Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 10.1s

Alternatives: 7

Speedup: 2.0×

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, 2.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* s (+ (+ (exp (/ x s)) 2.0) (exp (/ (fabs x) (- s)))))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * ((exp((x / s)) + 2.0e0) + exp((abs(x) / -s))))
end function

Julia

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

MATLAB

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

Derivation

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 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. exp-prod 99.1%

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

   3. exp-1-e 99.1%

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

   4. add-sqr-sqrt 46.8%

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

   5. fabs-sqr 46.8%

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

   6. add-sqr-sqrt 60.3%

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

5. Applied egg-rr 60.3%

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

   1. expm1-log1p-u 58.8%

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

   2. expm1-udef 58.9%

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

   3. pow-to-exp 58.8%

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

   4. e-exp-1 58.8%

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

   5. add-log-exp 58.9%

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

   6. *-un-lft-identity 58.9%

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

   7. +-commutative 58.9%

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

7. Applied egg-rr 58.9%

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

   1. expm1-def 58.9%

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

   2. expm1-log1p 60.3%

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

   3. associate-/l/ 60.3%

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

   4. *-commutative 60.3%

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

   5. associate-+r+ 60.3%

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

9. Simplified 60.3%

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

10. Final simplification 60.3%

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

Alternative 2: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;\left|x\right| \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(t_0\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= (fabs x) 1.9999999494757503e-5)
     (/ (exp (+ (/ x s) (* -2.0 (log1p t_0)))) s)
     (/ (/ 1.0 s) (+ t_0 3.0)))))

C

x = abs(x);
float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (fabsf(x) <= 1.9999999494757503e-5f) {
		tmp = expf(((x / s) + (-2.0f * log1pf(t_0)))) / s;
	} else {
		tmp = (1.0f / s) / (t_0 + 3.0f);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.9999999494757503e-5))
		tmp = Float32(exp(Float32(Float32(x / s) + Float32(Float32(-2.0) * log1p(t_0)))) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(t_0 + Float32(3.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1.99999995e-5

   1. Initial program 98.5%

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

      1. associate-*l* 98.5%

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

      2. +-commutative 98.5%

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

      3. +-commutative 98.5%

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

   3. Simplified 98.5%

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

      1. add-exp-log 94.5%

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

      2. log-div 94.5%

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

      3. add-log-exp 95.1%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 47.4%

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

      6. sqr-neg 47.4%

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

      7. sqrt-unprod 47.1%

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

      8. add-sqr-sqrt 47.1%

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

      9. add-sqr-sqrt 22.6%

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

      10. fabs-sqr 22.6%

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

      11. add-sqr-sqrt 72.0%

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

      12. *-commutative 72.0%

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

   5. Applied egg-rr 94.3%

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

      1. associate--r+ 94.4%

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

      2. exp-diff 94.4%

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

      3. cancel-sign-sub-inv 94.4%

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

      4. metadata-eval 94.4%

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

      5. rem-exp-log 98.8%

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

   7. Simplified 98.8%

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

   if 1.99999995e-5 < (fabs.f32 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 100.0%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{3}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}} + 3} \]

      2. sqrt-unprod 100.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s \cdot s}}}} + 3} \]

      3. sqr-neg 100.0%

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

      4. sqrt-unprod -0.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}} + 3} \]

      5. add-sqr-sqrt 4.6%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{-s}}} + 3} \]

      6. expm1-log1p-u 4.6%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)\right)} + 3} \]

      7. expm1-udef 4.6%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)} - 1\right)} + 3} \]

   6. Applied egg-rr 48.6%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\left(e^{\frac{x}{s}} + 1\right) - 1\right)} + 3} \]
   7. Step-by-step derivation

      1. associate--l+ 48.6%

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

      2. metadata-eval 48.6%

         \[\leadsto \frac{\frac{1}{s}}{\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 3} \]

      3. +-rgt-identity 48.6%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

   8. Simplified 48.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \]

Alternative 3: 96.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{1}{s \cdot \left(3 + e^{\frac{\left|x\right|}{s}}\right)} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 1.0 (* s (+ 3.0 (exp (/ (fabs x) s))))))

C

x = abs(x);
float code(float x, float s) {
	return 1.0f / (s * (3.0f + expf((fabsf(x) / s))));
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * (3.0e0 + exp((abs(x) / s))))
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(3.0) + exp(Float32(abs(x) / s)))))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = single(1.0) / (s * (single(3.0) + exp((abs(x) / s))));
end

Derivation

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 99.0%

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

4. Taylor expanded in s around inf 95.1%

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{3}} \]

5. Taylor expanded in s around 0 95.5%

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

6. Final simplification 95.5%

   \[\leadsto \frac{1}{s \cdot \left(3 + e^{\frac{\left|x\right|}{s}}\right)} \]

