Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 14.3s

Alternatives: 12

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

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	float t_1 = t_0 + 1.0f;
	return t_0 / (t_1 * (s * 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 = t_0 + 1.0e0
    code = t_0 / (t_1 * (s * t_1))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 2: 99.6% 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 99.6%

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

   1. *-commutative 99.6%

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

   2. fabs-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. fabs-neg 99.6%

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

   5. distribute-lft-in 99.6%

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

   6. *-rgt-identity 99.6%

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

   7. +-commutative 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{x + -2 \cdot \left(s \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}{s}}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (exp (/ (+ x (* -2.0 (* s (log1p (exp (/ x s)))))) s)) s))

C

float code(float x, float s) {
	return expf(((x + (-2.0f * (s * log1pf(expf((x / s)))))) / s)) / s;
}

Julia

function code(x, s)
	return Float32(exp(Float32(Float32(x + Float32(Float32(-2.0) * Float32(s * log1p(exp(Float32(x / s)))))) / s)) / s)
end

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. pow2 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. distribute-frac-neg2 99.6%

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

   7. pow2 99.6%

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

   8. associate-/r* 99.6%

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

6. Applied egg-rr 61.5%

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

   1. *-lft-identity 61.5%

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

   2. associate-/l/ 61.8%

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

   3. exp-to-pow 61.9%

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

   4. log1p-undefine 61.9%

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

   5. *-commutative 61.9%

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

   6. rem-exp-log 60.2%

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

   7. exp-sum 60.2%

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

   8. exp-diff 88.6%

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

   9. associate--r+ 88.7%

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

   10. exp-diff 88.9%

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

8. Simplified 90.6%

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

9. Taylor expanded in s around 0 99.5%

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

    1. log1p-define 99.5%

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

11. Simplified 99.5%

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

12. Final simplification 99.5%

    \[\leadsto \frac{e^{\frac{x + -2 \cdot \left(s \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}{s}}}{s} \]
13. Add Preprocessing

Alternative 4: 97.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{x + -2 \cdot \left(s \cdot \log 2 + x \cdot \left(0.5 + \frac{x}{s} \cdot 0.125\right)\right)}{s}}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (exp
   (/ (+ x (* -2.0 (+ (* s (log 2.0)) (* x (+ 0.5 (* (/ x s) 0.125)))))) s))
  s))

C

float code(float x, float s) {
	return expf(((x + (-2.0f * ((s * logf(2.0f)) + (x * (0.5f + ((x / s) * 0.125f)))))) / s)) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp(((x + ((-2.0e0) * ((s * log(2.0e0)) + (x * (0.5e0 + ((x / s) * 0.125e0)))))) / s)) / s
end function

Julia

function code(x, s)
	return Float32(exp(Float32(Float32(x + Float32(Float32(-2.0) * Float32(Float32(s * log(Float32(2.0))) + Float32(x * Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.125))))))) / s)) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = exp(((x + (single(-2.0) * ((s * log(single(2.0))) + (x * (single(0.5) + ((x / s) * single(0.125))))))) / s)) / s;
end

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. pow2 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. distribute-frac-neg2 99.6%

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

   7. pow2 99.6%

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

   8. associate-/r* 99.6%

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

6. Applied egg-rr 61.5%

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

   1. *-lft-identity 61.5%

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

   2. associate-/l/ 61.8%

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

   3. exp-to-pow 61.9%

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

   4. log1p-undefine 61.9%

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

   5. *-commutative 61.9%

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

   6. rem-exp-log 60.2%

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

   7. exp-sum 60.2%

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

   8. exp-diff 88.6%

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

   9. associate--r+ 88.7%

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

   10. exp-diff 88.9%

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

8. Simplified 90.6%

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

9. Taylor expanded in s around 0 99.5%

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

    1. log1p-define 99.5%

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

11. Simplified 99.5%

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

12. Taylor expanded in x around 0 96.7%

    \[\leadsto \frac{e^{\frac{x + -2 \cdot \color{blue}{\left(s \cdot \log 2 + x \cdot \left(0.5 + 0.125 \cdot \frac{x}{s}\right)\right)}}{s}}}{s} \]

13. Final simplification 96.7%

    \[\leadsto \frac{e^{\frac{x + -2 \cdot \left(s \cdot \log 2 + x \cdot \left(0.5 + \frac{x}{s} \cdot 0.125\right)\right)}{s}}}{s} \]
14. Add Preprocessing

Alternative 5: 59.9% accurate, 5.5× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (* (/ -1.0 (- -1.0 (pow E (/ x s)))) (/ 0.5 s)))

C

float code(float x, float s) {
	return (-1.0f / (-1.0f - powf(((float) M_E), (x / s)))) * (0.5f / s);
}

