Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 10.8s

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: 99.6% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 99.3%

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

5. Final simplification 99.3%

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

Alternative 3: 95.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 95.7%

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

   1. *-lft-identity N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

7. Applied rewrites 95.7%

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

8. Final simplification 95.7%

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

Alternative 4: 95.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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 / (((t_0 + 1.0e0) * s) * 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 95.7%

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

6. Final simplification 95.7%

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

Alternative 5: 94.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{e^{\frac{\frac{1}{s}}{\frac{1}{\left|x\right|}}} \cdot s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 0.25 (* (exp (/ (/ 1.0 s) (/ 1.0 (fabs x)))) s)))

C

float code(float x, float s) {
	return 0.25f / (expf(((1.0f / s) / (1.0f / fabsf(x)))) * s);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / (exp(((1.0e0 / s) / (1.0e0 / abs(x)))) * s)
end function

Julia

function code(x, s)
	return Float32(Float32(0.25) / Float32(exp(Float32(Float32(Float32(1.0) / s) / Float32(Float32(1.0) / abs(x)))) * s))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.25) / (exp(((single(1.0) / s) / (single(1.0) / abs(x)))) * s);
end

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 95.5%

   \[\leadsto \frac{\color{blue}{0.25} \cdot e^{\frac{-\left|x\right|}{s}}}{s} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f32 N/A

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

   7. div-inv N/A

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

   8. lift-exp.f32 N/A

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

   9. exp-neg N/A

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

   10. mul-1-neg N/A

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

   11. pow-exp N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-pow.f32 N/A

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

9. Applied rewrites 95.5%

   \[\leadsto \color{blue}{\frac{0.25}{e^{\frac{\left|x\right|}{s}} \cdot s}} \]
10. Step-by-step derivation

    1. lift-/.f32 N/A

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

    2. clear-num N/A

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

    3. associate-/r/ N/A

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

    4. lift-fabs.f32 N/A

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

    5. remove-double-div N/A

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

    6. fabs-div N/A

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

    7. metadata-eval N/A

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

    8. un-div-inv N/A

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

    9. lower-/.f32 N/A

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

    10. lower-/.f32 N/A

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

    11. fabs-div N/A

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

    12. metadata-eval N/A

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

    13. lift-fabs.f32 N/A

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

    14. lower-/.f32 95.5

        \[\leadsto \frac{0.25}{e^{\frac{\frac{1}{s}}{\color{blue}{\frac{1}{\left|x\right|}}}} \cdot s} \]

11. Applied rewrites 95.5%

    \[\leadsto \frac{0.25}{e^{\color{blue}{\frac{\frac{1}{s}}{\frac{1}{\left|x\right|}}}} \cdot s} \]
12. Add Preprocessing

Alternative 6: 94.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{e^{\frac{\left|x\right|}{s}} \cdot s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 (* (exp (/ (fabs x) s)) s)))

C

float code(float x, float s) {
	return 0.25f / (expf((fabsf(x) / s)) * s);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / (exp((abs(x) / s)) * s)
end function

Julia

function code(x, s)
	return Float32(Float32(0.25) / Float32(exp(Float32(abs(x) / s)) * s))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.25) / (exp((abs(x) / s)) * s);
end

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 95.5%

   \[\leadsto \frac{\color{blue}{0.25} \cdot e^{\frac{-\left|x\right|}{s}}}{s} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f32 N/A

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

   7. div-inv N/A

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

   8. lift-exp.f32 N/A

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

   9. exp-neg N/A

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

   10. mul-1-neg N/A

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

   11. pow-exp N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-pow.f32 N/A

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

9. Applied rewrites 95.5%

   \[\leadsto \color{blue}{\frac{0.25}{e^{\frac{\left|x\right|}{s}} \cdot s}} \]
10. Add Preprocessing

Alternative 7: 76.4% accurate, 7.9× speedup

Math

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

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)) * 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 99.7%

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

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

7. Applied rewrites 74.9%

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

8. Final simplification 74.9%

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

Alternative 8: 51.1% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\left(\frac{\left|x\right|}{s} + 1\right) \cdot s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 (* (+ (/ (fabs x) s) 1.0) s)))

C

float code(float x, float s) {
	return 0.25f / (((fabsf(x) / s) + 1.0f) * s);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / (((abs(x) / s) + 1.0e0) * s)
end function

Julia

function code(x, s)
	return Float32(Float32(0.25) / Float32(Float32(Float32(abs(x) / s) + Float32(1.0)) * s))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.25) / (((abs(x) / s) + single(1.0)) * s);
end

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 95.5%

   \[\leadsto \frac{\color{blue}{0.25} \cdot e^{\frac{-\left|x\right|}{s}}}{s} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f32 N/A

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

   7. div-inv N/A

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

   8. lift-exp.f32 N/A

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

   9. exp-neg N/A

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

   10. mul-1-neg N/A

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

   11. pow-exp N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-pow.f32 N/A

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

9. Applied rewrites 95.5%

   \[\leadsto \color{blue}{\frac{0.25}{e^{\frac{\left|x\right|}{s}} \cdot s}} \]

10. Taylor expanded in s around inf

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

    1. +-commutative N/A

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

    2. lower-+.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-fabs.f32 50.7

       \[\leadsto \frac{0.25}{\left(\frac{\color{blue}{\left|x\right|}}{s} + 1\right) \cdot s} \]

12. Applied rewrites 50.7%

    \[\leadsto \frac{0.25}{\color{blue}{\left(\frac{\left|x\right|}{s} + 1\right)} \cdot s} \]
13. Add Preprocessing

Alternative 9: 27.4% 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 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 25.4

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

5. Applied rewrites 25.4%

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

Reproduce

herbie shell --seed 2024244 
(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))))))