \cdot 2}}} \]

    4. div-sub N/A

       \[\leadsto \frac{e^{\color{blue}{\frac{0}{s} - \frac{\left|x\right|}{s}}}}{\frac{1}{\frac{\frac{1}{2}}{s \cdot 2}}} \]

    5. clear-num N/A

       \[\leadsto \frac{e^{\frac{0}{s} - \color{blue}{\frac{1}{\frac{s}{\left|x\right|}}}}}{\frac{1}{\frac{\frac{1}{2}}{s \cdot 2}}} \]

    6. frac-sub N/A

       \[\leadsto \frac{e^{\color{blue}{\frac{0 \cdot \frac{s}{\left|x\right|} - s \cdot 1}{s \cdot \frac{s}{\left|x\right|}}}}}{\frac{1}{\frac{\frac{1}{2}}{s \cdot 2}}} \]

    7. lower-/.f32 N/A

       \[\leadsto \frac{e^{\color{blue}{\frac{0 \cdot \frac{s}{\left|x\right|} - s \cdot 1}{s \cdot \frac{s}{\left|x\right|}}}}}{\frac{1}{\frac{\frac{1}{2}}{s \cdot 2}}} \]

    8. *-rgt-identity N/A

       \[\leadsto \frac{e^{\frac{0 \cdot \frac{s}{\left|x\right|} - \color{blue}{s}}{s \cdot \frac{s}{\left|x\right|}}}}{\frac{1}{\frac{\frac{1}{2}}{s \cdot 2}}} \]

    9. lower--.f32 N/A

       \[\leadsto \frac{e^{\frac{\color{blue}{0 \cdot \frac{s}{\left|x\right|} - s}}{s \cdot \frac{s}{\left|x\right|}}}}{\frac{1}{\frac{\frac{1}{2}}{s \cdot 2}}} \]

    10. lower-*.f32 N/A

        \[\leadsto \frac{e^{\frac{\color{blue}{0 \cdot \frac{s}{\left|x\right|}} - s}{s \cdot \frac{s}{\left|x\right|}}}}{\frac{1}{\frac{\frac{1}{2}}{s \cdot 2}}} \]

    11. lower-/.f32 N/A

        \[\leadsto \frac{e^{\frac{0 \cdot \color{blue}{\frac{s}{\left|x\right|}} - s}{s \cdot \frac{s}{\left|x\right|}}}}{\frac{1}{\frac{\frac{1}{2}}{s \cdot 2}}} \]

    12. lower-*.f32 N/A

        \[\leadsto \frac{e^{\frac{0 \cdot \frac{s}{\left|x\right|} - s}{\color{blue}{s \cdot \frac{s}{\left|x\right|}}}}}{\frac{1}{\frac{\frac{1}{2}}{s \cdot 2}}} \]

    13. lower-/.f32 93.5

        \[\leadsto \frac{e^{\frac{0 \cdot \frac{s}{\left|x\right|} - s}{s \cdot \color{blue}{\frac{s}{\left|x\right|}}}}}{\frac{1}{\frac{\frac{1}{2}}{s \cdot 2}}} \]

11. Applied rewrites 93.5%

    \[\leadsto \frac{e^{\color{blue}{\frac{0 \cdot \frac{s}{\left|x\right|} - s}{s \cdot \frac{s}{\left|x\right|}}}}}{\frac{1}{\frac{\frac{1}{2}}{s \cdot 2}}} \]

12. Final simplification 93.5%

    \[\leadsto \frac{e^{\frac{\frac{s}{\left|x\right|} \cdot 0 - s}{\frac{s}{\left|x\right|} \cdot s}}}{\frac{1}{\frac{\frac{1}{2}}{2 \cdot s}}} \]
13. Add Preprocessing

Alternative 9: 94.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 93.5%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. lift-pow.f32 N/A

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

   5. sqr-pow N/A

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

   6. associate-/r* N/A

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

   7. clear-num N/A

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

   8. lower-/.f32 N/A

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

   9. div-inv N/A

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

9. Applied rewrites 93.5%

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

10. Final simplification 93.5%

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

Alternative 10: 94.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 93.5

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

5. Applied rewrites 93.5%

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

Alternative 11: 27.0% 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 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 27.7

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

5. Applied rewrites 27.7%

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

Reproduce

herbie shell --seed 2024255 
(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))))))