Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 11.7s

Alternatives: 19

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-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 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 28.8% accurate, 0.9× speedup

Math

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

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)
     (/ (fma (* x x) (/ (/ -0.0625 s) s) 0.25) s)
     (/ (+ (/ 1.0 (* (/ s (* -0.0625 x)) (/ s x))) 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 = fmaf((x * x), ((-0.0625f / s) / s), 0.25f) / s;
	} else {
		tmp = ((1.0f / ((s / (-0.0625f * x)) * (s / x))) + 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(0.019999999552965164))
		tmp = Float32(fma(Float32(x * x), Float32(Float32(Float32(-0.0625) / s) / s), Float32(0.25)) / s);
	else
		tmp = Float32(Float32(Float32(Float32(1.0) / Float32(Float32(s / Float32(Float32(-0.0625) * x)) * Float32(s / x))) + 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 99.8%

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

   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

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}}{s} \]
   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 3.1%

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

   9. Applied rewrites 4.5%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{x \cdot x}{1}, \color{blue}{\frac{\frac{-0.0625}{s}}{s}}, 0.25\right)}{s} \]

   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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in s around inf

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}}{s} \]
   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 81.6%

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

   9. Applied rewrites 88.4%

      \[\leadsto \frac{\frac{1}{\frac{s}{x} \cdot \frac{s}{-0.0625 \cdot x}} + 0.25}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.4%

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

Alternative 3: 28.8% accurate, 0.9× speedup

Math

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

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)
     (/ (fma (* x x) (/ (/ -0.0625 s) s) 0.25) s)
     (/ (+ (* (/ x s) (/ (* -0.0625 x) s)) 0.25) s))))

C

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

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(0.019999999552965164))
		tmp = Float32(fma(Float32(x * x), Float32(Float32(Float32(-0.0625) / s) / s), Float32(0.25)) / s);
	else
		tmp = Float32(Float32(Float32(Float32(x / s) * Float32(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 99.8%

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

   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

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}}{s} \]
   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 3.1%

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

   9. Applied rewrites 4.5%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{x \cdot x}{1}, \color{blue}{\frac{\frac{-0.0625}{s}}{s}}, 0.25\right)}{s} \]

   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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in s around inf

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}}{s} \]
   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 81.6%

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

   9. Applied rewrites 88.4%

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

4. Final simplification 27.4%

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

Alternative 4: 28.6% accurate, 0.9× speedup

Math

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

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)
     (/ (fma (* x x) (/ -0.0625 (* s s)) 0.25) s)
     (/ (+ (* (/ x s) (/ (* -0.0625 x) s)) 0.25) s))))

C

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

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(0.019999999552965164))
		tmp = Float32(fma(Float32(x * x), Float32(Float32(-0.0625) / Float32(s * s)), Float32(0.25)) / s);
	else
		tmp = Float32(Float32(Float32(Float32(x / s) * Float32(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 99.8%

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

   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

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}}{s} \]
   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 3.1%

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

   9. Applied rewrites 4.5%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-x \cdot x}{-1}, \color{blue}{\frac{-0.0625}{s \cdot s}}, 0.25\right)}{s} \]

   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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in s around inf

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}}{s} \]
   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 81.6%

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

   9. Applied rewrites 88.4%

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

4. Final simplification 27.4%

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

Alternative 5: 28.6% accurate, 0.9× speedup

Math

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

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)
     (/ (fma (* 0.0625 (* x x)) (/ -1.0 (* s s)) 0.25) s)
     (/ (+ (* (/ x s) (/ (* -0.0625 x) s)) 0.25) s))))

C

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

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(0.019999999552965164))
		tmp = Float32(fma(Float32(Float32(0.0625) * Float32(x * x)), Float32(Float32(-1.0) / Float32(s * s)), Float32(0.25)) / s);
	else
		tmp = Float32(Float32(Float32(Float32(x / s) * Float32(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 99.8%

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

   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

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}}{s} \]
   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 3.1%

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

   9. Applied rewrites 4.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625 \cdot \left(x \cdot x\right), \color{blue}{\frac{1}{\left(-s\right) \cdot s}}, 0.25\right)}{s} \]

   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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in s around inf

