Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 12.4s

Alternatives: 15

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

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

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 99.7%

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

4. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 0.5:\\ \;\;\;\;\frac{t\_0}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s}}{s}\\ \end{array} \end{array} \]

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)) 0.5)
     (/ t_0 (* 4.0 s))
     (/ (+ 0.25 (/ (* (/ x s) (* -0.0625 x)) s)) 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)) <= 0.5f) {
		tmp = t_0 / (4.0f * s);
	} else {
		tmp = (0.25f + (((x / s) * (-0.0625f * x)) / s)) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 - (-1.0e0)
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.5e0) then
        tmp = t_0 / (4.0e0 * s)
    else
        tmp = (0.25e0 + (((x / s) * ((-0.0625e0) * x)) / s)) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(0.5))
		tmp = Float32(t_0 / Float32(Float32(4.0) * s));
	else
		tmp = Float32(Float32(Float32(0.25) + Float32(Float32(Float32(x / s) * Float32(Float32(-0.0625) * x)) / s)) / 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(0.5))
		tmp = t_0 / (single(4.0) * s);
	else
		tmp = (single(0.25) + (((x / s) * (single(-0.0625) * x)) / s)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. lower-*.f32 99.6

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

   5. Applied rewrites 99.6%

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

   if 0.5 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.0%

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

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

      6. associate-/l/ N/A

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

      7. associate-/r* N/A

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

      8. 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 99.4%

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

   5. Taylor expanded in s around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. +-commutative N/A

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

      6. lower-+.f32 N/A

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

   7. Applied rewrites 89.6%

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

   9. Applied rewrites 90.6%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 0.5:\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 0.5:\\ \;\;\;\;\frac{1}{\left(\left(2 - \frac{\left|x\right|}{s}\right) \cdot s\right) \cdot \left(2 - \frac{\left|x\right| - 0.5 \cdot \frac{x \cdot x}{s}}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s}}{s}\\ \end{array} \end{array} \]

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)) 0.5)
     (/
      1.0
      (*
       (* (- 2.0 (/ (fabs x) s)) s)
       (- 2.0 (/ (- (fabs x) (* 0.5 (/ (* x x) s))) s))))
     (/ (+ 0.25 (/ (* (/ x s) (* -0.0625 x)) s)) 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)) <= 0.5f) {
		tmp = 1.0f / (((2.0f - (fabsf(x) / s)) * s) * (2.0f - ((fabsf(x) - (0.5f * ((x * x) / s))) / s)));
	} else {
		tmp = (0.25f + (((x / s) * (-0.0625f * x)) / s)) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 - (-1.0e0)
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.5e0) then
        tmp = 1.0e0 / (((2.0e0 - (abs(x) / s)) * s) * (2.0e0 - ((abs(x) - (0.5e0 * ((x * x) / s))) / s)))
    else
        tmp = (0.25e0 + (((x / s) * ((-0.0625e0) * x)) / s)) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(0.5))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(2.0) - Float32(abs(x) / s)) * s) * Float32(Float32(2.0) - Float32(Float32(abs(x) - Float32(Float32(0.5) * Float32(Float32(x * x) / s))) / s))));
	else
		tmp = Float32(Float32(Float32(0.25) + Float32(Float32(Float32(x / s) * Float32(Float32(-0.0625) * x)) / s)) / 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(0.5))
		tmp = single(1.0) / (((single(2.0) - (abs(x) / s)) * s) * (single(2.0) - ((abs(x) - (single(0.5) * ((x * x) / s))) / s)));
	else
		tmp = (single(0.25) + (((x / s) * (single(-0.0625) * x)) / s)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-fabs.f32 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in s around inf

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

   11. Applied rewrites 37.8%

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

   12. Taylor expanded in s around inf

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

       1. +-commutative N/A

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

       2. lower-+.f32 N/A

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

   14. Applied rewrites 72.6%

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

   if 0.5 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.0%

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

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

      6. associate-/l/ N/A

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

      7. associate-/r* N/A

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

      8. 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 99.4%

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

   5. Taylor expanded in s around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. +-commutative N/A

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

      6. lower-+.f32 N/A

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

   7. Applied rewrites 89.6%

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

   9. Applied rewrites 90.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 0.5:\\ \;\;\;\;\frac{1}{\left(\left(2 - \frac{\left|x\right|}{s}\right) \cdot s\right) \cdot \left(2 - \frac{\left|x\right| - 0.5 \cdot \frac{x \cdot x}{s}}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 0.5:\\ \;\;\;\;\frac{1}{2 \cdot \left(\left(2 - \frac{\left|x\right| - 0.5 \cdot \frac{x \cdot x}{s}}{s}\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s}}{s}\\ \end{array} \end{array} \]

