Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 18.1s

Alternatives: 8

Speedup: 1.0×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.5:\\ \;\;\;\;\frac{e^{\frac{x\_m}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x\_m}{-s}}}{s \cdot 4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 0.5)
   (/ (exp (+ (/ x_m s) (* -2.0 (log1p (exp (/ x_m s)))))) s)
   (/ (exp (/ x_m (- s))) (* s 4.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 0.5f) {
		tmp = expf(((x_m / s) + (-2.0f * log1pf(expf((x_m / s)))))) / s;
	} else {
		tmp = expf((x_m / -s)) / (s * 4.0f);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(0.5))
		tmp = Float32(exp(Float32(Float32(x_m / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x_m / s)))))) / s);
	else
		tmp = Float32(exp(Float32(x_m / Float32(-s))) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.5

   1. Initial program 99.1%

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

      1. fabs-neg 99.1%

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

      2. distribute-frac-neg 99.1%

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

      3. distribute-frac-neg2 99.1%

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

      4. fabs-neg 99.1%

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

      5. *-commutative 99.1%

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

      6. fabs-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. fabs-neg 99.1%

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

   3. Simplified 99.2%

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

   6. Applied egg-rr 84.1%

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

      1. *-lft-identity 84.1%

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

      2. *-commutative 84.1%

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

      3. exp-to-pow 84.0%

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

      4. log1p-undefine 84.1%

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

      5. *-commutative 84.1%

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

      6. rem-exp-log 79.0%

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

      7. prod-exp 79.3%

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

      8. exp-diff 93.5%

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

      9. associate--r+ 93.7%

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

      10. exp-diff 94.2%

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

      11. cancel-sign-sub-inv 94.2%

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

      12. metadata-eval 94.2%

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

      13. rem-exp-log 99.3%

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

   8. Simplified 99.3%

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

   if 0.5 < (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 s around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt 53.2%

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

      4. fabs-sqr 53.2%

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

      5. add-sqr-sqrt 54.7%

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

      6. add-sqr-sqrt 53.2%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 53.2%

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

      9. fabs-sqr 53.2%

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

      10. add-sqr-sqrt 53.2%

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

      11. add-sqr-sqrt 53.2%

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

      12. fabs-sqr 53.2%

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

      13. add-sqr-sqrt 100.0%

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

      14. sqr-neg 100.0%

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

      15. distribute-frac-neg 100.0%

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

      16. distribute-frac-neg 100.0%

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

      17. sqrt-unprod -0.0%

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

      18. add-sqr-sqrt 3.1%

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

   9. Applied egg-rr 54.7%

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

       1. rec-exp 54.7%

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

       2. distribute-frac-neg 54.7%

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

   11. Simplified 54.7%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{-s}}}{s \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x_m) (- s)))))
   (/ t_0 (* (+ t_0 1.0) (fma s t_0 s)))))

C

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

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(abs(x_m) / Float32(-s)))
	return Float32(t_0 / Float32(Float32(t_0 + Float32(1.0)) * fma(s, t_0, 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. distribute-lft-in 99.6%

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

   3. *-rgt-identity 99.6%

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

   4. fabs-neg 99.6%

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

   5. +-commutative 99.6%

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

   6. fma-define 99.7%

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

   7. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x_m) (- s)))))
   (/ t_0 (* (+ t_0 1.0) (+ s (/ s (exp (/ (fabs x_m) s))))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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 4: 94.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{\mathsf{fma}\left(x\_m \cdot s, 0.5, 0.5\right) + \left(x\_m \cdot s\right) \cdot -0.25}{s}}{e^{\frac{x\_m}{s}} + 1} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/
  (/ (+ (fma (* x_m s) 0.5 0.5) (* (* x_m s) -0.25)) s)
  (+ (exp (/ x_m s)) 1.0)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return ((fmaf((x_m * s), 0.5f, 0.5f) + ((x_m * s) * -0.25f)) / s) / (expf((x_m / s)) + 1.0f);
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(fma(Float32(x_m * s), Float32(0.5), Float32(0.5)) + Float32(Float32(x_m * s) * Float32(-0.25))) / s) / Float32(exp(Float32(x_m / s)) + Float32(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. 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

