Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 9.5s

Alternatives: 9

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x_m) s)))) (/ (/ t_0 s) (pow (+ t_0 1.0) 2.0))))

C

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

Fortran

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	t_0 = exp((-x_m / s));
	tmp = (t_0 / s) / ((t_0 + single(1.0)) ^ 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. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-fabs.f32 N/A

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

   11. lower-neg.f32 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-fabs.f32 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. pow2 N/A

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

   4. sqrt-pow1 N/A

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

   5. metadata-eval N/A

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

   6. unpow1 64.1

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

7. Applied rewrites 64.1%

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

8. Taylor expanded in x around inf

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. associate-*r/ N/A

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

   6. mul-1-neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f32 N/A

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

   12. lower-exp.f32 N/A

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

   13. mul-1-neg N/A

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

   14. distribute-neg-frac2 N/A

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

   15. lower-/.f32 N/A

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

   16. lower-fabs.f32 N/A

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

   17. lower-neg.f32 64.0

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

10. Applied rewrites 64.0%

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

12. Applied rewrites 66.4%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x_m) s)))) (/ t_0 (* (pow (+ t_0 1.0) 2.0) s))))

C

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

Fortran

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	t_0 = exp((-x_m / s));
	tmp = t_0 / (((t_0 + single(1.0)) ^ single(2.0)) * 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 x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-fabs.f32 N/A

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

   11. lower-neg.f32 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-fabs.f32 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. pow2 N/A

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

   4. sqrt-pow1 N/A

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

   5. metadata-eval N/A

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

   6. unpow1 64.1

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

7. Applied rewrites 64.1%

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

8. Taylor expanded in x around inf

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. associate-*r/ N/A

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

   6. mul-1-neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f32 N/A

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

   12. lower-exp.f32 N/A

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

   13. mul-1-neg N/A

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

   14. distribute-neg-frac2 N/A

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

   15. lower-/.f32 N/A

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

   16. lower-fabs.f32 N/A

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

   17. lower-neg.f32 64.0

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

10. Applied rewrites 64.0%

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

12. Applied rewrites 66.4%

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

Alternative 3: 97.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ \frac{t\_0}{\left(\left(2 - \frac{x\_m - 0.5 \cdot \left(\frac{x\_m}{s} \cdot x\_m\right)}{s}\right) \cdot s\right) \cdot \left(1 + t\_0\right)} \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s))))
   (/
    t_0
    (* (* (- 2.0 (/ (- x_m (* 0.5 (* (/ x_m s) x_m))) s)) s) (+ 1.0 t_0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-fabsf(x_m) / s));
	return t_0 / (((2.0f - ((x_m - (0.5f * ((x_m / s) * x_m))) / s)) * s) * (1.0f + t_0));
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((-abs(x_m) / s))
    code = t_0 / (((2.0e0 - ((x_m - (0.5e0 * ((x_m / s) * x_m))) / s)) * s) * (1.0e0 + t_0))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(Float32(-abs(x_m)) / s))
	return Float32(t_0 / Float32(Float32(Float32(Float32(2.0) - Float32(Float32(x_m - Float32(Float32(0.5) * Float32(Float32(x_m / s) * x_m))) / s)) * s) * Float32(Float32(1.0) + t_0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	t_0 = exp((-abs(x_m) / s));
	tmp = t_0 / (((single(2.0) - ((x_m - (single(0.5) * ((x_m / s) * x_m))) / s)) * s) * (single(1.0) + t_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. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 73.0%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(\left(2 - \frac{\mathsf{fma}\left(-0.5, \frac{x \cdot x}{s}, \left|x\right|\right)}{s}\right) \cdot s\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 94.8%

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

Alternative 4: 95.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\_m\right|}{s}}\\ \frac{\frac{t\_0}{s}}{t\_0 + 1} \cdot 0.5 \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s)))) (* (/ (/ t_0 s) (+ t_0 1.0)) 0.5)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-fabsf(x_m) / s));
	return ((t_0 / s) / (t_0 + 1.0f)) * 0.5f;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((-abs(x_m) / s))
    code = ((t_0 / s) / (t_0 + 1.0e0)) * 0.5e0
end function

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(Float32(-abs(x_m)) / s))
	return Float32(Float32(Float32(t_0 / s) / Float32(t_0 + Float32(1.0))) * Float32(0.5))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	t_0 = exp((-abs(x_m) / s));
	tmp = ((t_0 / s) / (t_0 + single(1.0))) * single(0.5);
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

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. flip-+ N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 19.0%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 95.1%

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

8. Final simplification 95.1%

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

Alternative 5: 95.2% accurate, 1.5× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x_m)) s)))) (/ t_0 (* (* 2.0 s) (+ 1.0 t_0)))))

C

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

Fortran

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	t_0 = exp((-abs(x_m) / s));
	tmp = t_0 / ((single(2.0) * s) * (single(1.0) + t_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. Taylor expanded in s around inf

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

   1. lower-*.f32 95.1

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

5. Applied rewrites 95.1%

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

Alternative 6: 94.9% accurate, 2.9× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ (exp (/ (- x_m) s)) (* 4.0 s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf((-x_m / s)) / (4.0f * s);
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = exp((-x_m / s)) / (4.0e0 * s)
end function

