Logistic distribution

Percentage Accurate: 99.6% → 99.3%

Time: 16.9s

Alternatives: 9

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% 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: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{s}}{\left(1 + \left(1 + \mathsf{expm1}\left(\frac{-x}{s}\right)\right)\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/
  (/ 1.0 s)
  (* (+ 1.0 (+ 1.0 (expm1 (/ (- x) s)))) (+ 1.0 (exp (/ (fabs x) s))))))

C

x = abs(x);
float code(float x, float s) {
	return (1.0f / s) / ((1.0f + (1.0f + expm1f((-x / s)))) * (1.0f + expf((fabsf(x) / s))));
}

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(1.0) + Float32(Float32(1.0) + expm1(Float32(Float32(-x) / s)))) * Float32(Float32(1.0) + exp(Float32(abs(x) / s)))))
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around 0 99.8%

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

   1. associate-/r* 99.9%

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

   2. mul-1-neg 99.9%

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

   3. distribute-frac-neg 99.9%

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

6. Simplified 99.9%

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

   1. expm1-log1p-u 99.8%

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

   2. expm1-udef 99.8%

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

   3. log1p-udef 99.9%

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

   4. add-exp-log 99.9%

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

8. Applied egg-rr 99.9%

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

   1. associate--l+ 99.9%

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

   2. distribute-frac-neg 99.9%

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

   3. mul-1-neg 99.9%

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

   4. expm1-def 99.9%

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

   5. mul-1-neg 99.9%

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

   6. distribute-frac-neg 99.9%

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

10. Simplified 99.9%

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

11. Taylor expanded in x around 0 99.9%

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

    1. expm1-def 99.9%

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

    2. associate-*r/ 99.9%

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

    3. neg-mul-1 99.9%

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

    4. unpow1 99.9%

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

    5. sqr-pow 46.4%

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

    6. fabs-sqr 46.4%

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

    7. sqr-pow 97.8%

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

    8. unpow1 97.8%

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

13. Simplified 97.8%

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

14. Final simplification 97.8%

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

Alternative 2: 99.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around 0 99.8%

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

   1. associate-/r* 99.9%

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

   2. mul-1-neg 99.9%

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

   3. distribute-frac-neg 99.9%

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

6. Simplified 99.9%

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

   1. expm1-log1p-u 99.8%

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

   2. expm1-udef 99.8%

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

   3. log1p-udef 99.9%

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

   4. add-exp-log 99.9%

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

8. Applied egg-rr 99.9%

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

   1. associate--l+ 99.9%

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

   2. distribute-frac-neg 99.9%

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

   3. mul-1-neg 99.9%

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

   4. expm1-def 99.9%

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

   5. mul-1-neg 99.9%

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

   6. distribute-frac-neg 99.9%

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

10. Simplified 99.9%

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

11. Taylor expanded in x around 0 99.9%

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

    1. associate-*r/ 99.9%

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

    2. neg-mul-1 99.9%

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

    3. unpow1 99.9%

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

    4. sqr-pow 46.4%

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

    5. fabs-sqr 46.4%

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

    6. sqr-pow 97.7%

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

    7. unpow1 97.7%

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

13. Simplified 97.7%

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

14. Final simplification 97.7%

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

Alternative 3: 95.2% accurate, 3.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.5 (+ s (* s (pow E (/ x s))))))

C

x = abs(x);
float code(float x, float s) {
	return 0.5f / (s + (s * powf(((float) M_E), (x / s))));
}

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(0.5) / Float32(s + Float32(s * (Float32(exp(1)) ^ Float32(x / s)))))
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 96.0%

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

5. Taylor expanded in x around 0 96.0%

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

   1. expm1-log1p-u 96.0%

      \[\leadsto \frac{0.5}{s + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)\right)}} \]

   2. expm1-udef 82.0%

      \[\leadsto \frac{0.5}{s + \color{blue}{\left(e^{\mathsf{log1p}\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)} - 1\right)}} \]

7. Applied egg-rr 82.0%

   \[\leadsto \frac{0.5}{s + \color{blue}{\left(e^{\mathsf{log1p}\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)} - 1\right)}} \]
8. Step-by-step derivation

   1. expm1-def 96.0%

      \[\leadsto \frac{0.5}{s + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)\right)}} \]