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)) 0.5)
     (/ 1.0 (* 2.0 (* (- 2.0 (/ (- (fabs x) (* 0.5 (/ (* x x) s))) s)) s)))
     (/ (+ 0.25 (/ (* (/ x s) (* -0.0625 x)) s)) 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)) <= 0.5f) {
		tmp = 1.0f / (2.0f * ((2.0f - ((fabsf(x) - (0.5f * ((x * x) / s))) / s)) * s));
	} else {
		tmp = (0.25f + (((x / s) * (-0.0625f * x)) / s)) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 - (-1.0e0)
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.5e0) then
        tmp = 1.0e0 / (2.0e0 * ((2.0e0 - ((abs(x) - (0.5e0 * ((x * x) / s))) / s)) * s))
    else
        tmp = (0.25e0 + (((x / s) * ((-0.0625e0) * x)) / s)) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(0.5))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) * Float32(Float32(Float32(2.0) - Float32(Float32(abs(x) - Float32(Float32(0.5) * Float32(Float32(x * x) / s))) / s)) * s)));
	else
		tmp = Float32(Float32(Float32(0.25) + Float32(Float32(Float32(x / s) * Float32(Float32(-0.0625) * x)) / s)) / 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(0.5))
		tmp = single(1.0) / (single(2.0) * ((single(2.0) - ((abs(x) - (single(0.5) * ((x * x) / s))) / s)) * s));
	else
		tmp = (single(0.25) + (((x / s) * (single(-0.0625) * x)) / s)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-fabs.f32 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in s around inf

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

   11. Applied rewrites 37.8%

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

   12. Taylor expanded in s around inf

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

       1. +-commutative N/A

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

       2. lower-+.f32 N/A

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

   14. Applied rewrites 69.7%

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

   if 0.5 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.0%

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

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

      6. associate-/l/ N/A

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

      7. associate-/r* N/A

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

      8. 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 99.4%

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

   5. Taylor expanded in s around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. +-commutative N/A

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

      6. lower-+.f32 N/A

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

   7. Applied rewrites 89.6%

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

   9. Applied rewrites 90.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 0.5:\\ \;\;\;\;\frac{1}{2 \cdot \left(\left(2 - \frac{\left|x\right| - 0.5 \cdot \frac{x \cdot x}{s}}{s}\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - \frac{\left|x\right|}{s}\\ t_1 := e^{\frac{-\left|x\right|}{s}}\\ t_2 := t\_1 - -1\\ \mathbf{if}\;\frac{t\_1}{\left(t\_2 \cdot s\right) \cdot t\_2} \leq 0.5:\\ \;\;\;\;\frac{1}{\left(t\_0 \cdot s\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- 2.0 (/ (fabs x) s)))
        (t_1 (exp (/ (- (fabs x)) s)))
        (t_2 (- t_1 -1.0)))
   (if (<= (/ t_1 (* (* t_2 s) t_2)) 0.5)
     (/ 1.0 (* (* t_0 s) t_0))
     (/ (+ 0.25 (/ (* (/ x s) (* -0.0625 x)) s)) s))))