Alternative 4: 96.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ (/ 1.0 s) (+ (exp (/ x s)) 3.0)))

C

x = abs(x);
float code(float x, float s) {
	return (1.0f / s) / (expf((x / s)) + 3.0f);
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / (exp((x / s)) + 3.0e0)
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(exp(Float32(x / s)) + Float32(3.0)))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = (single(1.0) / s) / (exp((x / s)) + single(3.0));
end

Derivation

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 99.0%

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

4. Taylor expanded in s around inf 95.1%

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{3}} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 95.1%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}} + 3} \]

   2. sqrt-unprod 92.1%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s \cdot s}}}} + 3} \]

   3. sqr-neg 92.1%

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

   4. sqrt-unprod -0.0%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}} + 3} \]

   5. add-sqr-sqrt 26.1%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{-s}}} + 3} \]

   6. expm1-log1p-u 26.1%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)\right)} + 3} \]

   7. expm1-udef 26.1%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)} - 1\right)} + 3} \]

6. Applied egg-rr 58.7%

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\left(e^{\frac{x}{s}} + 1\right) - 1\right)} + 3} \]
7. Step-by-step derivation

   1. associate--l+ 58.7%

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

   2. metadata-eval 58.7%

      \[\leadsto \frac{\frac{1}{s}}{\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 3} \]

   3. +-rgt-identity 58.7%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

8. Simplified 58.7%

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

9. Final simplification 58.7%

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3} \]

Alternative 5: 64.4% accurate, 41.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25 + \left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{x}{s} \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999494757503e-5)
   (/ (+ 0.25 (* (* (/ x s) (/ x s)) -0.0625)) s)
   (/ 1.0 (* x (* (/ x s) 0.5)))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999494757503e-5f) {
		tmp = (0.25f + (((x / s) * (x / s)) * -0.0625f)) / s;
	} else {
		tmp = 1.0f / (x * ((x / s) * 0.5f));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.9999999494757503e-5) then
        tmp = (0.25e0 + (((x / s) * (x / s)) * (-0.0625e0))) / s
    else
        tmp = 1.0e0 / (x * ((x / s) * 0.5e0))
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999494757503e-5))
		tmp = Float32(Float32(Float32(0.25) + Float32(Float32(Float32(x / s) * Float32(x / s)) * Float32(-0.0625))) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(x / s) * Float32(0.5))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999494757503e-5))
		tmp = (single(0.25) + (((x / s) * (x / s)) * single(-0.0625))) / s;
	else
		tmp = single(1.0) / (x * ((x / s) * single(0.5)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999995e-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. associate-*l* 99.1%

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

      2. +-commutative 99.1%

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

      3. +-commutative 99.1%

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

   3. Simplified 99.1%

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

      1. *-un-lft-identity 99.1%

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

      2. times-frac 98.6%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 31.5%

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

      5. sqr-neg 31.5%

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

      6. sqrt-unprod 31.2%

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

      7. add-sqr-sqrt 31.2%

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

      8. add-sqr-sqrt 14.3%

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

      9. fabs-sqr 14.3%

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

      10. add-sqr-sqrt 84.0%

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

      11. pow2 84.0%

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

   5. Applied egg-rr 86.7%

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

   6. Taylor expanded in x around 0 31.3%

      \[\leadsto \frac{1}{s} \cdot \color{blue}{\left(0.25 + -0.0625 \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 31.3%

         \[\leadsto \frac{1}{s} \cdot \left(0.25 + \color{blue}{\frac{{x}^{2}}{{s}^{2}} \cdot -0.0625}\right) \]

      2. unpow2 31.3%

         \[\leadsto \frac{1}{s} \cdot \left(0.25 + \frac{\color{blue}{x \cdot x}}{{s}^{2}} \cdot -0.0625\right) \]

      3. unpow2 31.3%

         \[\leadsto \frac{1}{s} \cdot \left(0.25 + \frac{x \cdot x}{\color{blue}{s \cdot s}} \cdot -0.0625\right) \]

   8. Simplified 31.3%

      \[\leadsto \frac{1}{s} \cdot \color{blue}{\left(0.25 + \frac{x \cdot x}{s \cdot s} \cdot -0.0625\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 31.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(0.25 + \frac{x \cdot x}{s \cdot s} \cdot -0.0625\right)}{s}} \]

      2. *-un-lft-identity 31.3%

         \[\leadsto \frac{\color{blue}{0.25 + \frac{x \cdot x}{s \cdot s} \cdot -0.0625}}{s} \]

   10. Applied egg-rr 31.3%

       \[\leadsto \color{blue}{\frac{0.25 + \frac{x \cdot x}{s \cdot s} \cdot -0.0625}{s}} \]

   11. Taylor expanded in x around 0 31.3%

       \[\leadsto \frac{0.25 + \color{blue}{\frac{{x}^{2}}{{s}^{2}}} \cdot -0.0625}{s} \]
   12. Step-by-step derivation

