Logistic distribution

Percentage Accurate: 99.6% → 99.5%

Time: 22.3s

Alternatives: 9

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)}}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/
  (/ 1.0 (expm1 (log1p (+ 1.0 (exp (/ (fabs x) (- s)))))))
  (* s (+ 1.0 (exp (/ x s))))))

C

x = abs(x);
float code(float x, float s) {
	return (1.0f / expm1f(log1pf((1.0f + expf((fabsf(x) / -s)))))) / (s * (1.0f + expf((x / s))));
}

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(1.0) / expm1(log1p(Float32(Float32(1.0) + exp(Float32(abs(x) / Float32(-s))))))) / Float32(s * Float32(Float32(1.0) + exp(Float32(x / s)))))
end

Derivation

1. Initial program 99.2%

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

3. Simplified 99.3%

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

   1. fma-udef 99.2%

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

   2. add-sqr-sqrt 46.0%

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

   3. fabs-sqr 46.0%

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

   4. add-sqr-sqrt 60.1%

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

5. Applied egg-rr 60.1%

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

   1. fma-def 60.0%

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

7. Simplified 60.0%

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

8. Taylor expanded in s around 0 60.0%

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

   1. expm1-log1p-u 60.1%

      \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)}}}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

10. Applied egg-rr 60.1%

    \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)}}}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

11. Final simplification 60.1%

    \[\leadsto \frac{\frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)}}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* s (* (+ 1.0 (exp (/ x s))) (+ 1.0 (exp (/ (- (fabs x)) s)))))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * ((1.0e0 + exp((x / s))) * (1.0e0 + exp((-abs(x) / s)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.3%

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

   1. fma-udef 99.2%

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

   2. add-sqr-sqrt 46.0%

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

   3. fabs-sqr 46.0%

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

   4. add-sqr-sqrt 60.1%

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

5. Applied egg-rr 60.1%

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

   1. fma-def 60.0%

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

7. Simplified 60.0%

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

8. Taylor expanded in s around 0 60.0%

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

9. Taylor expanded in x around inf 60.0%

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

    1. mul-1-neg 60.0%

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

11. Simplified 60.0%

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

12. Final simplification 60.0%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (+ 1.0 (exp (/ (- (fabs x)) s))) (+ s (* s (exp (/ x s)))))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((1.0e0 + exp((-abs(x) / s))) * (s + (s * exp((x / s)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.3%

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

   1. fma-udef 99.2%

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

   2. add-sqr-sqrt 46.0%

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

   3. fabs-sqr 46.0%

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

   4. add-sqr-sqrt 60.1%

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

5. Applied egg-rr 60.1%

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

   1. fma-def 60.0%

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

7. Simplified 60.0%

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

8. Taylor expanded in x around inf 60.1%

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

   1. *-un-lft-identity 60.1%

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

   2. mul-1-neg 60.1%

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

10. Applied egg-rr 60.1%

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

    1. *-lft-identity 60.1%

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

    2. distribute-neg-frac 60.1%

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

12. Simplified 60.1%

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

13. Final simplification 60.1%

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

Alternative 4: 82.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \left(1 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}}{x + s \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 3.999999999279835e-23)
   (/ 0.25 s)
   (/ (/ 1.0 (+ 1.0 (+ 1.0 (* 0.5 (/ (* x x) (* s s)))))) (+ x (* s 2.0)))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 3.999999999279835e-23f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / (1.0f + (1.0f + (0.5f * ((x * x) / (s * s)))))) / (x + (s * 2.0f));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 3.999999999279835e-23) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / (1.0e0 + (1.0e0 + (0.5e0 * ((x * x) / (s * s)))))) / (x + (s * 2.0e0))
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(3.999999999279835e-23))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(0.5) * Float32(Float32(x * x) / Float32(s * s)))))) / Float32(x + Float32(s * Float32(2.0))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(3.999999999279835e-23))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / (single(1.0) + (single(1.0) + (single(0.5) * ((x * x) / (s * s)))))) / (x + (s * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 4e-23

   1. Initial program 98.9%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in s around inf 75.2%

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

   if 4e-23 < (fabs.f32 x)

   1. Initial program 99.2%

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

   3. Simplified 99.4%

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

      1. fma-udef 99.3%

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

      2. add-sqr-sqrt 45.6%

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

      3. fabs-sqr 45.6%

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

      4. add-sqr-sqrt 53.2%

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

   5. Applied egg-rr 53.2%

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

      1. fma-def 53.2%

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

   7. Simplified 53.2%

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

   8. Taylor expanded in s around inf 19.7%

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

      1. *-commutative 19.7%

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

   10. Simplified 19.7%

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

   11. Taylor expanded in s around inf 50.5%

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

       1. associate-+r+ 50.5%

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

       2. mul-1-neg 50.5%

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

       3. unsub-neg 50.5%

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

       4. unpow2 50.5%

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

       5. sqr-abs 50.5%

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

       6. unpow2 50.5%

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

   13. Simplified 50.5%

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

   14. Taylor expanded in s around inf 82.8%

       \[\leadsto \frac{\frac{1}{1 + \left(\color{blue}{1} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}}{x + s \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \left(1 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}}{x + s \cdot 2}\\ \end{array} \]

Alternative 5: 95.4% accurate, 5.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.5 (* s (+ 1.0 (exp (/ x s))))))

