Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 12.7s

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: 99.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* s (+ (+ (exp (/ (fabs x) (- s))) (exp (/ (fabs x) s))) 2.0))))

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * ((exp((abs(x) / -s)) + exp((abs(x) / s))) + 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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. expm1-log1p-u 97.1%

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

   2. expm1-udef 96.8%

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

   3. associate-/l/ 97.6%

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

5. Applied egg-rr 97.6%

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

   1. expm1-def 97.8%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (+ 1.0 (exp (/ (- x) s))) (fma s (exp (/ x s)) s))))

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. *-lft-identity 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-/l* 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. exp-neg 99.7%

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

   6. associate-/r/ 99.7%

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

   7. /-rgt-identity 99.7%

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

   8. associate-*l* 99.7%

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

3. Simplified 99.8%

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

   1. fma-udef 99.7%

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

5. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. fma-def 99.8%

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

   3. add-sqr-sqrt 51.8%

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

   4. fabs-sqr 51.8%

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

   5. add-sqr-sqrt 64.3%

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

7. Applied egg-rr 64.3%

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

   1. fma-udef 64.3%

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

9. Applied egg-rr 64.3%

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

10. Taylor expanded in x around inf 64.3%

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

    1. +-commutative 64.3%

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

    2. fma-udef 64.3%

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

    3. associate-*r/ 64.3%

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

    4. mul-1-neg 64.3%

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

    5. unpow1 64.3%

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

    6. sqr-pow 51.9%

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

    7. fabs-sqr 51.9%

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

    8. sqr-pow 99.8%

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

    9. unpow1 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 3: 51.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{\frac{-x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 9.99999993922529e-9)
   (/ 0.25 s)
   (* (/ 1.0 s) (/ 1.0 (/ (- x) s)))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 9.99999993922529e-9f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / s) * (1.0f / (-x / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 9.99999993922529e-9) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / s) * (1.0e0 / (-x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(9.99999993922529e-9))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) * Float32(Float32(1.0) / Float32(Float32(-x) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(9.99999993922529e-9))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / s) * (single(1.0) / (-x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 9.99999994e-9

   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 99.4%

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

   4. Taylor expanded in s around inf 57.1%

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

   if 9.99999994e-9 < (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 98.6%

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

   4. Taylor expanded in s around inf 59.2%

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

      1. mul-1-neg 59.2%

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

      2. unsub-neg 59.2%

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

   6. Simplified 59.2%

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

      1. div-inv 59.2%

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

      2. +-commutative 59.2%

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

   8. Applied egg-rr 59.2%

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

   9. Taylor expanded in s around 0 44.0%

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

       1. associate-*r/ 44.0%

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

       2. mul-1-neg 44.0%

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

       3. unpow1 44.0%

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

       4. sqr-pow 25.2%

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

       5. fabs-sqr 25.2%

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

       6. sqr-pow 44.0%

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

       7. unpow1 44.0%

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

   11. Simplified 44.0%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 4: 61.8% accurate, 5.7× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ (/ 1.0 (+ (exp (/ x s)) 3.0)) s))

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / (exp((x / s)) + 3.0e0)) / s
end function

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / (exp((x / s)) + single(3.0))) / 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 99.0%

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

   1. add-sqr-sqrt 98.6%

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

   2. pow2 98.6%

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

   3. associate-/l/ 99.4%

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

5. Applied egg-rr 99.4%

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

   1. unpow2 99.4%

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

   2. add-sqr-sqrt 99.8%

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

   3. associate-/r* 99.8%

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

7. Applied egg-rr 62.2%

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

8. Taylor expanded in x around 0 62.9%

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

9. Final simplification 62.9%

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

Alternative 5: 36.1% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{-1}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.99999993922529e-9) (/ 0.25 s) (* (/ s x) (/ -1.0 s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.99999993922529e-9f) {
		tmp = 0.25f / s;
	} else {
		tmp = (s / x) * (-1.0f / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 9.99999993922529e-9) then
        tmp = 0.25e0 / s
    else
        tmp = (s / x) * ((-1.0e0) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.99999993922529e-9))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(s / x) * Float32(Float32(-1.0) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.99999993922529e-9))
		tmp = single(0.25) / s;
	else
		tmp = (s / x) * (single(-1.0) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 9.99999994e-9

   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 99.1%

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

   4. Taylor expanded in s around inf 36.2%

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

   if 9.99999994e-9 < 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 98.7%

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

   4. Taylor expanded in s around inf 53.3%

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

      1. mul-1-neg 53.3%

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

      2. unsub-neg 53.3%

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

   6. Simplified 53.3%

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

      1. div-inv 53.3%

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

      2. +-commutative 53.3%

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

   8. Applied egg-rr 53.3%

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

   9. Taylor expanded in s around 0 42.4%

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

       1. mul-1-neg 42.4%

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

       2. distribute-frac-neg 42.4%

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

       3. unpow1 42.4%

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

       4. sqr-pow 42.4%

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

       5. fabs-sqr 42.4%

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

       6. sqr-pow 42.4%

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

       7. unpow1 42.4%

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

   11. Simplified 42.4%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{-1}{s}\\ \end{array} \]

Alternative 6: 28.4% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.99999993922529e-9) (/ 0.25 s) (/ 1.0 (+ x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.99999993922529e-9f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (x + x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 9.99999993922529e-9) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x + x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.99999993922529e-9))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.99999993922529e-9))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x + x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 9.99999994e-9

   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 99.1%

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

   4. Taylor expanded in s around inf 36.2%

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

   if 9.99999994e-9 < x

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. fabs-sqr 100.0%

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

      5. add-sqr-sqrt 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in s around inf 3.2%

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

   9. Taylor expanded in x around inf 11.0%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot x}} \]
   10. Step-by-step derivation

       1. *-commutative 11.0%

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

       2. rem-log-exp 78.6%

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

       3. exp-lft-sqr 78.6%

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

       4. prod-exp 78.6%

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

       5. rem-log-exp 11.0%

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

   11. Simplified 11.0%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x}\\ \end{array} \]

Alternative 7: 26.8% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 26.9%

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

5. Final simplification 26.9%

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

Reproduce

herbie shell --seed 2023208 
(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))))))