C

float code(float x, float s) {
	float t_0 = 2.0f - (fabsf(x) / s);
	float t_1 = expf((-fabsf(x) / s));
	float t_2 = t_1 - -1.0f;
	float tmp;
	if ((t_1 / ((t_2 * s) * t_2)) <= 0.5f) {
		tmp = 1.0f / ((t_0 * s) * t_0);
	} else {
		tmp = (0.25f + (((x / s) * (-0.0625f * x)) / s)) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: t_2
    real(4) :: tmp
    t_0 = 2.0e0 - (abs(x) / s)
    t_1 = exp((-abs(x) / s))
    t_2 = t_1 - (-1.0e0)
    if ((t_1 / ((t_2 * s) * t_2)) <= 0.5e0) then
        tmp = 1.0e0 / ((t_0 * s) * t_0)
    else
        tmp = (0.25e0 + (((x / s) * ((-0.0625e0) * x)) / s)) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(2.0) - Float32(abs(x) / s))
	t_1 = exp(Float32(Float32(-abs(x)) / s))
	t_2 = Float32(t_1 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_1 / Float32(Float32(t_2 * s) * t_2)) <= Float32(0.5))
		tmp = Float32(Float32(1.0) / Float32(Float32(t_0 * s) * t_0));
	else
		tmp = Float32(Float32(Float32(0.25) + Float32(Float32(Float32(x / s) * Float32(Float32(-0.0625) * x)) / s)) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = single(2.0) - (abs(x) / s);
	t_1 = exp((-abs(x) / s));
	t_2 = t_1 - single(-1.0);
	tmp = single(0.0);
	if ((t_1 / ((t_2 * s) * t_2)) <= single(0.5))
		tmp = single(1.0) / ((t_0 * s) * t_0);
	else
		tmp = (single(0.25) + (((x / s) * (single(-0.0625) * x)) / s)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-fabs.f32 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in s around inf

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

   11. Applied rewrites 37.8%

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

   12. Taylor expanded in s around inf

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

       1. mul-1-neg N/A

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

       2. sub-neg N/A

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

       3. lower--.f32 N/A

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

       4. lower-/.f32 N/A

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

       5. lower-fabs.f32 54.1

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

   14. Applied rewrites 54.1%

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

   if 0.5 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.0%

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

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

      6. associate-/l/ N/A

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

      7. associate-/r* N/A

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

      8. 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 99.4%

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

   5. Taylor expanded in s around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. +-commutative N/A

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

      6. lower-+.f32 N/A

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

   7. Applied rewrites 89.6%

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

   9. Applied rewrites 90.6%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 0.5:\\ \;\;\;\;\frac{1}{\left(\left(2 - \frac{\left|x\right|}{s}\right) \cdot s\right) \cdot \left(2 - \frac{\left|x\right|}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 0.5:\\ \;\;\;\;\frac{\frac{1}{s}}{2 \cdot \left(2 - \frac{\left|x\right|}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s}}{s}\\ \end{array} \end{array} \]

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)) 0.5)
     (/ (/ 1.0 s) (* 2.0 (- 2.0 (/ (fabs x) s))))
     (/ (+ 0.25 (/ (* (/ x s) (* -0.0625 x)) s)) 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)) <= 0.5f) {
		tmp = (1.0f / s) / (2.0f * (2.0f - (fabsf(x) / s)));
	} else {
		tmp = (0.25f + (((x / s) * (-0.0625f * x)) / s)) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 - (-1.0e0)
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.5e0) then
        tmp = (1.0e0 / s) / (2.0e0 * (2.0e0 - (abs(x) / s)))
    else
        tmp = (0.25e0 + (((x / s) * ((-0.0625e0) * x)) / s)) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(0.5))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(2.0) * Float32(Float32(2.0) - Float32(abs(x) / s))));
	else
		tmp = Float32(Float32(Float32(0.25) + Float32(Float32(Float32(x / s) * Float32(Float32(-0.0625) * x)) / s)) / 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(0.5))
		tmp = (single(1.0) / s) / (single(2.0) * (single(2.0) - (abs(x) / s)));
	else
		tmp = (single(0.25) + (((x / s) * (single(-0.0625) * x)) / s)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-fabs.f32 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in s around inf

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

   11. Applied rewrites 37.8%

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

       1. lift-/.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. lift-*.f32 N/A

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

       4. associate-*l* N/A

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

       5. associate-/r* N/A

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

       6. lower-/.f32 N/A

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

   13. Applied rewrites 37.8%

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

   if 0.5 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.0%

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

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

      6. associate-/l/ N/A

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

      7. associate-/r* N/A

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

      8. 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 99.4%

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

   5. Taylor expanded in s around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. +-commutative N/A

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

      6. lower-+.f32 N/A

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

   7. Applied rewrites 89.6%

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

   9. Applied rewrites 90.6%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 0.5:\\ \;\;\;\;\frac{\frac{1}{s}}{2 \cdot \left(2 - \frac{\left|x\right|}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 0.5:\\ \;\;\;\;\frac{\frac{1}{s}}{2 \cdot \left(2 - \frac{\left|x\right|}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.0625 \cdot \left(x \cdot x\right)}{s}}{s} + 0.25}{s}\\ \end{array} \end{array} \]