C

x = abs(x);
float code(float x, float s) {
	return 0.5f / (s * (1.0f + expf((x / s))));
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (s * (1.0e0 + exp((x / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.3%

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

   1. fma-udef 99.2%

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

   2. add-sqr-sqrt 46.0%

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

   3. fabs-sqr 46.0%

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

   4. add-sqr-sqrt 60.1%

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

5. Applied egg-rr 60.1%

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

   1. fma-def 60.0%

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

7. Simplified 60.0%

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

8. Taylor expanded in s around 0 60.0%

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

9. Taylor expanded in s around inf 58.5%

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

10. Final simplification 58.5%

    \[\leadsto \frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

Alternative 6: 74.3% accurate, 36.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)}{x + s \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999858590343e-10)
   (/ 0.25 s)
   (/ (* 2.0 (* (/ s x) (/ s x))) (+ x (* s 2.0)))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 4.999999858590343e-10f) {
		tmp = 0.25f / s;
	} else {
		tmp = (2.0f * ((s / x) * (s / x))) / (x + (s * 2.0f));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.999999858590343e-10) then
        tmp = 0.25e0 / s
    else
        tmp = (2.0e0 * ((s / x) * (s / x))) / (x + (s * 2.0e0))
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999858590343e-10))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(Float32(s / x) * Float32(s / x))) / Float32(x + Float32(s * Float32(2.0))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999858590343e-10))
		tmp = single(0.25) / s;
	else
		tmp = (single(2.0) * ((s / x) * (s / x))) / (x + (s * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999986e-10

   1. Initial program 98.8%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in s around inf 36.5%

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

   if 4.99999986e-10 < x

   1. Initial program 100.0%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. fabs-sqr 99.9%

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

      4. add-sqr-sqrt 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. fma-def 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in s around inf 12.4%

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

      1. *-commutative 12.4%

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

   10. Simplified 12.4%

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

   11. Taylor expanded in s around inf 45.5%

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

       1. associate-+r+ 45.5%

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

       2. mul-1-neg 45.5%

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

       3. unsub-neg 45.5%

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

       4. unpow2 45.5%

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

       5. sqr-abs 45.5%

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

       6. unpow2 45.5%

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

   13. Simplified 45.5%

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

   14. Taylor expanded in x around inf 84.3%

       \[\leadsto \frac{\color{blue}{2 \cdot \frac{{s}^{2}}{{x}^{2}}}}{x + s \cdot 2} \]
   15. Step-by-step derivation

       1. unpow2 84.3%

          \[\leadsto \frac{2 \cdot \frac{\color{blue}{s \cdot s}}{{x}^{2}}}{x + s \cdot 2} \]

       2. unpow2 84.3%

          \[\leadsto \frac{2 \cdot \frac{s \cdot s}{\color{blue}{x \cdot x}}}{x + s \cdot 2} \]

       3. times-frac 84.3%

          \[\leadsto \frac{2 \cdot \color{blue}{\left(\frac{s}{x} \cdot \frac{s}{x}\right)}}{x + s \cdot 2} \]

   16. Simplified 84.3%

       \[\leadsto \frac{\color{blue}{2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)}}{x + s \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)}{x + s \cdot 2}\\ \end{array} \]

Alternative 7: 29.2% accurate, 88.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{0.5}{x + s \cdot 2} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.5 (+ x (* s 2.0))))

C

x = abs(x);
float code(float x, float s) {
	return 0.5f / (x + (s * 2.0f));
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (x + (s * 2.0e0))
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(0.5) / Float32(x + Float32(s * Float32(2.0))))
end

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.3%

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

   1. fma-udef 99.2%

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

   2. add-sqr-sqrt 46.0%

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

   3. fabs-sqr 46.0%

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

   4. add-sqr-sqrt 60.1%

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

5. Applied egg-rr 60.1%

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

   1. fma-def 60.0%

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

7. Simplified 60.0%

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

8. Taylor expanded in s around inf 29.9%

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

   1. *-commutative 29.9%

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

10. Simplified 29.9%

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

11. Taylor expanded in s around inf 29.9%

    \[\leadsto \frac{\color{blue}{0.5}}{x + s \cdot 2} \]

12. Final simplification 29.9%

    \[\leadsto \frac{0.5}{x + s \cdot 2} \]

Alternative 8: 30.4% accurate, 121.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.019999999552965164:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 0.019999999552965164) (/ 0.25 s) (/ 0.5 x)))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 0.019999999552965164f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.5f / x;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.019999999552965164e0) then
        tmp = 0.25e0 / s
    else
        tmp = 0.5e0 / x
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.019999999552965164))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.5) / x);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.019999999552965164))
		tmp = single(0.25) / s;
	else
		tmp = single(0.5) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0199999996

   1. Initial program 98.9%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in s around inf 35.4%

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

   if 0.0199999996 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. fabs-sqr 100.0%

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

      4. add-sqr-sqrt 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. fma-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in s around inf 10.6%

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

      1. *-commutative 10.6%

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

   10. Simplified 10.6%

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

   11. Taylor expanded in s around inf 10.8%

       \[\leadsto \frac{\color{blue}{0.5}}{x + s \cdot 2} \]

   12. Taylor expanded in x around inf 10.8%

       \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.019999999552965164:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 9: 27.0% accurate, 206.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.25 s))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.3%

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

4. Taylor expanded in s around inf 27.7%

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

5. Final simplification 27.7%

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

Reproduce

herbie shell --seed 2023275 
(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))))))