Julia

function code(x, s)
	return Float32(Float32(Float32(-1.0) / Float32(Float32(-1.0) - (Float32(exp(1)) ^ Float32(x / s)))) * Float32(Float32(0.5) / s))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(-1.0) / (single(-1.0) - (single(2.71828182845904523536) ^ (x / s)))) * (single(0.5) / s);
end

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

3. Simplified 99.6%

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

6. Applied egg-rr 62.5%

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

7. Taylor expanded in x around 0 58.4%

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

   1. *-un-lft-identity 58.4%

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

   2. exp-prod 58.4%

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

9. Applied egg-rr 58.4%

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

    1. exp-1-e 58.4%

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

11. Simplified 58.4%

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

12. Final simplification 58.4%

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

Alternative 6: 59.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s} \cdot \frac{1}{e^{\frac{x}{s}} + 1} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (* (/ 0.5 s) (/ 1.0 (+ (exp (/ x s)) 1.0))))

C

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

Fortran

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)) + 1.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

3. Simplified 99.6%

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

6. Applied egg-rr 62.5%

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

7. Taylor expanded in x around 0 58.4%

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

8. Final simplification 58.4%

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

Alternative 7: 59.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{s}}{e^{\frac{x}{s}} + 1} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 0.5 s) (+ (exp (/ x s)) 1.0)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

3. Simplified 99.6%

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

6. Applied egg-rr 62.5%

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

7. Taylor expanded in x around 0 58.4%

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

8. Taylor expanded in x around inf 58.4%

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

   1. associate-/r* 58.4%

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

10. Simplified 58.4%

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

11. Final simplification 58.4%

    \[\leadsto \frac{\frac{0.5}{s}}{e^{\frac{x}{s}} + 1} \]
12. Add Preprocessing

Alternative 8: 59.2% 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(Float32(-x) / 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 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 94.0%

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

   1. distribute-frac-neg2 94.0%

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

   2. rec-exp 94.0%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}{s \cdot 4} \]

   3. frac-2neg 94.0%

      \[\leadsto \frac{\frac{1}{e^{\color{blue}{\frac{-\left|x\right|}{-s}}}}}{s \cdot 4} \]

   4. add-sqr-sqrt -0.0%

      \[\leadsto \frac{\frac{1}{e^{\frac{-\left|x\right|}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}}}{s \cdot 4} \]

   5. sqrt-unprod 18.4%

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

   6. sqr-neg 18.4%

      \[\leadsto \frac{\frac{1}{e^{\frac{-\left|x\right|}{\sqrt{\color{blue}{s \cdot s}}}}}}{s \cdot 4} \]

   7. sqrt-unprod 21.3%

      \[\leadsto \frac{\frac{1}{e^{\frac{-\left|x\right|}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}}}{s \cdot 4} \]

   8. add-sqr-sqrt 21.3%

      \[\leadsto \frac{\frac{1}{e^{\frac{-\left|x\right|}{\color{blue}{s}}}}}{s \cdot 4} \]

   9. remove-double-neg 21.3%

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

   10. frac-2neg 21.3%

       \[\leadsto \frac{\frac{1}{e^{\color{blue}{\frac{\left|x\right|}{-s}}}}}{s \cdot 4} \]

   11. add-sqr-sqrt -0.0%

       \[\leadsto \frac{\frac{1}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}}}{s \cdot 4} \]

   12. add-sqr-sqrt -0.0%

       \[\leadsto \frac{\frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{-s} \cdot \sqrt{-s}}}}}{s \cdot 4} \]

   13. fabs-sqr -0.0%

       \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{-s} \cdot \sqrt{-s}}}}}{s \cdot 4} \]

   14. add-sqr-sqrt -0.0%

       \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{x}}{\sqrt{-s} \cdot \sqrt{-s}}}}}{s \cdot 4} \]

   15. sqrt-unprod 54.2%

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

   16. sqr-neg 54.2%

       \[\leadsto \frac{\frac{1}{e^{\frac{x}{\sqrt{\color{blue}{s \cdot s}}}}}}{s \cdot 4} \]

   17. sqrt-unprod 57.6%

       \[\leadsto \frac{\frac{1}{e^{\frac{x}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}}}{s \cdot 4} \]

   18. add-sqr-sqrt 57.6%

       \[\leadsto \frac{\frac{1}{e^{\frac{x}{\color{blue}{s}}}}}{s \cdot 4} \]

7. Applied egg-rr 57.6%

   \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{x}{s}}}}}{s \cdot 4} \]
8. Step-by-step derivation