       1. unpow2 31.3%

          \[\leadsto \frac{0.25 + \frac{\color{blue}{x \cdot x}}{{s}^{2}} \cdot -0.0625}{s} \]

       2. unpow2 31.3%

          \[\leadsto \frac{0.25 + \frac{x \cdot x}{\color{blue}{s \cdot s}} \cdot -0.0625}{s} \]

       3. times-frac 35.8%

          \[\leadsto \frac{0.25 + \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot -0.0625}{s} \]

   13. Simplified 35.8%

       \[\leadsto \frac{0.25 + \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot -0.0625}{s} \]

   if 1.99999995e-5 < 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%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

      3. exp-1-e 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. fabs-sqr 100.0%

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

      6. add-sqr-sqrt 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. pow-to-exp 100.0%

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

      4. e-exp-1 100.0%

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

      5. add-log-exp 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-/l/ 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 81.4%

       \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} + e^{\frac{\left|x\right|}{-s}}\right)} \]
   11. Step-by-step derivation

       1. unpow2 81.4%

          \[\leadsto \frac{1}{s \cdot \left(\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}\right)\right) + e^{\frac{\left|x\right|}{-s}}\right)} \]

       2. unpow2 81.4%

          \[\leadsto \frac{1}{s \cdot \left(\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)\right) + e^{\frac{\left|x\right|}{-s}}\right)} \]

       3. times-frac 81.4%

          \[\leadsto \frac{1}{s \cdot \left(\left(3 + \left(\frac{x}{s} + 0.5 \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)}\right)\right) + e^{\frac{\left|x\right|}{-s}}\right)} \]

   12. Simplified 81.4%

       \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)\right)\right)} + e^{\frac{\left|x\right|}{-s}}\right)} \]

   13. Taylor expanded in s around 0 66.5%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{{x}^{2}}{s}}} \]
   14. Step-by-step derivation

       1. *-commutative 66.5%

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

       2. unpow2 66.5%

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

       3. associate-*r/ 66.5%

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

       4. associate-*l* 66.5%

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

       5. *-commutative 66.5%

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

   15. Simplified 66.5%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25 + \left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{x}{s} \cdot 0.5\right)}\\ \end{array} \]

Alternative 6: 63.8% accurate, 55.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{x}{s} \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999494757503e-5) (/ 0.25 s) (/ 1.0 (* x (* (/ x s) 0.5)))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999494757503e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (x * ((x / s) * 0.5f));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.9999999494757503e-5) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x * ((x / s) * 0.5e0))
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999494757503e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(x / s) * Float32(0.5))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999494757503e-5))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x * ((x / s) * single(0.5)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999995e-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. associate-*l* 99.1%

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

      2. +-commutative 99.1%

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

      3. +-commutative 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in s around inf 35.8%

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

   if 1.99999995e-5 < 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%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

      3. exp-1-e 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. fabs-sqr 100.0%

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

      6. add-sqr-sqrt 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. pow-to-exp 100.0%

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

      4. e-exp-1 100.0%

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

      5. add-log-exp 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-/l/ 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 81.4%

       \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} + e^{\frac{\left|x\right|}{-s}}\right)} \]
   11. Step-by-step derivation

       1. unpow2 81.4%

          \[\leadsto \frac{1}{s \cdot \left(\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}\right)\right) + e^{\frac{\left|x\right|}{-s}}\right)} \]

       2. unpow2 81.4%

          \[\leadsto \frac{1}{s \cdot \left(\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)\right) + e^{\frac{\left|x\right|}{-s}}\right)} \]

       3. times-frac 81.4%

          \[\leadsto \frac{1}{s \cdot \left(\left(3 + \left(\frac{x}{s} + 0.5 \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)}\right)\right) + e^{\frac{\left|x\right|}{-s}}\right)} \]

   12. Simplified 81.4%

       \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)\right)\right)} + e^{\frac{\left|x\right|}{-s}}\right)} \]

   13. Taylor expanded in s around 0 66.5%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{{x}^{2}}{s}}} \]
   14. Step-by-step derivation

       1. *-commutative 66.5%

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

       2. unpow2 66.5%

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

       3. associate-*r/ 66.5%

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

       4. associate-*l* 66.5%

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

       5. *-commutative 66.5%

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

   15. Simplified 66.5%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{x}{s} \cdot 0.5\right)}\\ \end{array} \]

Alternative 7: 27.1% accurate, 206.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.25 s))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

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

MATLAB

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

Derivation

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. associate-*l* 99.3%

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

   2. +-commutative 99.3%

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

   3. +-commutative 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in s around inf 28.0%

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

5. Final simplification 28.0%

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

Reproduce

herbie shell --seed 2023224 
(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))))))