Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 11.9s

Alternatives: 8

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. fabs-neg 99.8%

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

   3. distribute-lft-in 99.8%

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

   4. *-rgt-identity 99.8%

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

   5. fabs-neg 99.8%

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

   6. distribute-rgt-in 99.8%

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

   7. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 63.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. exp-prod 99.8%

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

   3. rem-square-sqrt 49.5%

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

   4. fabs-sqr 49.5%

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

   5. rem-square-sqrt 97.4%

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

   6. exp-prod 97.4%

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

   7. neg-mul-1 97.4%

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

   8. distribute-neg-frac2 97.4%

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

7. Simplified 97.4%

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

8. Taylor expanded in x around 0 97.4%

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

   1. mul-1-neg 97.4%

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

   2. rem-square-sqrt 49.5%

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

   3. fabs-sqr 49.5%

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

   4. rem-square-sqrt 59.6%

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

10. Simplified 59.6%

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

11. Final simplification 59.6%

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

Alternative 3: 97.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot \left(\frac{1}{s} - \left(\frac{2 \cdot \log 2}{x} - \frac{\frac{x \cdot -0.25}{s} + -1}{s}\right)\right)}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (exp
   (*
    x
    (-
     (/ 1.0 s)
     (- (/ (* 2.0 (log 2.0)) x) (/ (+ (/ (* x -0.25) s) -1.0) s)))))
  s))

C

float code(float x, float s) {
	return expf((x * ((1.0f / s) - (((2.0f * logf(2.0f)) / x) - ((((x * -0.25f) / s) + -1.0f) / s))))) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp((x * ((1.0e0 / s) - (((2.0e0 * log(2.0e0)) / x) - ((((x * (-0.25e0)) / s) + (-1.0e0)) / s))))) / s
end function

Julia

function code(x, s)
	return Float32(exp(Float32(x * Float32(Float32(Float32(1.0) / s) - Float32(Float32(Float32(Float32(2.0) * log(Float32(2.0))) / x) - Float32(Float32(Float32(Float32(x * Float32(-0.25)) / s) + Float32(-1.0)) / s))))) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = exp((x * ((single(1.0) / s) - (((single(2.0) * log(single(2.0))) / x) - ((((x * single(-0.25)) / s) + single(-1.0)) / s))))) / s;
end

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.7%

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

   2. associate-/r/ 99.4%

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

6. Applied egg-rr 63.4%

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

7. Taylor expanded in s around 0 63.8%

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

   1. rem-exp-log 62.3%

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

   2. exp-to-pow 62.3%

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

   3. log1p-undefine 62.3%

      \[\leadsto \frac{e^{\frac{x}{s}}}{e^{\log s} \cdot e^{\color{blue}{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot 2}} \]

   4. *-commutative 62.3%

      \[\leadsto \frac{e^{\frac{x}{s}}}{e^{\log s} \cdot e^{\color{blue}{2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}} \]

   5. exp-sum 62.0%

      \[\leadsto \frac{e^{\frac{x}{s}}}{\color{blue}{e^{\log s + 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}} \]

   6. +-commutative 62.0%

      \[\leadsto \frac{e^{\frac{x}{s}}}{e^{\color{blue}{2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right) + \log s}}} \]

   7. exp-diff 87.0%

      \[\leadsto \color{blue}{e^{\frac{x}{s} - \left(2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right) + \log s\right)}} \]

   8. associate--r+ 87.2%

      \[\leadsto e^{\color{blue}{\left(\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right) - \log s}} \]

   9. exp-diff 87.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{e^{\log s}}} \]

9. Simplified 88.9%

   \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}} \]

10. Taylor expanded in x around -inf 99.6%

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

    1. associate-*r* 99.6%

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

    2. mul-1-neg 99.6%

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

    3. log1p-define 99.7%

       \[\leadsto \frac{e^{\left(-x\right) \cdot \left(2 \cdot \frac{\color{blue}{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{x} - \frac{1}{s}\right)}}{s} \]

    4. associate-*r/ 99.7%

       \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\color{blue}{\frac{2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}{x}} - \frac{1}{s}\right)}}{s} \]

12. Simplified 99.7%

    \[\leadsto \frac{e^{\color{blue}{\left(-x\right) \cdot \left(\frac{2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}{x} - \frac{1}{s}\right)}}}{s} \]

13. Taylor expanded in s around -inf 98.4%

    \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-0.25 \cdot x + 0.5 \cdot x}{s} - 1}{s} + 2 \cdot \frac{\log 2}{x}\right)} - \frac{1}{s}\right)}}{s} \]
14. Step-by-step derivation

