Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 13.7s

Alternatives: 8

Speedup: 2.9×

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, 1.5× speedup

Math

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

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 (/ (fabs 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((fabsf(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((abs(x_m) / 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(s + Float32(s / exp(Float32(abs(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((abs(x_m) / s)))));
end

Derivation

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. Step-by-step derivation

   1. *-commutative 99.5%

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

   2. distribute-lft-in 99.5%

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

   3. *-rgt-identity 99.5%

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

   4. fabs-neg 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. exp-neg 99.6%

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

   7. associate-*r/ 99.6%

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

   8. *-rgt-identity 99.6%

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

   9. *-lft-identity 99.6%

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

   10. metadata-eval 99.6%

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

   11. times-frac 99.6%

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

   12. neg-mul-1 99.6%

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

   13. neg-mul-1 99.6%

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

   14. fabs-neg 99.6%

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

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. Step-by-step derivation

   1. distribute-frac-neg 99.6%

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

   2. rec-exp 99.5%

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

   3. add-sqr-sqrt 48.3%

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

   4. fabs-sqr 48.3%

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

   5. add-sqr-sqrt 97.0%

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

6. Applied egg-rr 97.0%

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

   1. rec-exp 97.1%

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

   2. distribute-neg-frac 97.1%

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

8. Simplified 97.1%

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

   1. distribute-frac-neg 99.6%

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

   2. rec-exp 99.5%

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

   3. add-sqr-sqrt 48.3%

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

   4. fabs-sqr 48.3%

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

   5. add-sqr-sqrt 97.0%

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

10. Applied egg-rr 58.7%

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

    1. rec-exp 97.1%

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

    2. distribute-neg-frac 97.1%

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

12. Simplified 58.7%

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

13. Final simplification 58.7%

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

Alternative 2: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. *-commutative 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. sqrt-unprod 95.7%

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

   5. sqr-neg 95.7%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 23.3%

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

   8. *-un-lft-identity 23.3%

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

   9. distribute-rgt-in 23.3%

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

   10. +-commutative 23.3%

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

   11. *-commutative 23.3%

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

6. Applied egg-rr 60.5%

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

   1. distribute-frac-neg 99.6%

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

   2. rec-exp 99.5%

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

   3. add-sqr-sqrt 48.3%

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

   4. fabs-sqr 48.3%

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

   5. add-sqr-sqrt 97.0%

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

8. Applied egg-rr 99.4%

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

   1. rec-exp 97.1%

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

   2. distribute-neg-frac 97.1%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. frac-2neg 99.5%

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

   3. frac-2neg 99.5%

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

   4. add-sqr-sqrt 48.3%

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

   5. fabs-sqr 48.3%

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

   6. add-sqr-sqrt 60.5%

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

6. Applied egg-rr 60.5%

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

   1. distribute-frac-neg 99.6%

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

   2. rec-exp 99.5%

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

   3. add-sqr-sqrt 48.3%

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

   4. fabs-sqr 48.3%

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

   5. add-sqr-sqrt 97.0%

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

8. Applied egg-rr 99.4%

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

   1. rec-exp 97.1%

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

   2. distribute-neg-frac 97.1%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 4: 94.9% accurate, 5.7× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.5f / (s * (1.0f + 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
    code = 0.5e0 / (s * (1.0e0 + exp((x_m / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. *-commutative 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. sqrt-unprod 95.7%

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

   5. sqr-neg 95.7%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 23.3%

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

   8. *-un-lft-identity 23.3%

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

   9. distribute-rgt-in 23.3%

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

   10. +-commutative 23.3%

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

   11. *-commutative 23.3%

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

6. Applied egg-rr 60.5%

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

7. Taylor expanded in s around inf 58.5%

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

8. Taylor expanded in x around inf 58.5%

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

9. Final simplification 58.5%

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

Alternative 5: 94.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf((-x_m / s)) / (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)) / (s * 4.0e0)
end function

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. *-commutative 99.5%

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

   2. distribute-lft-in 99.5%

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

   3. *-rgt-identity 99.5%

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

   4. fabs-neg 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. exp-neg 99.6%

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

   7. associate-*r/ 99.6%

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

   8. *-rgt-identity 99.6%

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

   9. *-lft-identity 99.6%

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

   10. metadata-eval 99.6%

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

   11. times-frac 99.6%

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

   12. neg-mul-1 99.6%

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

   13. neg-mul-1 99.6%

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

   14. fabs-neg 99.6%

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

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. Step-by-step derivation

   1. distribute-frac-neg 99.6%

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

   2. rec-exp 99.5%

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

   3. add-sqr-sqrt 48.3%

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

   4. fabs-sqr 48.3%

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

   5. add-sqr-sqrt 97.0%

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

6. Applied egg-rr 97.0%

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

   1. rec-exp 97.1%

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

   2. distribute-neg-frac 97.1%

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

8. Simplified 97.1%

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

   1. distribute-frac-neg 99.6%

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

   2. rec-exp 99.5%

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

   3. add-sqr-sqrt 48.3%

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

   4. fabs-sqr 48.3%

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

   5. add-sqr-sqrt 97.0%

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

10. Applied egg-rr 58.7%

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

    1. rec-exp 97.1%

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

    2. distribute-neg-frac 97.1%

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

12. Simplified 58.7%

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

13. Taylor expanded in s around inf 57.6%

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

    1. *-commutative 57.6%

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

15. Simplified 57.6%

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

16. Final simplification 57.6%

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

Alternative 6: 50.7% accurate, 56.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{2 \cdot \left(s \cdot \left(\frac{x\_m}{s} + 2\right)\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. *-commutative 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. sqrt-unprod 95.7%

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

   5. sqr-neg 95.7%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 23.3%

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

   8. *-un-lft-identity 23.3%

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

   9. distribute-rgt-in 23.3%

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

   10. +-commutative 23.3%

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

   11. *-commutative 23.3%

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

6. Applied egg-rr 60.5%

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

7. Taylor expanded in s around inf 58.5%

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

8. Taylor expanded in x around 0 49.4%

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

9. Final simplification 49.4%

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

Alternative 7: 29.7% accurate, 68.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{2 \cdot \left(x\_m + s \cdot 2\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. *-commutative 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. sqrt-unprod 95.7%

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

   5. sqr-neg 95.7%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 23.3%

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

   8. *-un-lft-identity 23.3%

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

   9. distribute-rgt-in 23.3%

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

   10. +-commutative 23.3%

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

   11. *-commutative 23.3%

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

6. Applied egg-rr 60.5%

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

7. Taylor expanded in s around inf 58.5%

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

8. Taylor expanded in x around 0 27.1%

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

9. Final simplification 27.1%

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

Alternative 8: 27.3% 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 99.5%

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

   1. *-commutative 99.5%

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

   2. distribute-lft-in 99.5%

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

   3. *-rgt-identity 99.5%

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

   4. fabs-neg 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. exp-neg 99.6%

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

   7. associate-*r/ 99.6%

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

   8. *-rgt-identity 99.6%

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

   9. *-lft-identity 99.6%

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

   10. metadata-eval 99.6%

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

   11. times-frac 99.6%

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

   12. neg-mul-1 99.6%

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

   13. neg-mul-1 99.6%

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

   14. fabs-neg 99.6%

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

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 s around inf 25.1%

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

6. Final simplification 25.1%

   \[\leadsto \frac{0.25}{s} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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))))))