Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 13.9s

Alternatives: 10

Speedup: 2.0×

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: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((x_m / -s));
	return (t_0 / (t_0 + 1.0f)) / (s + (s / expf((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
    real(4) :: t_0
    t_0 = exp((x_m / -s))
    code = (t_0 / (t_0 + 1.0e0)) / (s + (s / exp((x_m / s))))
end function

Julia

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. fabs-neg 99.4%

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

   3. distribute-lft-in 99.5%

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

   4. *-rgt-identity 99.5%

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

   5. fabs-neg 99.5%

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

   6. distribute-rgt-in 99.5%

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

   7. cancel-sign-sub 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 99.5%

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

   1. associate-/r* 99.5%

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

7. Simplified 59.7%

   \[\leadsto \color{blue}{\frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{e^{\frac{x}{s}}}}} \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 2.0× 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(x_m / Float32(-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 99.4%

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

   1. *-commutative 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.5%

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

   1. associate-/r* 99.5%

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

   2. exp-prod 99.5%

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

   3. rem-square-sqrt 48.1%

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

   4. fabs-sqr 48.1%

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

   5. rem-square-sqrt 57.9%

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

   6. exp-prod 57.9%

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

   7. neg-mul-1 57.9%

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

   8. distribute-neg-frac2 57.9%

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

7. Simplified 59.6%

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

Alternative 3: 97.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{{\left(e^{{\left(\frac{x\_m}{s}\right)}^{2}}\right)}^{-0.25}}{s \cdot 4} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (pow (exp (pow (/ x_m s) 2.0)) -0.25) (* s 4.0)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return powf(expf(powf((x_m / s), 2.0f)), -0.25f) / (s * 4.0f);
}

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) ** 2.0e0)) ** (-0.25e0)) / (s * 4.0e0)
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32((exp((Float32(x_m / s) ^ Float32(2.0))) ^ Float32(-0.25)) / Float32(s * Float32(4.0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (exp(((x_m / s) ^ single(2.0))) ^ single(-0.25)) / (s * single(4.0));
end

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

3. Simplified 99.4%

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

   1. associate-/r* 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-lft-in 99.6%

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

   4. *-rgt-identity 99.6%

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

   5. fma-undefine 99.6%

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

   6. add-exp-log 97.8%

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

   7. associate-/r* 97.8%

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

6. Applied egg-rr 85.1%

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

7. Taylor expanded in x around 0 87.2%

   \[\leadsto e^{\color{blue}{-0.25 \cdot \frac{{x}^{2}}{{s}^{2}} - \log \left(4 \cdot s\right)}} \]
8. Step-by-step derivation

   1. unpow2 87.2%

      \[\leadsto e^{-0.25 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}} - \log \left(4 \cdot s\right)} \]

   2. unpow2 87.2%

      \[\leadsto e^{-0.25 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}} - \log \left(4 \cdot s\right)} \]

   3. times-frac 95.0%

      \[\leadsto e^{-0.25 \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} - \log \left(4 \cdot s\right)} \]

   4. unpow2 95.0%

      \[\leadsto e^{-0.25 \cdot \color{blue}{{\left(\frac{x}{s}\right)}^{2}} - \log \left(4 \cdot s\right)} \]

   5. *-commutative 95.0%

      \[\leadsto e^{-0.25 \cdot {\left(\frac{x}{s}\right)}^{2} - \log \color{blue}{\left(s \cdot 4\right)}} \]

9. Simplified 95.0%

   \[\leadsto e^{\color{blue}{-0.25 \cdot {\left(\frac{x}{s}\right)}^{2} - \log \left(s \cdot 4\right)}} \]
10. Step-by-step derivation

    1. exp-diff 95.1%

       \[\leadsto \color{blue}{\frac{e^{-0.25 \cdot {\left(\frac{x}{s}\right)}^{2}}}{e^{\log \left(s \cdot 4\right)}}} \]

    2. *-commutative 95.1%

       \[\leadsto \frac{e^{\color{blue}{{\left(\frac{x}{s}\right)}^{2} \cdot -0.25}}}{e^{\log \left(s \cdot 4\right)}} \]

    3. exp-prod 95.1%

       \[\leadsto \frac{\color{blue}{{\left(e^{{\left(\frac{x}{s}\right)}^{2}}\right)}^{-0.25}}}{e^{\log \left(s \cdot 4\right)}} \]