    1. +-commutative 98.4%

       \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\color{blue}{\left(2 \cdot \frac{\log 2}{x} + -1 \cdot \frac{-1 \cdot \frac{-0.25 \cdot x + 0.5 \cdot x}{s} - 1}{s}\right)} - \frac{1}{s}\right)}}{s} \]

    2. mul-1-neg 98.4%

       \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\left(2 \cdot \frac{\log 2}{x} + \color{blue}{\left(-\frac{-1 \cdot \frac{-0.25 \cdot x + 0.5 \cdot x}{s} - 1}{s}\right)}\right) - \frac{1}{s}\right)}}{s} \]

    3. unsub-neg 98.4%

       \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\color{blue}{\left(2 \cdot \frac{\log 2}{x} - \frac{-1 \cdot \frac{-0.25 \cdot x + 0.5 \cdot x}{s} - 1}{s}\right)} - \frac{1}{s}\right)}}{s} \]

    4. associate-*r/ 98.4%

       \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\left(\color{blue}{\frac{2 \cdot \log 2}{x}} - \frac{-1 \cdot \frac{-0.25 \cdot x + 0.5 \cdot x}{s} - 1}{s}\right) - \frac{1}{s}\right)}}{s} \]

    5. sub-neg 98.4%

       \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\left(\frac{2 \cdot \log 2}{x} - \frac{\color{blue}{-1 \cdot \frac{-0.25 \cdot x + 0.5 \cdot x}{s} + \left(-1\right)}}{s}\right) - \frac{1}{s}\right)}}{s} \]

    6. associate-*r/ 98.4%

       \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\left(\frac{2 \cdot \log 2}{x} - \frac{\color{blue}{\frac{-1 \cdot \left(-0.25 \cdot x + 0.5 \cdot x\right)}{s}} + \left(-1\right)}{s}\right) - \frac{1}{s}\right)}}{s} \]

    7. distribute-rgt-out 98.4%

       \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\left(\frac{2 \cdot \log 2}{x} - \frac{\frac{-1 \cdot \color{blue}{\left(x \cdot \left(-0.25 + 0.5\right)\right)}}{s} + \left(-1\right)}{s}\right) - \frac{1}{s}\right)}}{s} \]

    8. metadata-eval 98.4%

       \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\left(\frac{2 \cdot \log 2}{x} - \frac{\frac{-1 \cdot \left(x \cdot \color{blue}{0.25}\right)}{s} + \left(-1\right)}{s}\right) - \frac{1}{s}\right)}}{s} \]

    9. *-commutative 98.4%

       \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\left(\frac{2 \cdot \log 2}{x} - \frac{\frac{-1 \cdot \color{blue}{\left(0.25 \cdot x\right)}}{s} + \left(-1\right)}{s}\right) - \frac{1}{s}\right)}}{s} \]

    10. neg-mul-1 98.4%

        \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\left(\frac{2 \cdot \log 2}{x} - \frac{\frac{\color{blue}{-0.25 \cdot x}}{s} + \left(-1\right)}{s}\right) - \frac{1}{s}\right)}}{s} \]

    11. distribute-lft-neg-in 98.4%

        \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\left(\frac{2 \cdot \log 2}{x} - \frac{\frac{\color{blue}{\left(-0.25\right) \cdot x}}{s} + \left(-1\right)}{s}\right) - \frac{1}{s}\right)}}{s} \]

    12. metadata-eval 98.4%

        \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\left(\frac{2 \cdot \log 2}{x} - \frac{\frac{\color{blue}{-0.25} \cdot x}{s} + \left(-1\right)}{s}\right) - \frac{1}{s}\right)}}{s} \]

    13. *-commutative 98.4%

        \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\left(\frac{2 \cdot \log 2}{x} - \frac{\frac{\color{blue}{x \cdot -0.25}}{s} + \left(-1\right)}{s}\right) - \frac{1}{s}\right)}}{s} \]

    14. metadata-eval 98.4%

        \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\left(\frac{2 \cdot \log 2}{x} - \frac{\frac{x \cdot -0.25}{s} + \color{blue}{-1}}{s}\right) - \frac{1}{s}\right)}}{s} \]

15. Simplified 98.4%

    \[\leadsto \frac{e^{\left(-x\right) \cdot \left(\color{blue}{\left(\frac{2 \cdot \log 2}{x} - \frac{\frac{x \cdot -0.25}{s} + -1}{s}\right)} - \frac{1}{s}\right)}}{s} \]

16. Final simplification 98.4%

    \[\leadsto \frac{e^{x \cdot \left(\frac{1}{s} - \left(\frac{2 \cdot \log 2}{x} - \frac{\frac{x \cdot -0.25}{s} + -1}{s}\right)\right)}}{s} \]
17. Add Preprocessing

