Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 13.5s

Alternatives: 6

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: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{-s}\\ \frac{e^{\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{t\_0}\right), t\_0\right)}}{s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) (- s))))
   (/ (exp (fma -2.0 (log1p (exp t_0)) t_0)) s)))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / -s;
	return expf(fmaf(-2.0f, log1pf(expf(t_0)), t_0)) / s;
}

Julia

function code(x, s)
	t_0 = Float32(abs(x) / Float32(-s))
	return Float32(exp(fma(Float32(-2.0), log1p(exp(t_0)), t_0)) / s)
end

Derivation

1. Initial program 99.4%

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

   1. lift-fabs.f32 N/A

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

   2. remove-double-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-fabs.f32 N/A

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

   12. remove-double-neg N/A

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

   13. lift-neg.f32 N/A

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

   14. remove-double-neg N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{-\frac{\left|x\right|}{s}}\right), -\frac{\left|x\right|}{s}\right)}}{s}} \]

7. Final simplification 99.5%

   \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right), \frac{\left|x\right|}{-s}\right)}}{s} \]
8. Add Preprocessing

Alternative 2: 96.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. Initial program 99.5%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

   5. Applied rewrites 0.5%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left|x\right| \cdot \left(x \cdot x\right), \frac{0.041666666666666664}{s \cdot \left(s \cdot s\right)}, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(-s\right)} - \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{s}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \color{blue}{\frac{\left(\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right) - \frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}} \]
   7. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} - \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left|x\right|\right) \cdot \left(\frac{-1}{16} + \frac{5}{48}\right)}}{{s}^{4}} - \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{1}{24}}}{{s}^{4}} - \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}}{{s}^{4}} - \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} \]

      5. +-inverses 99.5

         \[\leadsto \color{blue}{0} \]

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{0} \]

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

   1. Initial program 99.2%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left|x\right| \cdot \left(x \cdot x\right), \frac{0.041666666666666664}{s \cdot \left(s \cdot s\right)}, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(-s\right)} - \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{s}} \]

   6. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} + \frac{-1}{16} \cdot \frac{{x}^{2}}{{s}^{2}}}{s}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \frac{{x}^{2}}{{s}^{2}} + \frac{1}{4}}}{s} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{{s}^{2}} \cdot \frac{-1}{16}} + \frac{1}{4}}{s} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{{s}^{2}} \cdot \frac{-1}{16} + \frac{1}{4}}{s} \]

      5. sqr-abs N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}}{s} \]

      8. unpow2 N/A

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

      9. sqr-abs N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{{x}^{2}}}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      11. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{{x}^{2}}{{s}^{2}}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      13. lower-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      14. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x \cdot x}{\color{blue}{s \cdot s}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      15. lower-*.f32 81.5

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

   8. Applied rewrites 81.5%

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

      1. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{s \cdot s} \cdot \frac{-1}{16} + \frac{1}{4}}{s} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\frac{x \cdot x}{\color{blue}{s \cdot s}} \cdot \frac{-1}{16} + \frac{1}{4}}{s} \]

      3. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot x}{s \cdot s}} \cdot \frac{-1}{16} + \frac{1}{4}}{s} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot x}{s \cdot s}} \cdot \frac{-1}{16} + \frac{1}{4}}{s} \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{s \cdot s} \cdot \frac{-1}{16} + \frac{1}{4}}{s} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{\frac{x \cdot x}{\color{blue}{s \cdot s}} \cdot \frac{-1}{16} + \frac{1}{4}}{s} \]

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot \frac{-1}{16}, \frac{1}{4}\right)}}{s} \]

      10. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x}{s}}, \frac{x}{s} \cdot \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      11. lower-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x}{s} \cdot \frac{-1}{16}}, \frac{1}{4}\right)}{s} \]

      12. lower-/.f32 93.2

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

   10. Applied rewrites 93.2%

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

4. Final simplification 97.8%

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

Alternative 3: 96.3% accurate, 1.0× speedup

Math

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

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))) 0.0) 0.0 (/ 0.25 s))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

   5. Applied rewrites 0.5%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left|x\right| \cdot \left(x \cdot x\right), \frac{0.041666666666666664}{s \cdot \left(s \cdot s\right)}, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(-s\right)} - \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{s}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \color{blue}{\frac{\left(\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right) - \frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}} \]
   7. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} - \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left|x\right|\right) \cdot \left(\frac{-1}{16} + \frac{5}{48}\right)}}{{s}^{4}} - \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{1}{24}}}{{s}^{4}} - \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}}{{s}^{4}} - \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} \]

      5. +-inverses 99.5

         \[\leadsto \color{blue}{0} \]

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{0} \]

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

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 91.7

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

   5. Applied rewrites 91.7%

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

4. Final simplification 97.4%

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

Alternative 4: 95.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 96.2

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

5. Applied rewrites 96.2%

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

6. Final simplification 96.2%

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

Alternative 5: 94.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 95.9

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

5. Applied rewrites 95.9%

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

6. Final simplification 95.9%

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

Alternative 6: 73.1% accurate, 373.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.0)

C

float code(float x, float s) {
	return 0.0f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(x, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

5. Applied rewrites 16.1%

   \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left|x\right| \cdot \left(x \cdot x\right), \frac{0.041666666666666664}{s \cdot \left(s \cdot s\right)}, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(-s\right)} - \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{s}} \]

6. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{\left(\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right) - \frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}} \]
7. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} - \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}} \]

   2. distribute-rgt-out N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left|x\right|\right) \cdot \left(\frac{-1}{16} + \frac{5}{48}\right)}}{{s}^{4}} - \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{1}{24}}}{{s}^{4}} - \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}}{{s}^{4}} - \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} \]

   5. +-inverses 73.7

      \[\leadsto \color{blue}{0} \]

8. Applied rewrites 73.7%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))))