Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 13.6s

Alternatives: 9

Speedup: N/A×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. distribute-lft-in 99.7%

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

   3. *-rgt-identity 99.7%

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

   4. fabs-neg 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. exp-neg 99.7%

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

   7. associate-*r/ 99.7%

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

   8. *-rgt-identity 99.7%

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

   9. *-lft-identity 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.7%

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

   12. neg-mul-1 99.7%

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

   13. neg-mul-1 99.7%

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

   14. fabs-neg 99.7%

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

3. Simplified 99.7%

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

   1. distribute-frac-neg 99.7%

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

   2. rec-exp 99.7%

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

   3. add-sqr-sqrt 46.2%

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

   4. fabs-sqr 46.2%

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

   5. add-sqr-sqrt 58.2%

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

6. Applied egg-rr 58.2%

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

7. Final simplification 58.2%

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

Alternative 2: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{x_m}{s}}\\ \mathbf{if}\;\left|x_m\right| \leq 3.5:\\ \;\;\;\;\frac{e^{\frac{x_m}{s} + -2 \cdot \mathsf{log1p}\left(t_0\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot t_0}}{4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m s))))
   (if (<= (fabs x_m) 3.5)
     (/ (exp (+ (/ x_m s) (* -2.0 (log1p t_0)))) s)
     (/ (/ 1.0 (* s t_0)) 4.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((x_m / s));
	float tmp;
	if (fabsf(x_m) <= 3.5f) {
		tmp = expf(((x_m / s) + (-2.0f * log1pf(t_0)))) / s;
	} else {
		tmp = (1.0f / (s * t_0)) / 4.0f;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(x_m / s))
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(3.5))
		tmp = Float32(exp(Float32(Float32(x_m / s) + Float32(Float32(-2.0) * log1p(t_0)))) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(s * t_0)) / Float32(4.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 3.5

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

   3. Simplified 99.3%

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

   6. Applied egg-rr 80.9%

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

      1. unpow2 80.9%

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

      2. frac-times 80.9%

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

      3. add-sqr-sqrt 81.5%

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

      4. pow2 81.5%

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

      5. +-commutative 81.5%

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

      6. clear-num 81.5%

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

      7. frac-2neg 81.5%

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

      8. metadata-eval 81.5%

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

   8. Applied egg-rr 81.5%

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

      1. associate-/r/ 82.2%

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

      2. distribute-rgt-neg-in 82.2%

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

   10. Simplified 82.2%

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

   11. Taylor expanded in x around inf 82.3%

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

       1. *-commutative 82.3%

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

       2. associate-/r* 82.3%

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

       3. exp-to-pow 82.2%

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

       4. log1p-def 82.4%

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

       5. *-commutative 82.4%

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

       6. exp-diff 99.4%

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

       7. cancel-sign-sub-inv 99.4%

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

       8. metadata-eval 99.4%

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

   13. Simplified 99.4%

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

   if 3.5 < (fabs.f32 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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

      2. associate-*r/ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-*r/ 100.0%

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

      6. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   9. Applied egg-rr 48.3%

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

       1. unpow-1 48.3%

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

   11. Simplified 48.3%

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

       1. distribute-frac-neg 100.0%

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

       2. rec-exp 100.0%

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

       3. add-sqr-sqrt 46.6%

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

       4. fabs-sqr 46.6%

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

       5. add-sqr-sqrt 48.3%

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

   13. Applied egg-rr 46.6%

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

       1. rec-exp 46.6%

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

       2. distribute-neg-frac 46.6%

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

   15. Simplified 46.6%

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

   16. Taylor expanded in x around 0 48.3%

       \[\leadsto \frac{\frac{1}{s \cdot e^{\frac{x}{s}}}}{\color{blue}{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.5:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot e^{\frac{x}{s}}}}{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. associate-/r* 99.7%

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

   2. associate-*r/ 99.7%

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

   3. mul-1-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-*r/ 99.7%

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

   6. mul-1-neg 99.7%

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

7. Simplified 99.7%

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

   1. clear-num 99.7%

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

   2. inv-pow 99.7%

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

9. Applied egg-rr 58.2%

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

    1. unpow-1 58.2%

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

11. Simplified 58.2%

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

    1. distribute-frac-neg 99.7%

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

    2. rec-exp 99.7%

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

    3. add-sqr-sqrt 46.2%

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

    4. fabs-sqr 46.2%

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

    5. add-sqr-sqrt 58.2%

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

13. Applied egg-rr 58.7%

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

    1. rec-exp 58.7%

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

    2. distribute-neg-frac 58.7%

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

15. Simplified 58.7%

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

16. Final simplification 58.7%

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

Alternative 4: 30.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left|x_m\right|}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 5.000000058430487e-8) (/ 0.25 s) (/ 0.5 (fabs x_m))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 5.000000058430487e-8f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.5f / fabsf(x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x_m) <= 5.000000058430487e-8) then
        tmp = 0.25e0 / s
    else
        tmp = 0.5e0 / abs(x_m)
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(5.000000058430487e-8))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.5) / abs(x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (abs(x_m) <= single(5.000000058430487e-8))
		tmp = single(0.25) / s;
	else
		tmp = single(0.5) / abs(x_m);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 5.00000006e-8