Alternative 4: 60.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{x}{-s}}}{s \cdot 4 + x \cdot \left(\frac{x \cdot 3}{s} - 4\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (exp (/ x (- s))) (+ (* s 4.0) (* x (- (/ (* x 3.0) s) 4.0)))))

C

float code(float x, float s) {
	return expf((x / -s)) / ((s * 4.0f) + (x * (((x * 3.0f) / s) - 4.0f)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp((x / -s)) / ((s * 4.0e0) + (x * (((x * 3.0e0) / s) - 4.0e0)))
end function

Julia

function code(x, s)
	return Float32(exp(Float32(x / Float32(-s))) / Float32(Float32(s * Float32(4.0)) + Float32(x * Float32(Float32(Float32(x * Float32(3.0)) / s) - Float32(4.0)))))
end

MATLAB

function tmp = code(x, s)
	tmp = exp((x / -s)) / ((s * single(4.0)) + (x * (((x * single(3.0)) / s) - single(4.0))));
end

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. exp-prod 99.8%

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

   3. rem-square-sqrt 49.5%

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

   4. fabs-sqr 49.5%

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

   5. rem-square-sqrt 97.4%

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

   6. exp-prod 97.4%

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

   7. neg-mul-1 97.4%

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

   8. distribute-neg-frac2 97.4%

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

7. Simplified 97.4%

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

8. Taylor expanded in x around 0 97.4%

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

   1. mul-1-neg 97.4%

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

   2. rem-square-sqrt 49.5%

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

   3. fabs-sqr 49.5%

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

   4. rem-square-sqrt 59.6%

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

10. Simplified 59.6%

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

11. Taylor expanded in x around 0 55.9%

    \[\leadsto \frac{e^{\frac{-x}{s}}}{\color{blue}{4 \cdot s + x \cdot \left(x \cdot \left(-1.6666666666666667 \cdot \frac{x}{{s}^{2}} + 3 \cdot \frac{1}{s}\right) - 4\right)}} \]

12. Taylor expanded in x around 0 58.5%

    \[\leadsto \frac{e^{\frac{-x}{s}}}{4 \cdot s + x \cdot \left(\color{blue}{3 \cdot \frac{x}{s}} - 4\right)} \]
13. Step-by-step derivation

    1. associate-*r/ 58.5%

       \[\leadsto \frac{e^{\frac{-x}{s}}}{4 \cdot s + x \cdot \left(\color{blue}{\frac{3 \cdot x}{s}} - 4\right)} \]

    2. *-commutative 58.5%

       \[\leadsto \frac{e^{\frac{-x}{s}}}{4 \cdot s + x \cdot \left(\frac{\color{blue}{x \cdot 3}}{s} - 4\right)} \]

14. Simplified 58.5%

    \[\leadsto \frac{e^{\frac{-x}{s}}}{4 \cdot s + x \cdot \left(\color{blue}{\frac{x \cdot 3}{s}} - 4\right)} \]

15. Final simplification 58.5%

    \[\leadsto \frac{e^{\frac{x}{-s}}}{s \cdot 4 + x \cdot \left(\frac{x \cdot 3}{s} - 4\right)} \]
16. Add Preprocessing

Alternative 5: 60.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{x}{-s}}}{s \cdot 4 + x \cdot \left(3 \cdot \frac{x}{s} - 4\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (exp (/ x (- s))) (+ (* s 4.0) (* x (- (* 3.0 (/ x s)) 4.0)))))

C

float code(float x, float s) {
	return expf((x / -s)) / ((s * 4.0f) + (x * ((3.0f * (x / s)) - 4.0f)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp((x / -s)) / ((s * 4.0e0) + (x * ((3.0e0 * (x / s)) - 4.0e0)))
end function

Julia

function code(x, s)
	return Float32(exp(Float32(x / Float32(-s))) / Float32(Float32(s * Float32(4.0)) + Float32(x * Float32(Float32(Float32(3.0) * Float32(x / s)) - Float32(4.0)))))
end

MATLAB

function tmp = code(x, s)
	tmp = exp((x / -s)) / ((s * single(4.0)) + (x * ((single(3.0) * (x / s)) - single(4.0))));
end

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. exp-prod 99.8%

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

   3. rem-square-sqrt 49.5%

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

   4. fabs-sqr 49.5%

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

   5. rem-square-sqrt 97.4%

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

   6. exp-prod 97.4%

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

   7. neg-mul-1 97.4%

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

   8. distribute-neg-frac2 97.4%

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

7. Simplified 97.4%

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

8. Taylor expanded in x around 0 97.4%

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

   1. mul-1-neg 97.4%

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

   2. rem-square-sqrt 49.5%

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

   3. fabs-sqr 49.5%

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

   4. rem-square-sqrt 59.6%

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

10. Simplified 59.6%

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

11. Taylor expanded in x around 0 58.5%

    \[\leadsto \frac{e^{\frac{-x}{s}}}{\color{blue}{4 \cdot s + x \cdot \left(3 \cdot \frac{x}{s} - 4\right)}} \]

