Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 12.0s

Alternatives: 7

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	return Float32(Float32(-1.0) / Float32(fma(s, exp(t_0), s) * Float32(Float32(-1.0) - exp(Float32(-t_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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in s around 0

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. exp-neg N/A

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

   4. rgt-mult-inverse N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-fabs.f32 99.4

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 2: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0.009999999776482582:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{\frac{\mathsf{fma}\left(x, -x, \frac{\mathsf{fma}\left(\left|x\right| \cdot \left(x \cdot x\right), -3, -\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot -3\right)\right)}{-s}\right)}{-s}}{s} - -4\right)}\\ \end{array} \end{array} \]

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))) 0.009999999776482582)
     t_0
     (/
      1.0
      (*
       s
       (-
        (/
         (/
          (fma
           x
           (- x)
           (/
            (fma (* (fabs x) (* x x)) -3.0 (- (* (fabs x) (* (* x x) -3.0))))
            (- s)))
          (- s))
         s)
        -4.0))))))

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))) <= 0.009999999776482582f) {
		tmp = t_0;
	} else {
		tmp = 1.0f / (s * (((fmaf(x, -x, (fmaf((fabsf(x) * (x * x)), -3.0f, -(fabsf(x) * ((x * x) * -3.0f))) / -s)) / -s) / s) - -4.0f));
	}
	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(t_1 * Float32(s * t_1))) <= Float32(0.009999999776482582))
		tmp = t_0;
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(Float32(fma(x, Float32(-x), Float32(fma(Float32(abs(x) * Float32(x * x)), Float32(-3.0), Float32(-Float32(abs(x) * Float32(Float32(x * x) * Float32(-3.0))))) / Float32(-s))) / Float32(-s)) / s) - Float32(-4.0))));
	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))))) < 0.00999999978

   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 99.9%

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

   5. Taylor expanded in s around 0

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

      1. neg-mul-1 N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-fabs.f32 99.4

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

   7. Applied rewrites 99.4%

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

   if 0.00999999978 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.0%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in s around -inf

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

   9. Applied rewrites 90.5%

      \[\leadsto \frac{1}{\color{blue}{-s \cdot \left(\frac{\frac{\mathsf{fma}\left(x, -x, \frac{\mathsf{fma}\left(\left|x\right| \cdot \left(x \cdot x\right), -3, \left(-\left|x\right|\right) \cdot \left(\left(x \cdot x\right) \cdot -3\right)\right)}{-s}\right)}{-s} + 0}{-s} + -4\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0.009999999776482582:\\ \;\;\;\;e^{-\frac{\left|x\right|}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{\frac{\mathsf{fma}\left(x, -x, \frac{\mathsf{fma}\left(\left|x\right| \cdot \left(x \cdot x\right), -3, -\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot -3\right)\right)}{-s}\right)}{-s}}{s} - -4\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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))))))