Logistic distribution

Percentage Accurate: 99.6% → 99.3%

Time: 8.9s

Alternatives: 6

Speedup: N/A×

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: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ {\left(e^{-2}\right)}^{\left(0.5 \cdot \frac{\left|x\right|}{s}\right)} \cdot \frac{{\left(1 + e^{\frac{-\left|x\right|}{s}}\right)}^{-2}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (*
  (pow (exp -2.0) (* 0.5 (/ (fabs x) s)))
  (/ (pow (+ 1.0 (exp (/ (- (fabs x)) s))) -2.0) s)))

C

float code(float x, float s) {
	return powf(expf(-2.0f), (0.5f * (fabsf(x) / s))) * (powf((1.0f + expf((-fabsf(x) / s))), -2.0f) / s);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (exp((-2.0e0)) ** (0.5e0 * (abs(x) / s))) * (((1.0e0 + exp((-abs(x) / s))) ** (-2.0e0)) / s)
end function

Julia

function code(x, s)
	return Float32((exp(Float32(-2.0)) ^ Float32(Float32(0.5) * Float32(abs(x) / s))) * Float32((Float32(Float32(1.0) + exp(Float32(Float32(-abs(x)) / s))) ^ Float32(-2.0)) / s))
end

MATLAB

function tmp = code(x, s)
	tmp = (exp(single(-2.0)) ^ (single(0.5) * (abs(x) / s))) * (((single(1.0) + exp((-abs(x) / s))) ^ single(-2.0)) / s);
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-exp.f32 98.9

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

6. Applied rewrites 98.9%

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

   1. lift-pow.f32 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f32 N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. prod-exp N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. div-inv N/A

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

   11. lower-*.f32 N/A

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

   12. metadata-eval 99.0

       \[\leadsto \frac{{\left(1 + e^{\frac{-\left|x\right|}{s}}\right)}^{-2}}{s} \cdot {\left(e^{-2}\right)}^{\left(\frac{\left|x\right|}{s} \cdot \color{blue}{0.5}\right)} \]

8. Applied rewrites 99.0%

   \[\leadsto \frac{{\left(1 + e^{\frac{-\left|x\right|}{s}}\right)}^{-2}}{s} \cdot \color{blue}{{\left(e^{-2}\right)}^{\left(\frac{\left|x\right|}{s} \cdot 0.5\right)}} \]

9. Final simplification 99.0%

   \[\leadsto {\left(e^{-2}\right)}^{\left(0.5 \cdot \frac{\left|x\right|}{s}\right)} \cdot \frac{{\left(1 + e^{\frac{-\left|x\right|}{s}}\right)}^{-2}}{s} \]
10. Add Preprocessing

Alternative 2: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 98.9%

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

Alternative 3: 96.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 98.9%

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

5. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 95.7

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

7. Applied rewrites 95.7%

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

Alternative 4: 94.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 98.9%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 94.3%

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

Alternative 5: 75.6% 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 98.9%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 98.9%

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

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

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

7. Applied rewrites 78.6%

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

8. Final simplification 78.6%

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

Alternative 6: 26.6% 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 98.9%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 27.7

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

5. Applied rewrites 27.7%

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

Reproduce

herbie shell --seed 2024295 
(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))))))