    4. add-exp-log 96.6%

       \[\leadsto \frac{{\left(e^{{\left(\frac{x}{s}\right)}^{2}}\right)}^{-0.25}}{\color{blue}{s \cdot 4}} \]

11. Applied egg-rr 96.6%

    \[\leadsto \color{blue}{\frac{{\left(e^{{\left(\frac{x}{s}\right)}^{2}}\right)}^{-0.25}}{s \cdot 4}} \]
12. Add Preprocessing

Alternative 4: 96.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((x_m / -s));
	return (t_0 / (t_0 + 1.0f)) / (s + (s / (1.0f + (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
    real(4) :: t_0
    t_0 = exp((x_m / -s))
    code = (t_0 / (t_0 + 1.0e0)) / (s + (s / (1.0e0 + (x_m / s))))
end function

Julia

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. fabs-neg 99.4%

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

   3. distribute-lft-in 99.5%

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

   4. *-rgt-identity 99.5%

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

   5. fabs-neg 99.5%

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

   6. distribute-rgt-in 99.5%

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

   7. cancel-sign-sub 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 99.5%

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

   1. associate-/r* 99.5%

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

7. Simplified 59.7%

   \[\leadsto \color{blue}{\frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{e^{\frac{x}{s}}}}} \]

8. Taylor expanded in x around 0 56.3%

   \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{\color{blue}{1 + \frac{x}{s}}}} \]
9. Step-by-step derivation

   1. +-commutative 56.3%

      \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{\color{blue}{\frac{x}{s} + 1}}} \]

10. Simplified 56.3%

    \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{\color{blue}{\frac{x}{s} + 1}}} \]

11. Final simplification 56.3%

    \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{1 + \frac{x}{s}}} \]
12. Add Preprocessing

Alternative 5: 95.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ e^{-0.25 \cdot \left(\frac{x\_m}{s} \cdot \frac{x\_m}{s}\right) - \log \left(s \cdot 4\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (exp (- (* -0.25 (* (/ x_m s) (/ x_m s))) (log (* s 4.0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf(((-0.25f * ((x_m / s) * (x_m / s))) - logf((s * 4.0f))));
}

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((((-0.25e0) * ((x_m / s) * (x_m / s))) - log((s * 4.0e0))))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return exp(Float32(Float32(Float32(-0.25) * Float32(Float32(x_m / s) * Float32(x_m / s))) - log(Float32(s * Float32(4.0)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = exp(((single(-0.25) * ((x_m / s) * (x_m / s))) - log((s * single(4.0)))));
end

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

3. Simplified 99.4%

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

   1. associate-/r* 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-lft-in 99.6%

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

   4. *-rgt-identity 99.6%

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

   5. fma-undefine 99.6%

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

   6. add-exp-log 97.8%

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

   7. associate-/r* 97.8%

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

6. Applied egg-rr 85.1%

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

7. Taylor expanded in x around 0 87.2%

   \[\leadsto e^{\color{blue}{-0.25 \cdot \frac{{x}^{2}}{{s}^{2}} - \log \left(4 \cdot s\right)}} \]
8. Step-by-step derivation

   1. unpow2 87.2%

      \[\leadsto e^{-0.25 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}} - \log \left(4 \cdot s\right)} \]

   2. unpow2 87.2%

      \[\leadsto e^{-0.25 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}} - \log \left(4 \cdot s\right)} \]

   3. times-frac 95.0%

      \[\leadsto e^{-0.25 \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} - \log \left(4 \cdot s\right)} \]

   4. unpow2 95.0%

      \[\leadsto e^{-0.25 \cdot \color{blue}{{\left(\frac{x}{s}\right)}^{2}} - \log \left(4 \cdot s\right)} \]

   5. *-commutative 95.0%

      \[\leadsto e^{-0.25 \cdot {\left(\frac{x}{s}\right)}^{2} - \log \color{blue}{\left(s \cdot 4\right)}} \]

9. Simplified 95.0%

   \[\leadsto e^{\color{blue}{-0.25 \cdot {\left(\frac{x}{s}\right)}^{2} - \log \left(s \cdot 4\right)}} \]
10. Step-by-step derivation

    1. unpow2 95.0%

       \[\leadsto e^{-0.25 \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} - \log \left(s \cdot 4\right)} \]

11. Applied egg-rr 95.0%

    \[\leadsto e^{-0.25 \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} - \log \left(s \cdot 4\right)} \]
12. Add Preprocessing

