Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 12.0s

Alternatives: 14

Speedup: 1.0×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-exp.f32 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 0:\\ \;\;\;\;\frac{1}{\left(4 + \frac{x}{s \cdot s} \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s} \cdot x}{s} + 4\right) \cdot s}\\ \end{array} \end{array} \]

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)
     (/ 1.0 (* (+ 4.0 (* (/ x (* s s)) x)) s))
     (/ 1.0 (* (+ (/ (* (/ x s) x) s) 4.0) 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 = 1.0f / ((4.0f + ((x / (s * s)) * x)) * s);
	} else {
		tmp = 1.0f / (((((x / s) * x) / s) + 4.0f) * 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 = 1.0e0 / ((4.0e0 + ((x / (s * s)) * x)) * s)
    else
        tmp = 1.0e0 / (((((x / s) * x) / s) + 4.0e0) * 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(1.0) / Float32(Float32(Float32(4.0) + Float32(Float32(x / Float32(s * s)) * x)) * s));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(x / s) * x) / s) + Float32(4.0)) * 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 = single(1.0) / ((single(4.0) + ((x / (s * s)) * x)) * s);
	else
		tmp = single(1.0) / (((((x / s) * x) / s) + single(4.0)) * 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 99.9%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

   7. Applied rewrites 68.6%

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

   9. Applied rewrites 82.7%

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

   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 98.5%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

   7. Applied rewrites 82.9%

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

   9. Applied rewrites 85.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 0:\\ \;\;\;\;\frac{1}{\left(4 + \frac{x}{s \cdot s} \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s} \cdot x}{s} + 4\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 1.9999999556392617 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{x}{s \cdot s} \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \end{array} \]

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)) 1.9999999556392617e+22)
     (/ 1.0 (* (+ 4.0 (* (/ x (* s s)) x)) 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)) <= 1.9999999556392617e+22f) {
		tmp = 1.0f / ((4.0f + ((x / (s * s)) * x)) * 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)) <= 1.9999999556392617e+22) then
        tmp = 1.0e0 / ((4.0e0 + ((x / (s * s)) * x)) * 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(1.9999999556392617e+22))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) + Float32(Float32(x / Float32(s * s)) * x)) * 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(1.9999999556392617e+22))
		tmp = single(1.0) / ((single(4.0) + ((x / (s * s)) * x)) * 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))))) < 1.99999996e22

   1. Initial program 99.6%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

   7. Applied rewrites 73.1%

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

   9. Applied rewrites 83.7%

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

   if 1.99999996e22 < (/.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 97.8%

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

   3. 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 67.3%

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

   7. Applied rewrites 75.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 1.9999999556392617 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{x}{s \cdot s} \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 4.0000000801635094 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{x}{s \cdot s} \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (- t_0 -1.0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 4.0000000801635094e+20)
     (/ 1.0 (* (+ 4.0 (* (/ x (* s s)) 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)) <= 4.0000000801635094e+20f) {
		tmp = 1.0f / ((4.0f + ((x / (s * s)) * x)) * s);
	} else {
		tmp = 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)) <= 4.0000000801635094e+20) then
        tmp = 1.0e0 / ((4.0e0 + ((x / (s * s)) * x)) * s)
    else
        tmp = 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(4.0000000801635094e+20))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) + Float32(Float32(x / Float32(s * s)) * x)) * s));
	else
		tmp = Float32(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(4.0000000801635094e+20))
		tmp = single(1.0) / ((single(4.0) + ((x / (s * s)) * x)) * s);
	else
		tmp = 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))))) < 4.00000008e20

   1. Initial program 99.6%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

   7. Applied rewrites 73.4%

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

   9. Applied rewrites 84.2%

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

   if 4.00000008e20 < (/.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 97.6%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 65.7

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

   5. Applied rewrites 65.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 4.0000000801635094 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{x}{s \cdot s} \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\left(\frac{x}{s \cdot s} \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (- t_0 -1.0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 4.999999873689376e-5)
     (/ 1.0 (* (* (/ x (* s s)) 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)) <= 4.999999873689376e-5f) {
		tmp = 1.0f / (((x / (s * s)) * x) * s);
	} else {
		tmp = 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)) <= 4.999999873689376e-5) then
        tmp = 1.0e0 / (((x / (s * s)) * x) * s)
    else
        tmp = 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(4.999999873689376e-5))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x / Float32(s * s)) * x) * s));
	else
		tmp = Float32(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(4.999999873689376e-5))
		tmp = single(1.0) / (((x / (s * s)) * x) * s);
	else
		tmp = 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))))) < 4.99999987e-5

   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

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 82.3%

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

   if 4.99999987e-5 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 98.5%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 82.7

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

   5. Applied rewrites 82.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\left(\frac{x}{s \cdot s} \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. distribute-frac-neg N/A

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

   11. exp-neg N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-exp.f32 N/A

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

   15. lower-/.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 7: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-fabs.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. mul-1-neg N/A

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

   12. distribute-neg-frac2 N/A

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

   13. mul-1-neg N/A

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

   14. lower-/.f32 N/A

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

   15. lower-fabs.f32 N/A

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

   16. mul-1-neg N/A

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

   17. lower-neg.f32 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 8: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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)) ** (-2.0e0)) / s) * t_0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 9: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-exp.f32 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. div-inv N/A

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

   4. associate-/l* N/A

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

   5. lift-pow.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. pow-exp N/A

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

   8. neg-mul-1 N/A

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

   9. exp-neg N/A

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

   10. lift-exp.f32 N/A

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

   11. times-frac N/A

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

   12. div-inv N/A

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

8. Applied rewrites 99.4%

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

9. Final simplification 99.4%

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

Alternative 10: 97.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((-abs(x) / s))
    code = t_0 / (((s / (1.0e0 - (((((x * x) * (-0.5e0)) / s) - abs(x)) / s))) + s) * (t_0 - (-1.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. distribute-frac-neg N/A

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

   11. exp-neg N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-exp.f32 N/A

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

   15. lower-/.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 95.8%

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

8. Final simplification 95.8%

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

Alternative 11: 96.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. distribute-frac-neg N/A

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

   11. exp-neg N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-exp.f32 N/A

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

   15. lower-/.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-fabs.f32 95.4

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

7. Applied rewrites 95.4%

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

8. Final simplification 95.4%

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

Alternative 12: 96.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 94.4

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

7. Applied rewrites 94.4%

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

Alternative 13: 94.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 92.8%

   \[\leadsto \frac{\color{blue}{0.25}}{s} \cdot e^{\frac{-\left|x\right|}{s}} \]
8. Add Preprocessing

Alternative 14: 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 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 27.1

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

5. Applied rewrites 27.1%

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

Reproduce

herbie shell --seed 2024270 
(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))))))