6. Applied egg-rr 82.6%

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

   1. add-log-exp 61.5%

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

   2. log1p-undefine 61.5%

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

   3. log-pow 61.2%

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

   4. sum-log 61.5%

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

   5. *-commutative 61.5%

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

   6. log-div 61.6%

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

   7. associate-/r* 61.6%

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

   8. add-exp-log 63.8%

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

   9. unpow2 63.8%

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

   10. associate-/r* 63.7%

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

8. Applied egg-rr 63.7%

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

9. Taylor expanded in s around inf 34.8%

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

    1. cancel-sign-sub-inv 34.8%

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

    2. +-commutative 34.8%

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

    3. *-commutative 34.8%

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

    4. fma-define 34.8%

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

    5. div-inv 34.8%

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

    6. add-exp-log 34.8%

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

    7. neg-log 34.8%

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

    8. add-sqr-sqrt 34.8%

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

    9. sqrt-unprod 34.8%

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

    10. sqr-neg 34.8%

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

    11. sqrt-unprod -0.0%

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

    12. add-sqr-sqrt 32.3%

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

    13. add-exp-log 32.3%

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

    14. metadata-eval 32.3%

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

    15. div-inv 32.3%

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

    16. add-exp-log 32.3%

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

    17. neg-log 32.3%

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

    18. add-sqr-sqrt 32.3%

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

    19. sqrt-unprod 32.3%

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

11. Applied egg-rr 59.0%

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

12. Final simplification 59.0%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot s, 0.5, 0.5\right) + \left(x \cdot s\right) \cdot -0.25}{s}}{e^{\frac{x}{s}} + 1} \]
13. Add Preprocessing

Alternative 5: 94.7% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

6. Applied egg-rr 82.6%

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

   1. add-log-exp 61.5%

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

   2. log1p-undefine 61.5%

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

   3. log-pow 61.2%

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

   4. sum-log 61.5%

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

   5. *-commutative 61.5%

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

   6. log-div 61.6%

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

   7. associate-/r* 61.6%

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

   8. add-exp-log 63.8%

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

   9. unpow2 63.8%

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

   10. associate-/r* 63.7%

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

8. Applied egg-rr 63.7%

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

9. Taylor expanded in x around 0 59.2%

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

10. Final simplification 59.2%

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

Alternative 6: 94.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 93.3%

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

   1. *-commutative 93.3%

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

7. Simplified 93.3%

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

   1. distribute-frac-neg 93.3%

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

   2. exp-neg 93.3%

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

   3. add-sqr-sqrt 45.8%

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

   4. fabs-sqr 45.8%

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

   5. add-sqr-sqrt 58.3%

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

   6. add-sqr-sqrt 45.8%

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

   7. sqrt-unprod 93.3%

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

   8. add-sqr-sqrt 45.8%

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

   9. fabs-sqr 45.8%

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

   10. add-sqr-sqrt 47.0%

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

   11. add-sqr-sqrt 45.8%

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

   12. fabs-sqr 45.8%

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

   13. add-sqr-sqrt 93.3%

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

   14. sqr-neg 93.3%

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

   15. distribute-frac-neg 93.3%

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

   16. distribute-frac-neg 93.3%

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

   17. sqrt-unprod -0.0%

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

   18. add-sqr-sqrt 23.7%

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

9. Applied egg-rr 58.3%

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

    1. rec-exp 58.3%

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

    2. distribute-frac-neg 58.3%

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

11. Simplified 58.3%

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

12. Final simplification 58.3%

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

Alternative 7: 64.6% accurate, 36.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\left(0.25 + \frac{x\_m \cdot -0.125}{s}\right) - 0.5 \cdot \left(\frac{x\_m}{s} \cdot -0.25\right)}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (- (+ 0.25 (/ (* x_m -0.125) s)) (* 0.5 (* (/ x_m s) -0.25))) s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return ((0.25f + ((x_m * -0.125f) / s)) - (0.5f * ((x_m / s) * -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 + ((x_m * (-0.125e0)) / s)) - (0.5e0 * ((x_m / s) * (-0.25e0)))) / s
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(Float32(0.25) + Float32(Float32(x_m * Float32(-0.125)) / s)) - Float32(Float32(0.5) * Float32(Float32(x_m / s) * Float32(-0.25)))) / s)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = ((single(0.25) + ((x_m * single(-0.125)) / s)) - (single(0.5) * ((x_m / s) * 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. 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

6. Applied egg-rr 82.6%

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

   1. add-log-exp 61.5%

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

   2. log1p-undefine 61.5%

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

   3. log-pow 61.2%

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

   4. sum-log 61.5%

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

   5. *-commutative 61.5%

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

   6. log-div 61.6%

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

   7. associate-/r* 61.6%

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

   8. add-exp-log 63.8%

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

   9. unpow2 63.8%

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

   10. associate-/r* 63.7%

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

8. Applied egg-rr 63.7%

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

9. Taylor expanded in s around inf 34.8%

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

10. Taylor expanded in s around -inf 62.9%

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

    1. mul-1-neg 62.9%

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

    2. distribute-rgt-out-- 62.9%

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

    3. metadata-eval 62.9%

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

    4. *-commutative 62.9%

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

    5. associate-*r/ 62.9%

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

    6. associate-*r/ 62.9%

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

    7. *-commutative 62.9%

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

12. Simplified 62.9%

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

13. Final simplification 62.9%

    \[\leadsto \frac{\left(0.25 + \frac{x \cdot -0.125}{s}\right) - 0.5 \cdot \left(\frac{x}{s} \cdot -0.25\right)}{s} \]
14. Add Preprocessing

Alternative 8: 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 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 26.4%

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

6. Final simplification 26.4%

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

Reproduce

herbie shell --seed 2024071 
(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))))))