   2. expm1-log1p 96.0%

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

   3. unpow1 96.0%

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

   4. sqr-pow 44.4%

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

   5. fabs-sqr 44.4%

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

   6. sqr-pow 60.4%

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

   7. unpow1 60.4%

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

   8. *-lft-identity 60.4%

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

   9. *-lft-identity 60.4%

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

9. Simplified 60.4%

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

    1. *-un-lft-identity 60.4%

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

    2. exp-prod 60.4%

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

    3. exp-1-e 60.4%

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

11. Applied egg-rr 60.4%

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

12. Final simplification 60.4%

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

Alternative 4: 95.1% accurate, 3.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ (/ 0.5 s) (+ 1.0 (exp (/ (fabs x) s)))))

C

x = abs(x);
float code(float x, float s) {
	return (0.5f / s) / (1.0f + 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 = (0.5e0 / s) / (1.0e0 + exp((abs(x) / s)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 96.0%

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

5. Taylor expanded in s around 0 96.0%

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

   1. associate-/r* 96.0%

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

7. Simplified 96.0%

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

8. Final simplification 96.0%

   \[\leadsto \frac{\frac{0.5}{s}}{1 + e^{\frac{\left|x\right|}{s}}} \]

Alternative 5: 95.2% accurate, 5.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.5 (+ s (* s (exp (/ x s))))))

C

x = abs(x);
float code(float x, float s) {
	return 0.5f / (s + (s * expf((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 = 0.5e0 / (s + (s * exp((x / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 96.0%

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

5. Taylor expanded in x around 0 96.0%

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

   1. expm1-log1p-u 96.0%

      \[\leadsto \frac{0.5}{s + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)\right)}} \]

   2. expm1-udef 82.0%

      \[\leadsto \frac{0.5}{s + \color{blue}{\left(e^{\mathsf{log1p}\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)} - 1\right)}} \]

7. Applied egg-rr 82.0%

   \[\leadsto \frac{0.5}{s + \color{blue}{\left(e^{\mathsf{log1p}\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)} - 1\right)}} \]
8. Step-by-step derivation

   1. expm1-def 96.0%

      \[\leadsto \frac{0.5}{s + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)\right)}} \]

   2. expm1-log1p 96.0%

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

   3. unpow1 96.0%

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

   4. sqr-pow 44.4%

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

   5. fabs-sqr 44.4%

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

   6. sqr-pow 60.4%

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

   7. unpow1 60.4%

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

   8. *-lft-identity 60.4%

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

   9. *-lft-identity 60.4%

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

9. Simplified 60.4%

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

10. Final simplification 60.4%

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

Alternative 6: 92.1% accurate, 32.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.1999999952050366 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 5.000000156871975e-23)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   (if (<= x 1.1999999952050366e-11)
     (/ (/ 1.0 s) (+ 4.0 (* (* x x) (/ 1.0 (* s s)))))
     0.0)))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 5.000000156871975e-23f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else if (x <= 1.1999999952050366e-11f) {
		tmp = (1.0f / s) / (4.0f + ((x * x) * (1.0f / (s * s))));
	} else {
		tmp = 0.0f;
	}
	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 <= 5.000000156871975e-23) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else if (x <= 1.1999999952050366e-11) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) * (1.0e0 / (s * s))))
    else
        tmp = 0.0e0
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(5.000000156871975e-23))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	elseif (x <= Float32(1.1999999952050366e-11))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) * Float32(Float32(1.0) / Float32(s * s)))));
	else
		tmp = Float32(0.0);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.000000156871975e-23))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	elseif (x <= single(1.1999999952050366e-11))
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) * (single(1.0) / (s * s))));
	else
		tmp = single(0.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < 5.00000016e-23

   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 99.8%

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

   4. Taylor expanded in s around 0 99.8%

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

      1. associate-/r* 99.9%

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

      2. mul-1-neg 99.9%

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

      3. distribute-frac-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in s around -inf 28.6%

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

      1. associate-+r+ 28.6%

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

   9. Simplified 72.4%

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

   10. Taylor expanded in x around 0 72.4%

       \[\leadsto \frac{\frac{1}{s}}{4 + \color{blue}{\frac{{x}^{2}}{{s}^{2}}}} \]
   11. Step-by-step derivation

       1. unpow2 72.4%

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

       2. unpow2 72.4%

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

       3. times-frac 78.9%

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

   12. Simplified 78.9%

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

   if 5.00000016e-23 < x < 1.2e-11

   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 99.5%

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

   4. Taylor expanded in s around 0 99.8%

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

      1. associate-/r* 99.8%

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

      2. mul-1-neg 99.8%

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

      3. distribute-frac-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in s around -inf 40.9%