   1. rec-exp 57.6%

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

   2. distribute-neg-frac2 57.6%

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

9. Simplified 57.6%

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

10. Final simplification 57.6%

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

Alternative 9: 49.0% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.499999956129175 \cdot 10^{-15}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 4.0000001015105716 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{x}{s} \cdot -0.125 - \frac{0.5 \cdot \left(x \cdot -0.25\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{1}{\frac{x}{s} + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.499999956129175e-15)
   (/ 0.25 s)
   (if (<= x 4.0000001015105716e+26)
     (/ (- (* (/ x s) -0.125) (/ (* 0.5 (* x -0.25)) s)) s)
     (* (/ 0.5 s) (/ 1.0 (+ (/ x s) 2.0))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.499999956129175e-15f) {
		tmp = 0.25f / s;
	} else if (x <= 4.0000001015105716e+26f) {
		tmp = (((x / s) * -0.125f) - ((0.5f * (x * -0.25f)) / s)) / s;
	} else {
		tmp = (0.5f / s) * (1.0f / ((x / s) + 2.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.499999956129175e-15) then
        tmp = 0.25e0 / s
    else if (x <= 4.0000001015105716e+26) then
        tmp = (((x / s) * (-0.125e0)) - ((0.5e0 * (x * (-0.25e0))) / s)) / s
    else
        tmp = (0.5e0 / s) * (1.0e0 / ((x / s) + 2.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.499999956129175e-15))
		tmp = Float32(Float32(0.25) / s);
	elseif (x <= Float32(4.0000001015105716e+26))
		tmp = Float32(Float32(Float32(Float32(x / s) * Float32(-0.125)) - Float32(Float32(Float32(0.5) * Float32(x * Float32(-0.25))) / s)) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / s) * Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.499999956129175e-15))
		tmp = single(0.25) / s;
	elseif (x <= single(4.0000001015105716e+26))
		tmp = (((x / s) * single(-0.125)) - ((single(0.5) * (x * single(-0.25))) / s)) / s;
	else
		tmp = (single(0.5) / s) * (single(1.0) / ((x / s) + single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < 2.49999996e-15

   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%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in s around inf 31.5%

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

   if 2.49999996e-15 < x < 4.0000001e26

   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

   6. Applied egg-rr 10.4%

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

   7. Taylor expanded in s around inf 45.5%

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

   8. Taylor expanded in s around -inf 82.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{0.5 \cdot \frac{-0.5 \cdot x - -0.25 \cdot x}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s}} \]
   9. Step-by-step derivation

      1. mul-1-neg 82.0%

         \[\leadsto \color{blue}{-\frac{0.5 \cdot \frac{-0.5 \cdot x - -0.25 \cdot x}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s}} \]

      2. associate-*r/ 82.1%

         \[\leadsto -\frac{\color{blue}{\frac{0.5 \cdot \left(-0.5 \cdot x - -0.25 \cdot x\right)}{s}} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

      3. distribute-rgt-out-- 82.1%

         \[\leadsto -\frac{\frac{0.5 \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.25\right)\right)}}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

      4. metadata-eval 82.1%

         \[\leadsto -\frac{\frac{0.5 \cdot \left(x \cdot \color{blue}{-0.25}\right)}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

      5. associate-*r/ 85.9%

         \[\leadsto -\frac{\frac{0.5 \cdot \left(x \cdot -0.25\right)}{s} - \left(0.25 + \color{blue}{\frac{-0.125 \cdot x}{s}}\right)}{s} \]

      6. *-commutative 85.9%

         \[\leadsto -\frac{\frac{0.5 \cdot \left(x \cdot -0.25\right)}{s} - \left(0.25 + \frac{\color{blue}{x \cdot -0.125}}{s}\right)}{s} \]

   10. Simplified 85.9%

       \[\leadsto \color{blue}{-\frac{\frac{0.5 \cdot \left(x \cdot -0.25\right)}{s} - \left(0.25 + \frac{x \cdot -0.125}{s}\right)}{s}} \]

   11. Taylor expanded in x around inf 77.1%

       \[\leadsto -\frac{\frac{0.5 \cdot \left(x \cdot -0.25\right)}{s} - \color{blue}{-0.125 \cdot \frac{x}{s}}}{s} \]

