Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 11.1s

Alternatives: 7

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 96.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. associate-/r* 99.9%

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

3. Simplified 99.8%

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

   1. remove-double-neg 99.8%

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

   2. frac-2neg 99.8%

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

   3. add-sqr-sqrt -0.0%

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

   4. add-sqr-sqrt 99.8%

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

   5. add-sqr-sqrt 99.8%

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

   6. sqrt-unprod 97.6%

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

   7. sqr-neg 97.6%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 96.2%

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

   10. distribute-frac-neg 96.2%

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

   11. exp-neg 96.2%

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

   12. remove-double-neg 96.2%

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

   13. frac-2neg 96.2%

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

   14. add-sqr-sqrt -0.0%

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

5. Applied egg-rr 97.4%

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

   1. rec-exp 97.4%

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

7. Simplified 97.4%

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

8. Final simplification 97.4%

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

Alternative 4: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0008500000112690032:\\ \;\;\;\;\frac{1}{\frac{s}{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 0.0008500000112690032)
   (/ 1.0 (/ s (exp (+ (/ x s) (* -2.0 (log1p (exp (/ x s))))))))
   (/ (exp (/ (fabs x) (- s))) (* s 4.0))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 0.0008500000112690032f) {
		tmp = 1.0f / (s / expf(((x / s) + (-2.0f * log1pf(expf((x / s)))))));
	} else {
		tmp = expf((fabsf(x) / -s)) / (s * 4.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.0008500000112690032))
		tmp = Float32(Float32(1.0) / Float32(s / exp(Float32(Float32(x / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x / s))))))));
	else
		tmp = Float32(exp(Float32(abs(x) / Float32(-s))) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 8.50000011e-4

   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%

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

   5. Applied egg-rr 79.7%

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

      1. frac-times 79.6%

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

      2. *-commutative 79.6%

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

      3. associate-/l* 79.7%

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

      4. associate-/l* 79.7%

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

      5. un-div-inv 79.7%

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

      6. add-exp-log 79.6%

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

      7. prod-exp 99.5%

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

      8. pow-flip 99.5%

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

      9. log-pow 99.5%

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

      10. metadata-eval 99.5%

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

      11. log1p-def 99.6%

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

   7. Applied egg-rr 99.6%

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

   if 8.50000011e-4 < (fabs.f32 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0008500000112690032:\\ \;\;\;\;\frac{1}{\frac{s}{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot 4}\\ \end{array} \]

Alternative 5: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0008500000112690032:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 0.0008500000112690032)
   (/ (exp (+ (/ x s) (* -2.0 (log1p (exp (/ x s)))))) s)
   (/ (exp (/ (fabs x) (- s))) (* s 4.0))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 0.0008500000112690032f) {
		tmp = expf(((x / s) + (-2.0f * log1pf(expf((x / s)))))) / s;
	} else {
		tmp = expf((fabsf(x) / -s)) / (s * 4.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.0008500000112690032))
		tmp = Float32(exp(Float32(Float32(x / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x / s)))))) / s);
	else
		tmp = Float32(exp(Float32(abs(x) / Float32(-s))) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 8.50000011e-4

   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. associate-/r* 99.6%

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

   3. Simplified 99.5%

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

      1. remove-double-neg 99.5%

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

      2. frac-2neg 99.5%

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

      3. add-sqr-sqrt -0.0%

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

      4. add-sqr-sqrt 99.5%

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

      5. add-sqr-sqrt 99.5%

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

      6. sqrt-unprod 94.3%

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

      7. sqr-neg 94.3%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt 90.8%

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

      10. distribute-frac-neg 90.8%

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

      11. exp-neg 90.8%

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

      12. remove-double-neg 90.8%

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

      13. frac-2neg 90.8%

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

      14. add-sqr-sqrt -0.0%

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

   5. Applied egg-rr 93.7%

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

      1. rec-exp 93.6%

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

   7. Simplified 93.6%

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

   9. Applied egg-rr 95.3%

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

       1. expm1-def 95.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\right)\right)} \]

       2. expm1-log1p 99.6%

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

   11. Simplified 99.6%

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

   if 8.50000011e-4 < (fabs.f32 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0008500000112690032:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot 4}\\ \end{array} \]

Alternative 6: 94.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 99.8%

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

4. Taylor expanded in s around inf 96.3%

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

   1. *-commutative 96.3%

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

6. Simplified 96.3%

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

7. Final simplification 96.3%

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

Alternative 7: 27.5% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 25.1%

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

5. Final simplification 25.1%

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

Reproduce

herbie shell --seed 2023326 
(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))))))