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

      1. associate-+r+ 40.9%

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

   9. Simplified 87.1%

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

       1. div-inv 92.6%

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

   11. Applied egg-rr 92.6%

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

   if 1.2e-11 < x

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in s around inf 98.1%

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

   5. Taylor expanded in x around 0 98.1%

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

      1. add-exp-log 98.1%

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

      2. *-commutative 98.1%

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

      3. log-prod 98.1%

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

      4. add-log-exp 98.1%

         \[\leadsto \frac{0.5}{s + e^{\color{blue}{\frac{\left|x\right|}{s}} + \log s}} \]

   7. Applied egg-rr 98.1%

      \[\leadsto \frac{0.5}{s + \color{blue}{e^{\frac{\left|x\right|}{s} + \log s}}} \]

   8. Taylor expanded in s around inf 96.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.1999999952050366 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 86.8% accurate, 41.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1999999952050366 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 1.1999999952050366e-11)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   0.0))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 1.1999999952050366e-11f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else {
		tmp = 0.0f;
	}
	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 <= 1.1999999952050366e-11) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else
        tmp = 0.0e0
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.1999999952050366e-11))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	else
		tmp = Float32(0.0);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.1999999952050366e-11))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	else
		tmp = single(0.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.2e-11

   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 99.8%

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

   4. Taylor expanded in s around 0 99.8%

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

      1. associate-/r* 99.9%

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

      2. mul-1-neg 99.9%

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

      3. distribute-frac-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in s around -inf 29.5%

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

      1. associate-+r+ 29.5%

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

   9. Simplified 73.5%

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

   10. Taylor expanded in x around 0 73.5%

       \[\leadsto \frac{\frac{1}{s}}{4 + \color{blue}{\frac{{x}^{2}}{{s}^{2}}}} \]
   11. Step-by-step derivation

       1. unpow2 73.5%

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

       2. unpow2 73.5%

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

       3. times-frac 77.4%

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

   12. Simplified 77.4%

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

   if 1.2e-11 < x

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in s around inf 98.1%

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

   5. Taylor expanded in x around 0 98.1%

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

      1. add-exp-log 98.1%

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

      2. *-commutative 98.1%

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

      3. log-prod 98.1%

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

      4. add-log-exp 98.1%

         \[\leadsto \frac{0.5}{s + e^{\color{blue}{\frac{\left|x\right|}{s}} + \log s}} \]

   7. Applied egg-rr 98.1%

      \[\leadsto \frac{0.5}{s + \color{blue}{e^{\frac{\left|x\right|}{s} + \log s}}} \]

   8. Taylor expanded in s around inf 96.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1999999952050366 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 85.7% accurate, 121.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 4.99999991225835e-14) (/ 0.25 s) 0.0))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 4.99999991225835e-14f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.0f;
	}
	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 <= 4.99999991225835e-14) then
        tmp = 0.25e0 / s
    else
        tmp = 0.0e0
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.99999991225835e-14))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(0.0);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.99999991225835e-14))
		tmp = single(0.25) / s;
	else
		tmp = single(0.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999991e-14

   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 99.7%

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

   4. Taylor expanded in s around inf 38.5%

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

   if 4.99999991e-14 < x

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in s around inf 98.1%

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

   5. Taylor expanded in x around 0 98.2%

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

      1. add-exp-log 98.2%

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

      2. *-commutative 98.2%

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

      3. log-prod 98.1%

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

      4. add-log-exp 98.1%

         \[\leadsto \frac{0.5}{s + e^{\color{blue}{\frac{\left|x\right|}{s}} + \log s}} \]

   7. Applied egg-rr 98.1%

      \[\leadsto \frac{0.5}{s + \color{blue}{e^{\frac{\left|x\right|}{s} + \log s}}} \]

   8. Taylor expanded in s around inf 95.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 73.2% accurate, 620.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 0 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 0.0)

C

x = abs(x);
float code(float x, float s) {
	return 0.0f;
}

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 = 0.0e0
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(0.0)
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 96.0%

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

5. Taylor expanded in x around 0 96.0%

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

   1. add-exp-log 95.0%

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

   2. *-commutative 95.0%

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

   3. log-prod 95.0%

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

   4. add-log-exp 95.0%

      \[\leadsto \frac{0.5}{s + e^{\color{blue}{\frac{\left|x\right|}{s}} + \log s}} \]

7. Applied egg-rr 95.0%

   \[\leadsto \frac{0.5}{s + \color{blue}{e^{\frac{\left|x\right|}{s} + \log s}}} \]

8. Taylor expanded in s around inf 72.9%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 72.9%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023283 
(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))))))