Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 10.9s

Alternatives: 10

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

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f32 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. lift-pow.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. pow-exp N/A

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

   8. *-lft-identity N/A

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

   9. lift-exp.f32 N/A

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

   10. un-div-inv N/A

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

   11. lower-/.f32 N/A

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 5: 96.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return (powf((2.0f - (fabsf(x) / s)), -2.0f) * expf((-fabsf(x) / s))) / 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)) * exp((-abs(x) / s))) / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 96.9

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

7. Applied rewrites 96.9%

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

Alternative 6: 94.7% 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 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 94.8

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

5. Applied rewrites 94.8%

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

Alternative 7: 66.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot x}{s}\\ \frac{1}{\left(\left(1 - \frac{-0.5 \cdot t\_0 - \left|x\right|}{s}\right) \cdot \left(4 - \frac{\mathsf{fma}\left(-3, t\_0, 4 \cdot \left|x\right|\right)}{s}\right)\right) \cdot s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (* x x) s)))
   (/
    1.0
    (*
     (*
      (- 1.0 (/ (- (* -0.5 t_0) (fabs x)) s))
      (- 4.0 (/ (fma -3.0 t_0 (* 4.0 (fabs x))) s)))
     s))))

C

float code(float x, float s) {
	float t_0 = (x * x) / s;
	return 1.0f / (((1.0f - (((-0.5f * t_0) - fabsf(x)) / s)) * (4.0f - (fmaf(-3.0f, t_0, (4.0f * fabsf(x))) / s))) * s);
}

Julia

function code(x, s)
	t_0 = Float32(Float32(x * x) / s)
	return Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(-0.5) * t_0) - abs(x)) / s)) * Float32(Float32(4.0) - Float32(fma(Float32(-3.0), t_0, Float32(Float32(4.0) * abs(x))) / s))) * s))
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 99.6%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. unpow2 N/A

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

   13. sqr-abs N/A

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

   14. lower-*.f32 N/A

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

   15. lower-fabs.f32 74.9

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

7. Applied rewrites 74.9%

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

8. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 82.6%

    \[\leadsto \frac{1}{\left(\color{blue}{\left(4 - \frac{\mathsf{fma}\left(-3, \frac{x \cdot x}{s}, 4 \cdot \left|x\right|\right)}{s}\right)} \cdot \left(1 - \frac{\frac{x \cdot x}{s} \cdot -0.5 - \left|x\right|}{s}\right)\right) \cdot s} \]

11. Final simplification 82.6%

    \[\leadsto \frac{1}{\left(\left(1 - \frac{-0.5 \cdot \frac{x \cdot x}{s} - \left|x\right|}{s}\right) \cdot \left(4 - \frac{\mathsf{fma}\left(-3, \frac{x \cdot x}{s}, 4 \cdot \left|x\right|\right)}{s}\right)\right) \cdot s} \]
12. Add Preprocessing

Alternative 8: 28.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{-0.0625}{s} \cdot \frac{x \cdot x}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= s 5.000000015855384e-30)
   (/ (* (/ -0.0625 s) (/ (* x x) s)) s)
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if (s <= 5.000000015855384e-30f) {
		tmp = ((-0.0625f / s) * ((x * x) / s)) / s;
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (s <= 5.000000015855384e-30) then
        tmp = (((-0.0625e0) / s) * ((x * x) / s)) / s
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (s <= Float32(5.000000015855384e-30))
		tmp = Float32(Float32(Float32(Float32(-0.0625) / s) * Float32(Float32(x * x) / s)) / s);
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (s <= single(5.000000015855384e-30))
		tmp = ((single(-0.0625) / s) * ((x * x) / s)) / s;
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 5.00000002e-30

   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

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]

   5. Applied rewrites 3.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.0625 \cdot \left(x \cdot x\right)}{s}}{s} + 0.25}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 3.9%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot x, \frac{x}{s \cdot s}, 0.25\right)}{s} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{-1}{16} \cdot \frac{{x}^{2}}{{s}^{2}}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 17.7%

       \[\leadsto \frac{\frac{-0.0625}{s} \cdot \frac{x \cdot x}{s}}{s} \]

   if 5.00000002e-30 < s

   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. Taylor expanded in s around inf

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

      1. lower-/.f32 37.8

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

   5. Applied rewrites 37.8%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.1% 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 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 99.6%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. unpow2 N/A

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

   13. sqr-abs N/A

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

   14. lower-*.f32 N/A

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

   15. lower-fabs.f32 74.9

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

7. Applied rewrites 74.9%

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

8. Taylor expanded in s around -inf

   \[\leadsto \frac{1}{\color{blue}{\left(4 + -1 \cdot \frac{-4 \cdot \left|x\right| + \left(-1 \cdot \frac{-4 \cdot {\left(\left|x\right|\right)}^{2} + \left(4 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{s} + 4 \cdot \left|x\right|\right)}{s}\right)} \cdot s} \]
9. 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

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

10. Applied rewrites 75.6%

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

11. Final simplification 75.6%

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

Alternative 10: 27.2% 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 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 29.9

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

5. Applied rewrites 29.9%

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

Reproduce

herbie shell --seed 2024284 
(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))))))