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in s around inf 57.4%

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

   if 5.00000006e-8 < (fabs.f32 x)

   1. Initial program 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in s around inf 99.1%

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

   6. Taylor expanded in s around inf 11.2%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + 4 \cdot s}} \]

   7. Taylor expanded in s around 0 10.3%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left|x\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.9% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 94.7%

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

   1. distribute-lft-in 94.7%

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

   2. *-rgt-identity 94.7%

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

   3. fma-udef 94.7%

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

   4. *-rgt-identity 94.7%

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

   5. fma-udef 94.7%

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

   6. *-rgt-identity 94.7%

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

   7. add-sqr-sqrt 43.0%

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

   8. fabs-sqr 43.0%

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

   9. add-sqr-sqrt 55.7%

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

7. Applied egg-rr 55.7%

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

8. Final simplification 55.7%

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

Alternative 6: 94.9% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 94.7%

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

   1. distribute-rgt-in 94.7%

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

   2. *-un-lft-identity 94.7%

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

   3. fma-udef 94.7%

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

   4. *-rgt-identity 94.7%

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

   5. associate-+l+ 94.7%

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

   6. add-sqr-sqrt 43.0%

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

   7. fabs-sqr 43.0%

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

   8. add-sqr-sqrt 55.7%

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

   9. fma-udef 55.7%

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

   10. *-rgt-identity 55.7%

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

7. Applied egg-rr 55.7%

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

   1. associate-+r+ 55.7%

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

   2. distribute-rgt1-in 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-lft-out 55.7%

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

   5. distribute-rgt-out 55.7%

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

   6. count-2 55.7%

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

9. Simplified 55.7%

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

10. Final simplification 55.7%

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

Alternative 7: 94.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{1}{s \cdot e^{\frac{x_m}{s}}}}{4} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ (/ 1.0 (* s (exp (/ x_m s)))) 4.0))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (1.0f / (s * expf((x_m / 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 = (1.0e0 / (s * exp((x_m / s)))) / 4.0e0
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(1.0) / Float32(s * exp(Float32(x_m / s)))) / Float32(4.0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(1.0) / (s * exp((x_m / s)))) / single(4.0);
end

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. associate-/r* 99.7%

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

   2. associate-*r/ 99.7%

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

   3. mul-1-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-*r/ 99.7%

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

   6. mul-1-neg 99.7%

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

7. Simplified 99.7%

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

   1. clear-num 99.7%

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

   2. inv-pow 99.7%

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

9. Applied egg-rr 58.2%

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

    1. unpow-1 58.2%

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

11. Simplified 58.2%

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

    1. distribute-frac-neg 99.7%

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

    2. rec-exp 99.7%

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

    3. add-sqr-sqrt 46.2%

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

    4. fabs-sqr 46.2%

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

    5. add-sqr-sqrt 58.2%

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

13. Applied egg-rr 58.7%

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

    1. rec-exp 58.7%

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

    2. distribute-neg-frac 58.7%

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

15. Simplified 58.7%

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

16. Taylor expanded in x around 0 54.9%

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

17. Final simplification 54.9%

    \[\leadsto \frac{\frac{1}{s \cdot e^{\frac{x}{s}}}}{4} \]
18. Add Preprocessing

Alternative 8: 29.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\mathsf{fma}\left(x_m, 2, s \cdot 4\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 1.0 (fma x_m 2.0 (* s 4.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 1.0f / fmaf(x_m, 2.0f, (s * 4.0f));
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(1.0) / fma(x_m, Float32(2.0), Float32(s * Float32(4.0))))
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 94.7%

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

6. Taylor expanded in s around inf 28.9%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + 4 \cdot s}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 27.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 \cdot \left|x\right| + 4 \cdot s}\right)\right)} \]

   2. expm1-udef 63.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{2 \cdot \left|x\right| + 4 \cdot s}\right)} - 1} \]

   3. *-commutative 63.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left|x\right| \cdot 2} + 4 \cdot s}\right)} - 1 \]

   4. fma-def 63.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 4 \cdot s\right)}}\right)} - 1 \]

   5. add-sqr-sqrt 30.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 4 \cdot s\right)}\right)} - 1 \]

   6. fabs-sqr 30.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 4 \cdot s\right)}\right)} - 1 \]

   7. add-sqr-sqrt 61.8%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\color{blue}{x}, 2, 4 \cdot s\right)}\right)} - 1 \]

   8. *-commutative 61.8%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{s \cdot 4}\right)}\right)} - 1 \]

8. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(x, 2, s \cdot 4\right)}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 26.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(x, 2, s \cdot 4\right)}\right)\right)} \]

   2. expm1-log1p 28.7%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x, 2, s \cdot 4\right)}} \]

10. Simplified 28.7%

    \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x, 2, s \cdot 4\right)}} \]

11. Final simplification 28.7%

    \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, s \cdot 4\right)} \]
12. Add Preprocessing

Alternative 9: 27.4% 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 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 26.0%

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

6. Final simplification 26.0%

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

Reproduce

herbie shell --seed 2024010 
(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))))))