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = exp((-x_m / 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 x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-fabs.f32 N/A

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

   11. lower-neg.f32 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-fabs.f32 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. pow2 N/A

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

   4. sqrt-pow1 N/A

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

   5. metadata-eval N/A

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

   6. unpow1 64.1

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

7. Applied rewrites 64.1%

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

8. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{-x}{s}}}{4 \cdot s} \]
9. Step-by-step derivation

10. Applied rewrites 63.0%

    \[\leadsto \frac{e^{\frac{-x}{s}}}{4 \cdot s} \]
11. Add Preprocessing

Alternative 7: 33.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.000000106112566 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\_m\right) \cdot \left(\frac{\frac{0.125 \cdot \frac{\left|x\_m\right|}{s} - 0.25}{s}}{x\_m} - \frac{0.125}{s \cdot s}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 3.000000106112566e-7)
   (/ 0.25 s)
   (*
    (- x_m)
    (- (/ (/ (- (* 0.125 (/ (fabs x_m) s)) 0.25) s) x_m) (/ 0.125 (* s s))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 3.000000106112566e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = -x_m * (((((0.125f * (fabsf(x_m) / s)) - 0.25f) / s) / x_m) - (0.125f / (s * s)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 3.000000106112566e-7) then
        tmp = 0.25e0 / s
    else
        tmp = -x_m * (((((0.125e0 * (abs(x_m) / s)) - 0.25e0) / s) / x_m) - (0.125e0 / (s * s)))
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(3.000000106112566e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(-x_m) * Float32(Float32(Float32(Float32(Float32(Float32(0.125) * Float32(abs(x_m) / s)) - Float32(0.25)) / s) / x_m) - Float32(Float32(0.125) / Float32(s * s))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(3.000000106112566e-7))
		tmp = single(0.25) / s;
	else
		tmp = -x_m * (((((single(0.125) * (abs(x_m) / s)) - single(0.25)) / s) / x_m) - (single(0.125) / (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.0000001e-7

   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 40.5

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

   5. Applied rewrites 40.5%

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

   if 3.0000001e-7 < x

   1. Initial program 99.9%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. flip-+ N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in s around -inf

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

   7. Applied rewrites 3.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot -0.5 - \left|x\right| \cdot x}{x}, 0.25, 0.0625 \cdot \frac{-2 \cdot \left(\left(x \cdot x\right) \cdot -2 - \left|x\right| \cdot x\right)}{x}\right)}{-s} - 0.25}{-s}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 3.6%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.25, \left|x\right|, 0.125 \cdot \left(x + \left|x\right|\right)\right)}{-s} - 0.25}{-s} \]

   11. Taylor expanded in x around -inf

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

   13. Applied rewrites 15.3%

       \[\leadsto \left(-x\right) \cdot \color{blue}{\left(\frac{\frac{0.125 \cdot \frac{\left|x\right|}{s} - 0.25}{s}}{x} - \frac{0.125}{s \cdot s}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 56.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\left(\frac{0.125 \cdot \frac{\left|x\_m\right|}{s}}{x\_m} - \left(\frac{0.25}{x\_m} + \frac{0.125}{s}\right)\right) \cdot x\_m}{-s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/
  (* (- (/ (* 0.125 (/ (fabs x_m) s)) x_m) (+ (/ 0.25 x_m) (/ 0.125 s))) x_m)
  (- s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return ((((0.125f * (fabsf(x_m) / s)) / x_m) - ((0.25f / x_m) + (0.125f / s))) * x_m) / -s;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = ((((0.125e0 * (abs(x_m) / s)) / x_m) - ((0.25e0 / x_m) + (0.125e0 / s))) * x_m) / -s
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.125) * Float32(abs(x_m) / s)) / x_m) - Float32(Float32(Float32(0.25) / x_m) + Float32(Float32(0.125) / s))) * x_m) / Float32(-s))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = ((((single(0.125) * (abs(x_m) / s)) / x_m) - ((single(0.25) / x_m) + (single(0.125) / s))) * x_m) / -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

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. flip-+ N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 19.0%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 28.2%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot -0.5 - \left|x\right| \cdot x}{x}, 0.25, 0.0625 \cdot \frac{-2 \cdot \left(\left(x \cdot x\right) \cdot -2 - \left|x\right| \cdot x\right)}{x}\right)}{-s} - 0.25}{-s}} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 29.9%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.25, \left|x\right|, 0.125 \cdot \left(x + \left|x\right|\right)\right)}{-s} - 0.25}{-s} \]

11. Taylor expanded in x around inf

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

13. Applied rewrites 43.8%

    \[\leadsto \frac{\left(\frac{0.125 \cdot \frac{\left|x\right|}{s}}{x} - \left(\frac{0.25}{x} + \frac{0.125}{s}\right)\right) \cdot x}{-\color{blue}{s}} \]
14. Add Preprocessing

Alternative 9: 26.8% accurate, 31.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.25}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.25 s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.25f / s;
}

Fortran

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

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(0.25) / s)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, 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 30.4

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

5. Applied rewrites 30.4%

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

Reproduce

herbie shell --seed 2024337 
(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))))))