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}}{s} \]
   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 81.6%

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

   9. Applied rewrites 88.4%

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

4. Final simplification 27.4%

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

Alternative 6: 28.3% accurate, 0.9× speedup

Math

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

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)) 0.019999999552965164)
     (/ (fma (* 0.0625 (* x x)) (/ -1.0 (* s s)) 0.25) 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)) <= 0.019999999552965164f) {
		tmp = fmaf((0.0625f * (x * x)), (-1.0f / (s * s)), 0.25f) / s;
	} else {
		tmp = ((((-0.0625f * (x * x)) / s) / s) + 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(0.019999999552965164))
		tmp = Float32(fma(Float32(Float32(0.0625) * Float32(x * x)), Float32(Float32(-1.0) / Float32(s * s)), Float32(0.25)) / s);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(-0.0625) * Float32(x * x)) / s) / 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 99.8%

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

   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

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}}{s} \]
   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 3.1%

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

   9. Applied rewrites 4.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625 \cdot \left(x \cdot x\right), \color{blue}{\frac{1}{\left(-s\right) \cdot s}}, 0.25\right)}{s} \]

   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 88.2%

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

4. Final simplification 27.4%

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

Alternative 7: 28.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (- t_0 -1.0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 0.019999999552965164)
     (/ (fma (* 0.0625 (* x x)) (/ -1.0 (* s s)) 0.25) 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 = fmaf((0.0625f * (x * x)), (-1.0f / (s * s)), 0.25f) / s;
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   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

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}}{s} \]
   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 3.1%

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

   9. Applied rewrites 4.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625 \cdot \left(x \cdot x\right), \color{blue}{\frac{1}{\left(-s\right) \cdot s}}, 0.25\right)}{s} \]

   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

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

      1. lower-/.f32 86.0

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

   5. Applied rewrites 86.0%

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

4. Final simplification 26.8%

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

Alternative 8: 28.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (- t_0 -1.0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 0.019999999552965164)
     (/ (fma (* x x) (/ -0.0625 (* s s)) 0.25) 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 = fmaf((x * x), (-0.0625f / (s * s)), 0.25f) / s;
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   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

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}}{s} \]
   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 3.1%

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

   8. Taylor expanded in s around inf

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}}{s} \]
   9. 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

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

   10. Applied rewrites 4.5%

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

   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

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

      1. lower-/.f32 86.0

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

   5. Applied rewrites 86.0%

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

4. Final simplification 26.8%

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

Alternative 9: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-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 99.7

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

4. Applied rewrites 99.7%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

6. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f32 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 10: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{{\left(1 + e^{\frac{-\left|x\right|}{s}}\right)}^{-2} \cdot e^{\frac{-\left|x\right|}{s}}}{s}} \]
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. *-lft-identity N/A

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

   3. pow-exp N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-pow.f32 99.6

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

   6. lift-exp.f32 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f32 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 11: 99.5% 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 99.6%

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 12: 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 99.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. 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 99.6%

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

5. Final simplification 99.6%

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

Alternative 13: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-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 99.7

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

4. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{-\left|x\right|}{s}\right)}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lift-pow.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. pow-exp N/A

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

   8. *-lft-identity N/A

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

   9. lift-exp.f32 N/A

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

   10. 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. Applied rewrites 99.2%

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

7. Final simplification 99.2%

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

Alternative 14: 96.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

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 95.2%

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

8. Final simplification 95.2%

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

Alternative 15: 96.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 94.8

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

7. Applied rewrites 94.8%

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

Alternative 16: 94.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return powf(expf(-1.0f), (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((-1.0e0)) ** (abs(x) / s)) / (4.0e0 * s)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. 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. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

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 93.9

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

7. Applied rewrites 93.9%

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

Alternative 17: 94.6% 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 99.6%

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

   1. lift-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 99.7

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

4. Applied rewrites 99.7%

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

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 93.9

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

7. Applied rewrites 93.9%

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

8. Final simplification 93.9%

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

Alternative 18: 94.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 93.9

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

5. Applied rewrites 93.9%

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

Alternative 19: 28.2% 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 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 26.8

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

5. Applied rewrites 26.8%

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

Reproduce

herbie shell --seed 2024259 
(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))))))