Alternative 6: 94.9% accurate, 5.3× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (expf((x_m / -s)) / 2.0f) / (s + (s / (1.0f + (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 = (exp((x_m / -s)) / 2.0e0) / (s + (s / (1.0e0 + (x_m / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. fabs-neg 99.4%

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

   3. distribute-lft-in 99.5%

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

   4. *-rgt-identity 99.5%

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

   5. fabs-neg 99.5%

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

   6. distribute-rgt-in 99.5%

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

   7. cancel-sign-sub 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 99.5%

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

   1. associate-/r* 99.5%

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

7. Simplified 59.7%

   \[\leadsto \color{blue}{\frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{e^{\frac{x}{s}}}}} \]

8. Taylor expanded in x around 0 56.3%

   \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{\color{blue}{1 + \frac{x}{s}}}} \]
9. Step-by-step derivation

   1. +-commutative 56.3%

      \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{\color{blue}{\frac{x}{s} + 1}}} \]

10. Simplified 56.3%

    \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{\color{blue}{\frac{x}{s} + 1}}} \]

11. Taylor expanded in x around 0 55.9%

    \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{\color{blue}{1} + 1}}{s + \frac{s}{\frac{x}{s} + 1}} \]

12. Final simplification 55.9%

    \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{2}}{s + \frac{s}{1 + \frac{x}{s}}} \]
13. Add Preprocessing

Alternative 7: 87.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2000000326942414 \cdot 10^{-26}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x\_m}{-s}} \cdot 0.5}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 1.2000000326942414e-26)
   (/ 0.25 s)
   (/ (* (exp (/ x_m (- s))) 0.5) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 1.2000000326942414e-26f) {
		tmp = 0.25f / s;
	} else {
		tmp = (expf((x_m / -s)) * 0.5f) / 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 <= 1.2000000326942414e-26) then
        tmp = 0.25e0 / s
    else
        tmp = (exp((x_m / -s)) * 0.5e0) / 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(1.2000000326942414e-26))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(exp(Float32(x_m / Float32(-s))) * Float32(0.5)) / 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(1.2000000326942414e-26))
		tmp = single(0.25) / s;
	else
		tmp = (exp((x_m / -s)) * single(0.5)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.20000003e-26

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in s around inf 33.1%

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

   if 1.20000003e-26 < x

   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. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. fabs-neg 99.6%

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

      3. distribute-lft-in 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. fabs-neg 99.7%

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

      6. distribute-rgt-in 99.7%

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

      7. cancel-sign-sub 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{e^{\frac{x}{s}}}}} \]

   8. Taylor expanded in x around 0 97.1%

      \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{\color{blue}{1 + \frac{x}{s}}}} \]
   9. Step-by-step derivation

      1. +-commutative 97.1%

         \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{\color{blue}{\frac{x}{s} + 1}}} \]

   10. Simplified 97.1%

       \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{\color{blue}{\frac{x}{s} + 1}}} \]

   11. Taylor expanded in x around 0 95.6%

       \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{\color{blue}{1} + 1}}{s + \frac{s}{\frac{x}{s} + 1}} \]

   12. Taylor expanded in x around inf 88.8%

       \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot \frac{x}{s}}}{s}} \]
   13. Step-by-step derivation

       1. associate-*r/ 88.8%

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

       2. associate-*r/ 88.8%

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

       3. mul-1-neg 88.8%

          \[\leadsto \frac{0.5 \cdot e^{\frac{\color{blue}{-x}}{s}}}{s} \]

   14. Simplified 88.8%

       \[\leadsto \color{blue}{\frac{0.5 \cdot e^{\frac{-x}{s}}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2000000326942414 \cdot 10^{-26}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{-s}} \cdot 0.5}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2000000326942414 \cdot 10^{-26}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(s, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 1.2000000326942414e-26) (/ 0.25 s) (+ 1.0 (fma s 4.0 -1.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 1.2000000326942414e-26f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f + fmaf(s, 4.0f, -1.0f);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(1.2000000326942414e-26))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) + fma(s, Float32(4.0), Float32(-1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 1.20000003e-26

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in s around inf 33.1%

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

   if 1.20000003e-26 < x

   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. Step-by-step derivation

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. fma-undefine 99.7%

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

      6. add-exp-log 99.0%

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

      7. associate-/r* 99.0%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in x around 0 15.4%

      \[\leadsto \color{blue}{e^{-\log \left(4 \cdot s\right)}} \]
   8. Step-by-step derivation