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)) 0.5)
     (/ (/ 1.0 s) (* 2.0 (- 2.0 (/ (fabs x) s))))
     (/ (+ (/ (/ (* -0.0625 (* x x)) s) 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)) <= 0.5f) {
		tmp = (1.0f / s) / (2.0f * (2.0f - (fabsf(x) / s)));
	} else {
		tmp = ((((-0.0625f * (x * x)) / s) / 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)) <= 0.5e0) then
        tmp = (1.0e0 / s) / (2.0e0 * (2.0e0 - (abs(x) / s)))
    else
        tmp = (((((-0.0625e0) * (x * x)) / s) / 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(0.5))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(2.0) * Float32(Float32(2.0) - Float32(abs(x) / s))));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(-0.0625) * Float32(x * x)) / s) / 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(0.5))
		tmp = (single(1.0) / s) / (single(2.0) * (single(2.0) - (abs(x) / s)));
	else
		tmp = ((((single(-0.0625) * (x * x)) / s) / 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))))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-fabs.f32 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in s around inf

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

   11. Applied rewrites 37.8%

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

       1. lift-/.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. lift-*.f32 N/A

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

       4. associate-*l* N/A

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

       5. associate-/r* N/A

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

       6. lower-/.f32 N/A

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

   13. Applied rewrites 37.8%

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

   if 0.5 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 89.6%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 0.5:\\ \;\;\;\;\frac{\frac{1}{s}}{2 \cdot \left(2 - \frac{\left|x\right|}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.0625 \cdot \left(x \cdot x\right)}{s}}{s} + 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

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(Float32(t_0 / (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 99.7%

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

   1. lift-/.f32 N/A

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

   2. 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. associate-/r* N/A

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

   6. associate-/l/ N/A

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

   7. associate-/r* N/A

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

   8. 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 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 10: 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 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 11: 97.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return (powf((1.0f + (1.0f / (1.0f - (((-0.5f * ((x * x) / s)) - fabsf(x)) / s)))), -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 + (1.0e0 / (1.0e0 - ((((-0.5e0) * ((x * x) / s)) - abs(x)) / s)))) ** (-2.0e0)) * exp((-abs(x) / s))) / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. unpow2 N/A

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

   13. sqr-abs N/A

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

   14. lower-*.f32 N/A

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

   15. lower-fabs.f32 96.6

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

9. Applied rewrites 96.6%

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

10. Final simplification 96.6%

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

Alternative 12: 96.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = 2.0e0 - ((abs(x) - (0.5e0 * ((x * x) / s))) / s)
    code = ((1.0e0 / (t_0 * t_0)) * exp((-abs(x) / s))) / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

7. Applied rewrites 96.2%

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

   1. lift-pow.f32 N/A

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

   2. sqr-pow N/A

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

   3. lower-*.f32 N/A

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

9. Applied rewrites 96.1%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. frac-2neg N/A

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

    4. metadata-eval N/A

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

    5. lift-/.f32 N/A

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

    6. frac-times N/A

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

11. Applied rewrites 96.2%

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

12. Final simplification 96.2%

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

Alternative 13: 96.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 93.5%

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

6. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 92.9

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

8. Applied rewrites 92.9%

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

9. Taylor expanded in s around inf

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

    1. mul-1-neg N/A

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

    2. sub-neg N/A

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

    3. lower--.f32 N/A

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

    4. lower-/.f32 N/A

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

    5. lower-fabs.f32 95.6

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

11. Applied rewrites 95.6%

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

12. Final simplification 95.6%

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

Alternative 14: 50.3% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. 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 93.5%

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

6. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 92.9

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

8. Applied rewrites 92.9%

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

9. Taylor expanded in s around inf

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

11. Applied rewrites 50.9%

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

12. Final simplification 50.9%

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

Alternative 15: 27.5% accurate, 31.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 31.2

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

5. Applied rewrites 31.2%

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

Reproduce

herbie shell --seed 2024277 
(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))))))