12. Final simplification 58.5%

    \[\leadsto \frac{e^{\frac{x}{-s}}}{s \cdot 4 + x \cdot \left(3 \cdot \frac{x}{s} - 4\right)} \]
13. Add Preprocessing

Alternative 6: 59.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{x}{-s}}}{s \cdot 4} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (exp (/ x (- s))) (* s 4.0)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. exp-prod 99.8%

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

   3. rem-square-sqrt 49.5%

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

   4. fabs-sqr 49.5%

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

   5. rem-square-sqrt 97.4%

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

   6. exp-prod 97.4%

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

   7. neg-mul-1 97.4%

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

   8. distribute-neg-frac2 97.4%

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

7. Simplified 97.4%

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

8. Taylor expanded in x around 0 97.4%

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

   1. mul-1-neg 97.4%

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

   2. rem-square-sqrt 49.5%

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

   3. fabs-sqr 49.5%

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

   4. rem-square-sqrt 59.6%

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

10. Simplified 59.6%

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

11. Taylor expanded in s around inf 57.0%

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

    1. *-commutative 57.0%

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

13. Simplified 57.0%

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

14. Final simplification 57.0%

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

Alternative 7: 53.7% accurate, 36.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{x \cdot -0.25}{s} + 0.25\right) + -0.25 \cdot \left(x \cdot \frac{-1}{s}\right)}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (+ (+ (/ (* x -0.25) s) 0.25) (* -0.25 (* x (/ -1.0 s)))) s))

C

float code(float x, float s) {
	return ((((x * -0.25f) / s) + 0.25f) + (-0.25f * (x * (-1.0f / s)))) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = ((((x * (-0.25e0)) / s) + 0.25e0) + ((-0.25e0) * (x * ((-1.0e0) / s)))) / s
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(Float32(Float32(x * Float32(-0.25)) / s) + Float32(0.25)) + Float32(Float32(-0.25) * Float32(x * Float32(Float32(-1.0) / s)))) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = ((((x * single(-0.25)) / s) + single(0.25)) + (single(-0.25) * (x * (single(-1.0) / s)))) / s;
end

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.7%

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

   2. associate-/r/ 99.4%

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

6. Applied egg-rr 63.4%

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

   1. *-un-lft-identity 63.4%

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

   2. exp-prod 63.4%

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

8. Applied egg-rr 63.4%

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

   1. exp-1-e 63.4%

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

10. Simplified 63.4%

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

11. Taylor expanded in s around -inf 27.6%

    \[\leadsto \color{blue}{-1 \cdot \frac{-0.25 \cdot \frac{x \cdot \log e}{s} - \left(0.25 + -0.25 \cdot \frac{x}{s}\right)}{s}} \]
12. Step-by-step derivation

    1. mul-1-neg 27.6%

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

    2. log-E 63.7%

       \[\leadsto -\frac{-0.25 \cdot \frac{x \cdot \color{blue}{1}}{s} - \left(0.25 + -0.25 \cdot \frac{x}{s}\right)}{s} \]

    3. associate-/l* 54.3%

       \[\leadsto -\frac{-0.25 \cdot \color{blue}{\left(x \cdot \frac{1}{s}\right)} - \left(0.25 + -0.25 \cdot \frac{x}{s}\right)}{s} \]

    4. associate-*r/ 54.3%

       \[\leadsto -\frac{-0.25 \cdot \left(x \cdot \frac{1}{s}\right) - \left(0.25 + \color{blue}{\frac{-0.25 \cdot x}{s}}\right)}{s} \]

    5. *-commutative 54.3%

       \[\leadsto -\frac{-0.25 \cdot \left(x \cdot \frac{1}{s}\right) - \left(0.25 + \frac{\color{blue}{x \cdot -0.25}}{s}\right)}{s} \]

13. Simplified 54.3%

    \[\leadsto \color{blue}{-\frac{-0.25 \cdot \left(x \cdot \frac{1}{s}\right) - \left(0.25 + \frac{x \cdot -0.25}{s}\right)}{s}} \]

14. Final simplification 54.3%

    \[\leadsto \frac{\left(\frac{x \cdot -0.25}{s} + 0.25\right) + -0.25 \cdot \left(x \cdot \frac{-1}{s}\right)}{s} \]
15. Add Preprocessing

Alternative 8: 27.1% accurate, 206.7× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 s))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in s around inf 25.0%

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

Reproduce

herbie shell --seed 2024135 
(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))))))