      1. *-commutative 15.4%

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

   9. Simplified 15.4%

      \[\leadsto \color{blue}{e^{-\log \left(s \cdot 4\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 11.1%

          \[\leadsto e^{\color{blue}{\sqrt{-\log \left(s \cdot 4\right)} \cdot \sqrt{-\log \left(s \cdot 4\right)}}} \]

       2. sqrt-unprod 12.2%

          \[\leadsto e^{\color{blue}{\sqrt{\left(-\log \left(s \cdot 4\right)\right) \cdot \left(-\log \left(s \cdot 4\right)\right)}}} \]

       3. sqr-neg 12.2%

          \[\leadsto e^{\sqrt{\color{blue}{\log \left(s \cdot 4\right) \cdot \log \left(s \cdot 4\right)}}} \]

       4. sqrt-unprod 1.1%

          \[\leadsto e^{\color{blue}{\sqrt{\log \left(s \cdot 4\right)} \cdot \sqrt{\log \left(s \cdot 4\right)}}} \]

       5. add-sqr-sqrt 11.1%

          \[\leadsto e^{\color{blue}{\log \left(s \cdot 4\right)}} \]

       6. expm1-log1p-u 11.1%

          \[\leadsto e^{\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot 4\right)\right)\right)}} \]

       7. add-exp-log 11.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot 4\right)\right)} \]

       8. expm1-undefine 72.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(s \cdot 4\right)} - 1} \]

   11. Applied egg-rr 72.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(s \cdot 4\right)} - 1} \]
   12. Step-by-step derivation

       1. log1p-undefine 72.7%

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

       2. rem-exp-log 72.7%

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

       3. associate-+r- 72.7%

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

       4. fma-neg 72.7%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(s, 4, -1\right)} \]

       5. metadata-eval 72.7%

          \[\leadsto 1 + \mathsf{fma}\left(s, 4, \color{blue}{-1}\right) \]

   13. Simplified 72.7%

       \[\leadsto \color{blue}{1 + \mathsf{fma}\left(s, 4, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 27.0% accurate, 206.7× 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 99.4%

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

   1. *-commutative 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around inf 26.6%

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

Alternative 10: 11.0% accurate, 206.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ s \cdot 4 \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = s * single(4.0);
end

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

3. Simplified 99.4%

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

   1. associate-/r* 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-lft-in 99.6%

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

   4. *-rgt-identity 99.6%

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

   5. fma-undefine 99.6%

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

   6. add-exp-log 97.8%

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

   7. associate-/r* 97.8%

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

6. Applied egg-rr 85.1%

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

7. Taylor expanded in x around 0 25.3%

   \[\leadsto \color{blue}{e^{-\log \left(4 \cdot s\right)}} \]
8. Step-by-step derivation

   1. *-commutative 25.3%

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

9. Simplified 25.3%

   \[\leadsto \color{blue}{e^{-\log \left(s \cdot 4\right)}} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 22.8%

       \[\leadsto e^{\color{blue}{\sqrt{-\log \left(s \cdot 4\right)} \cdot \sqrt{-\log \left(s \cdot 4\right)}}} \]

    2. sqrt-unprod 23.9%

       \[\leadsto e^{\color{blue}{\sqrt{\left(-\log \left(s \cdot 4\right)\right) \cdot \left(-\log \left(s \cdot 4\right)\right)}}} \]

    3. sqr-neg 23.9%

       \[\leadsto e^{\sqrt{\color{blue}{\log \left(s \cdot 4\right) \cdot \log \left(s \cdot 4\right)}}} \]

    4. sqrt-unprod 0.5%

       \[\leadsto e^{\color{blue}{\sqrt{\log \left(s \cdot 4\right)} \cdot \sqrt{\log \left(s \cdot 4\right)}}} \]

    5. add-sqr-sqrt 10.9%

       \[\leadsto e^{\color{blue}{\log \left(s \cdot 4\right)}} \]

    6. add-exp-log 10.9%

       \[\leadsto \color{blue}{s \cdot 4} \]

    7. *-un-lft-identity 10.9%

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

11. Applied egg-rr 10.9%

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

12. Taylor expanded in s around 0 10.9%

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

    1. *-commutative 10.9%

       \[\leadsto \color{blue}{s \cdot 4} \]

14. Simplified 10.9%

    \[\leadsto \color{blue}{s \cdot 4} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024136 
(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))))))