Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 9.7s

Alternatives: 9

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f32 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 96.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 0.0)
     t_0
     (/ -1.0 (* (+ (/ (/ (* x x) s) s) 4.0) (- s))))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((t_1 * s) * t_1)) <= 0.0f) {
		tmp = t_0;
	} else {
		tmp = -1.0f / (((((x * x) / s) / s) + 4.0f) * -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 = 1.0e0 + t_0
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.0e0) then
        tmp = t_0
    else
        tmp = (-1.0e0) / (((((x * x) / s) / s) + 4.0e0) * -s)
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((t_1 * s) * t_1)) <= single(0.0))
		tmp = t_0;
	else
		tmp = single(-1.0) / (((((x * x) / s) / s) + single(4.0)) * -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 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. neg-mul-1 N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower-/.f32 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-in N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f32 N/A

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

      18. lower-/.f32 N/A

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

      19. lower-fabs.f32 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto e^{\left(\frac{\left|x\right|}{s} - \log \color{blue}{\left(\frac{\mathsf{fma}\left(0.25, \frac{\left|x\right|}{s}, 0.25\right)}{s}\right)}\right) \cdot -1} \]
   8. Step-by-step derivation

   9. Applied rewrites 31.8%

      \[\leadsto e^{\left(\frac{\left|x\right|}{s} - \log \left(\frac{\frac{\left|x\right|}{s} \cdot 0.25 + 0.25}{s}\right)\right) \cdot -1} \]

   10. Taylor expanded in s around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f32 N/A

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

       5. lower-fabs.f32 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f32 100.0

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

   12. Applied rewrites 100.0%

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

   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. 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. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 92.3%

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

4. Final simplification 98.4%

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

Alternative 3: 94.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / (2.0f * ((1.0f + 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 / (2.0e0 * ((1.0e0 + t_0) * s))
end function

Julia

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

MATLAB

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

Derivation

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. Taylor expanded in s around inf

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

5. Applied rewrites 97.5%

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

6. Final simplification 97.5%

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

Alternative 4: 94.4% 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 99.8%

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

3. 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 97.3

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

5. Applied rewrites 97.3%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-exp.f32 97.3

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

7. Applied rewrites 97.3%

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

Alternative 5: 94.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. 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 97.3

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

5. Applied rewrites 97.3%

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

   1. lift-/.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. neg-sub0 N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{0 - \left|x\right|}}{s}}}{4 \cdot s} \]

   4. div-sub N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{0}{s} - \frac{\left|x\right|}{s}}}}{4 \cdot s} \]

   5. clear-num N/A

      \[\leadsto \frac{e^{\frac{0}{s} - \color{blue}{\frac{1}{\frac{s}{\left|x\right|}}}}}{4 \cdot s} \]

   6. frac-sub N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{0 \cdot \frac{s}{\left|x\right|} - s \cdot 1}{s \cdot \frac{s}{\left|x\right|}}}}}{4 \cdot s} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{0 \cdot \frac{s}{\left|x\right|} - s \cdot 1}{s \cdot \frac{s}{\left|x\right|}}}}}{4 \cdot s} \]

   8. *-rgt-identity N/A

      \[\leadsto \frac{e^{\frac{0 \cdot \frac{s}{\left|x\right|} - \color{blue}{s}}{s \cdot \frac{s}{\left|x\right|}}}}{4 \cdot s} \]

   9. lower--.f32 N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{0 \cdot \frac{s}{\left|x\right|} - s}}{s \cdot \frac{s}{\left|x\right|}}}}{4 \cdot s} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{e^{\frac{\color{blue}{0 \cdot \frac{s}{\left|x\right|}} - s}{s \cdot \frac{s}{\left|x\right|}}}}{4 \cdot s} \]

   11. lower-/.f32 N/A

       \[\leadsto \frac{e^{\frac{0 \cdot \color{blue}{\frac{s}{\left|x\right|}} - s}{s \cdot \frac{s}{\left|x\right|}}}}{4 \cdot s} \]

   12. associate-*r/ N/A

       \[\leadsto \frac{e^{\frac{0 \cdot \frac{s}{\left|x\right|} - s}{\color{blue}{\frac{s \cdot s}{\left|x\right|}}}}}{4 \cdot s} \]

   13. associate-*l/ N/A

       \[\leadsto \frac{e^{\frac{0 \cdot \frac{s}{\left|x\right|} - s}{\color{blue}{\frac{s}{\left|x\right|} \cdot s}}}}{4 \cdot s} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{e^{\frac{0 \cdot \frac{s}{\left|x\right|} - s}{\color{blue}{\frac{s}{\left|x\right|} \cdot s}}}}{4 \cdot s} \]

   15. lower-/.f32 97.3

       \[\leadsto \frac{e^{\frac{0 \cdot \frac{s}{\left|x\right|} - s}{\color{blue}{\frac{s}{\left|x\right|}} \cdot s}}}{4 \cdot s} \]

7. Applied rewrites 97.3%

   \[\leadsto \frac{e^{\color{blue}{\frac{0 \cdot \frac{s}{\left|x\right|} - s}{\frac{s}{\left|x\right|} \cdot s}}}}{4 \cdot s} \]
8. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{0 \cdot \frac{s}{\left|x\right|} - s}}{\frac{s}{\left|x\right|} \cdot s}}}{4 \cdot s} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{0 \cdot \frac{s}{\left|x\right|}} - s}{\frac{s}{\left|x\right|} \cdot s}}}{4 \cdot s} \]

   3. mul0-lft N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{0} - s}{\frac{s}{\left|x\right|} \cdot s}}}{4 \cdot s} \]

   4. neg-sub0 N/A

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

   5. lower-neg.f32 97.3

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

9. Applied rewrites 97.3%

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

Alternative 6: 94.4% 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 99.8%

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

3. 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 97.3

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

5. Applied rewrites 97.3%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 97.3

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

7. Applied rewrites 97.3%

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

8. Final simplification 97.3%

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

Alternative 7: 94.4% 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 99.8%

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

3. 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 97.3

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

5. Applied rewrites 97.3%

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

Alternative 8: 75.9% accurate, 7.6× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ -1.0 (* (+ (/ (/ (* x x) s) s) 4.0) (- s))))

C

float code(float x, float s) {
	return -1.0f / (((((x * x) / s) / s) + 4.0f) * -s);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (-1.0e0) / (((((x * x) / s) / s) + 4.0e0) * -s)
end function

Julia

function code(x, s)
	return Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(Float32(x * x) / s) / s) + Float32(4.0)) * Float32(-s)))
end

MATLAB

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

Derivation

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

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

   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. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-exp.f32 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 76.4%

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

8. Final simplification 76.4%

   \[\leadsto \frac{-1}{\left(\frac{\frac{x \cdot x}{s}}{s} + 4\right) \cdot \left(-s\right)} \]
9. Add Preprocessing

Alternative 9: 27.3% 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 99.8%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 22.1

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

5. Applied rewrites 22.1%

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

Reproduce

herbie shell --seed 2024331 
(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))))))