Logistic distribution

Percentage Accurate: 99.6% → 99.7%

Time: 14.4s

Alternatives: 14

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. 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-/l/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0.0010000000474974513:\\ \;\;\;\;\frac{t\_0}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{\frac{x \cdot x}{s}}{s} - -4\right)}\\ \end{array} \end{array} \]

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))) 0.0010000000474974513)
     (/ t_0 s)
     (/ 1.0 (* s (- (/ (/ (* x x) 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))) <= 0.0010000000474974513f) {
		tmp = t_0 / s;
	} else {
		tmp = 1.0f / (s * ((((x * x) / s) / s) - -4.0f));
	}
	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))) <= 0.0010000000474974513e0) then
        tmp = t_0 / s
    else
        tmp = 1.0e0 / (s * ((((x * x) / s) / s) - (-4.0e0)))
    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(0.0010000000474974513))
		tmp = Float32(t_0 / s);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(Float32(Float32(x * x) / s) / s) - Float32(-4.0))));
	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(0.0010000000474974513))
		tmp = t_0 / s;
	else
		tmp = single(1.0) / (s * ((((x * x) / s) / s) - single(-4.0)));
	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))))) < 0.00100000005

   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
   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-/l/ N/A

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

      4. lower-/.f32 N/A

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in s around 0

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

      1. neg-mul-1 N/A

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

      2. distribute-frac-neg2 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-fabs.f32 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f32 99.8

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

   9. Applied rewrites 99.8%

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

   if 0.00100000005 < (/.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 99.1%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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-/l/ N/A

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

      4. lower-/.f32 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 83.7%

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

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in s around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. neg-mul-1 N/A

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

       5. lower-*.f32 N/A

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

   12. Applied rewrites 91.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\left|x\right|}{-s}}}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{\left|x\right|}{-s}} + 1\right)\right)} \leq 0.0010000000474974513:\\ \;\;\;\;\frac{e^{\frac{\left|x\right|}{-s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{\frac{x \cdot x}{s}}{s} - -4\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(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))))))