   if 4.0000001e26 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr -0.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   8. Taylor expanded in x around 0 73.0%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.499999956129175 \cdot 10^{-15}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 4.0000001015105716 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{x}{s} \cdot -0.125 - \frac{0.5 \cdot \left(x \cdot -0.25\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{1}{\frac{x}{s} + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.5% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.0000001015105716 \cdot 10^{+26}:\\ \;\;\;\;\frac{\left(0.25 + \frac{x \cdot -0.125}{s}\right) - \frac{0.5 \cdot \left(x \cdot -0.25\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{1}{\frac{x}{s} + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.0000001015105716e+26)
   (/ (- (+ 0.25 (/ (* x -0.125) s)) (/ (* 0.5 (* x -0.25)) s)) s)
   (* (/ 0.5 s) (/ 1.0 (+ (/ x s) 2.0)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.0000001015105716e+26f) {
		tmp = ((0.25f + ((x * -0.125f) / s)) - ((0.5f * (x * -0.25f)) / s)) / s;
	} else {
		tmp = (0.5f / s) * (1.0f / ((x / s) + 2.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.0000001015105716e+26) then
        tmp = ((0.25e0 + ((x * (-0.125e0)) / s)) - ((0.5e0 * (x * (-0.25e0))) / s)) / s
    else
        tmp = (0.5e0 / s) * (1.0e0 / ((x / s) + 2.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.0000001015105716e+26))
		tmp = Float32(Float32(Float32(Float32(0.25) + Float32(Float32(x * Float32(-0.125)) / s)) - Float32(Float32(Float32(0.5) * Float32(x * Float32(-0.25))) / s)) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / s) * Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.0000001015105716e+26))
		tmp = ((single(0.25) + ((x * single(-0.125)) / s)) - ((single(0.5) * (x * single(-0.25))) / s)) / s;
	else
		tmp = (single(0.5) / s) * (single(1.0) / ((x / s) + single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.0000001e26

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   6. Applied egg-rr 67.5%

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

   7. Taylor expanded in s around inf 37.1%

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

   8. Taylor expanded in s around -inf 75.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{0.5 \cdot \frac{-0.5 \cdot x - -0.25 \cdot x}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s}} \]
   9. Step-by-step derivation

      1. mul-1-neg 75.1%

         \[\leadsto \color{blue}{-\frac{0.5 \cdot \frac{-0.5 \cdot x - -0.25 \cdot x}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s}} \]

      2. associate-*r/ 75.1%

         \[\leadsto -\frac{\color{blue}{\frac{0.5 \cdot \left(-0.5 \cdot x - -0.25 \cdot x\right)}{s}} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

      3. distribute-rgt-out-- 75.1%

         \[\leadsto -\frac{\frac{0.5 \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.25\right)\right)}}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

      4. metadata-eval 75.1%

         \[\leadsto -\frac{\frac{0.5 \cdot \left(x \cdot \color{blue}{-0.25}\right)}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

      5. associate-*r/ 76.8%

         \[\leadsto -\frac{\frac{0.5 \cdot \left(x \cdot -0.25\right)}{s} - \left(0.25 + \color{blue}{\frac{-0.125 \cdot x}{s}}\right)}{s} \]

      6. *-commutative 76.8%

         \[\leadsto -\frac{\frac{0.5 \cdot \left(x \cdot -0.25\right)}{s} - \left(0.25 + \frac{\color{blue}{x \cdot -0.125}}{s}\right)}{s} \]

   10. Simplified 76.8%

       \[\leadsto \color{blue}{-\frac{\frac{0.5 \cdot \left(x \cdot -0.25\right)}{s} - \left(0.25 + \frac{x \cdot -0.125}{s}\right)}{s}} \]

   if 4.0000001e26 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr -0.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   8. Taylor expanded in x around 0 73.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.0000001015105716 \cdot 10^{+26}:\\ \;\;\;\;\frac{\left(0.25 + \frac{x \cdot -0.125}{s}\right) - \frac{0.5 \cdot \left(x \cdot -0.25\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{1}{\frac{x}{s} + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.7% accurate, 56.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s} \cdot \frac{1}{\frac{x}{s} + 2} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (* (/ 0.5 s) (/ 1.0 (+ (/ x s) 2.0))))

C

float code(float x, float s) {
	return (0.5f / s) * (1.0f / ((x / s) + 2.0f));
}

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(Float32(0.5) / s) * Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.5) / s) * (single(1.0) / ((x / s) + single(2.0)));
end

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

3. Simplified 99.6%

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

6. Applied egg-rr 62.5%

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

7. Taylor expanded in x around 0 58.4%

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

8. Taylor expanded in x around 0 41.6%

   \[\leadsto \frac{1}{\color{blue}{2 + \frac{x}{s}}} \cdot \frac{0.5}{s} \]

9. Final simplification 41.6%

   \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\frac{x}{s} + 2} \]
10. Add Preprocessing

Alternative 12: 26.7% 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 99.6%

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

   1. *-commutative 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 24.2%

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

6. Final simplification 24.2%

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

Reproduce

herbie shell --seed 2024095 
(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))))))