Logistic distribution

Percentage Accurate: 99.5% → 99.2%

Time: 18.7s

Alternatives: 14

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 0.8× speedup

Math

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

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((exp(Float32(-1.0)) ^ Float32(abs(x) / s)) * Float32((Float32(exp(Float32(Float32(-abs(x)) / s)) - Float32(-1.0)) ^ Float32(-2.0)) / s))
end

MATLAB

function tmp = code(x, s)
	tmp = (exp(single(-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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-exp.f32 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 0.019999999552965164:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (- t_0 -1.0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 0.019999999552965164)
     t_0
     (/ (+ (/ (* (/ x s) (* -0.0625 x)) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = t_0 - -1.0f;
	float tmp;
	if ((t_0 / ((t_1 * s) * t_1)) <= 0.019999999552965164f) {
		tmp = t_0;
	} else {
		tmp = ((((x / s) * (-0.0625f * x)) / s) + 0.25f) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 - (-1.0e0)
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.019999999552965164e0) then
        tmp = t_0
    else
        tmp = ((((x / s) * ((-0.0625e0) * x)) / s) + 0.25e0) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(0.019999999552965164))
		tmp = t_0;
	else
		tmp = Float32(Float32(Float32(Float32(Float32(x / s) * Float32(Float32(-0.0625) * x)) / s) + Float32(0.25)) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = t_0 - single(-1.0);
	tmp = single(0.0);
	if ((t_0 / ((t_1 * s) * t_1)) <= single(0.019999999552965164))
		tmp = t_0;
	else
		tmp = ((((x / s) * (single(-0.0625) * x)) / s) + single(0.25)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0199999996

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f32 N/A

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

      5. lower-fabs.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f32 99.5

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

   7. Applied rewrites 99.5%

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

   if 0.0199999996 < (/.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 99.3%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 92.5%

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

   7. Applied rewrites 94.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 0.019999999552965164:\\ \;\;\;\;e^{\frac{-\left|x\right|}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 35.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := \frac{x \cdot x}{s}\\ t_2 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_2 \cdot s\right) \cdot t\_2} \leq 0.019999999552965164:\\ \;\;\;\;\frac{1}{\left(\frac{\mathsf{fma}\left(5, t\_1, \mathsf{fma}\left(t\_1, -4, 0\right)\right)}{s} + 4\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (/ (* x x) s)) (t_2 (- t_0 -1.0)))
   (if (<= (/ t_0 (* (* t_2 s) t_2)) 0.019999999552965164)
     (/ 1.0 (* (+ (/ (fma 5.0 t_1 (fma t_1 -4.0 0.0)) s) 4.0) s))
     (/ (+ (/ (* (/ x s) (* -0.0625 x)) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = (x * x) / s;
	float t_2 = t_0 - -1.0f;
	float tmp;
	if ((t_0 / ((t_2 * s) * t_2)) <= 0.019999999552965164f) {
		tmp = 1.0f / (((fmaf(5.0f, t_1, fmaf(t_1, -4.0f, 0.0f)) / s) + 4.0f) * s);
	} else {
		tmp = ((((x / s) * (-0.0625f * x)) / s) + 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(x * x) / s)
	t_2 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_2 * s) * t_2)) <= Float32(0.019999999552965164))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(fma(Float32(5.0), t_1, fma(t_1, Float32(-4.0), Float32(0.0))) / s) + Float32(4.0)) * s));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(x / s) * Float32(Float32(-0.0625) * x)) / s) + Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0199999996

   1. Initial program 99.9%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around -inf

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

   7. Applied rewrites 4.6%

      \[\leadsto \frac{1}{\color{blue}{\left(-s\right) \cdot \left(\frac{\mathsf{fma}\left(5, \frac{x \cdot x}{s}, \mathsf{fma}\left(\frac{x \cdot x}{s}, -4, 0\right)\right)}{-s} - 4\right)}} \]

   if 0.0199999996 < (/.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 99.3%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 92.5%

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

   7. Applied rewrites 94.8%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 0.019999999552965164:\\ \;\;\;\;\frac{1}{\left(\frac{\mathsf{fma}\left(5, \frac{x \cdot x}{s}, \mathsf{fma}\left(\frac{x \cdot x}{s}, -4, 0\right)\right)}{s} + 4\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. pow-exp N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-pow.f32 99.8

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

   6. lift-exp.f32 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f32 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 5: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-pow.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. pow-exp N/A

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

   6. *-lft-identity N/A

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lift-/.f32 N/A

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

   11. exp-neg N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-*.f32 N/A

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 7: 96.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. pow-exp N/A

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

   6. *-lft-identity N/A

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

   7. lift-exp.f32 N/A

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

   8. *-lft-identity N/A

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

   9. lift-*.f32 N/A

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

   10. associate-*l* N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. div-sub N/A

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

   10. unsub-neg N/A

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

   11. mul-1-neg N/A

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

   12. +-commutative N/A

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

   13. lower-+.f32 N/A

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

9. Applied rewrites 97.3%

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

10. Final simplification 97.3%

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

Alternative 8: 96.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = t_0 / ((single(2.0) - (abs(x) / s)) * ((t_0 - single(-1.0)) * 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. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 97.1

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

5. Applied rewrites 97.1%

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

6. Final simplification 97.1%

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

Alternative 9: 95.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 95.7%

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

6. Final simplification 95.7%

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

Alternative 10: 94.8% accurate, 2.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in s around inf

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

   1. lower-*.f32 95.3

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

7. Applied rewrites 95.3%

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

   1. lift-/.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. lift-neg.f32 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lower-*.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. metadata-eval N/A

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

   11. frac-2neg N/A

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

   12. lower-/.f32 95.3

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

9. Applied rewrites 95.3%

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

10. Final simplification 95.3%

    \[\leadsto \frac{{\mathsf{E}\left(\right)}^{\left(\frac{-1}{s} \cdot \left|x\right|\right)}}{4 \cdot s} \]
11. Add Preprocessing

Alternative 11: 94.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{{\mathsf{E}\left(\right)}^{\left(\frac{-\left|x\right|}{s}\right)}}{4 \cdot s} \end{array} \]

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in s around inf

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

   1. lower-*.f32 95.3

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

7. Applied rewrites 95.3%

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

   1. lift-exp.f32 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f32 95.3

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

9. Applied rewrites 95.3%

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

Alternative 12: 94.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in s around inf

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

   1. lower-*.f32 95.3

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

7. Applied rewrites 95.3%

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

   1. lift-pow.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. pow-exp N/A

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

   4. *-lft-identity N/A

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

   5. lift-exp.f32 95.3

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

9. Applied rewrites 95.3%

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

Alternative 13: 94.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 95.3%

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

Alternative 14: 27.2% 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 29.0

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

5. Applied rewrites 29.0%

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

Reproduce

herbie shell --seed 2024241 
(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))))))