Logistic distribution

Percentage Accurate: 99.5% → 99.3%

Time: 12.1s

Alternatives: 10

Speedup: 1.2×

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: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004000000189989805:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{-x}{s}}}{s}}{4}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 0.004000000189989805)
   (/ (exp (+ (/ x s) (* -2.0 (log1p (exp (/ x s)))))) s)
   (/ (/ (exp (/ (- x) s)) s) 4.0)))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 0.004000000189989805f) {
		tmp = expf(((x / s) + (-2.0f * log1pf(expf((x / s)))))) / s;
	} else {
		tmp = (expf((-x / s)) / s) / 4.0f;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.004000000189989805))
		tmp = Float32(exp(Float32(Float32(x / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x / s)))))) / s);
	else
		tmp = Float32(Float32(exp(Float32(Float32(-x) / s)) / s) / Float32(4.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.00400000019

   1. Initial program 98.4%

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

      1. associate-*l* 98.7%

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

   3. Simplified 98.7%

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

      1. *-un-lft-identity 98.7%

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

      2. *-commutative 98.7%

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

      3. times-frac 98.5%

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

   5. Applied egg-rr 82.2%

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

      1. associate-*r/ 82.4%

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

      2. add-exp-log 82.4%

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

      3. prod-exp 97.8%

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

      4. pow-flip 97.8%

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

      5. log-pow 98.7%

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

      6. metadata-eval 98.7%

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

      7. +-commutative 98.7%

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

      8. log1p-udef 98.7%

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

   7. Applied egg-rr 98.7%

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

   if 0.00400000019 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

      2. associate-*r/ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-*r/ 100.0%

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

      6. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in s around inf 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt 53.1%

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

      4. fabs-sqr 53.1%

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

      5. add-sqr-sqrt 54.6%

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

   9. Applied egg-rr 54.6%

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

       1. rec-exp 54.6%

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

       2. distribute-neg-frac 54.6%

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

   11. Simplified 54.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004000000189989805:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{-x}{s}}}{s}}{4}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ t_0 1.0)))
   (/ t_0 (* s (* t_1 t_1)))))

C

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

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

Julia

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

MATLAB

x = abs(x)
function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = t_0 + single(1.0);
	tmp = t_0 / (s * (t_1 * t_1));
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* 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(1.0) / Float32(fma(s, exp(Float32(abs(x) / Float32(-s))), s) * Float32(Float32(1.0) + exp(Float32(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 99.4%

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

4. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{\frac{t_0}{s}}{{\left(t_0 + 1\right)}^{2}} \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s)))) (/ (/ t_0 s) (pow (+ t_0 1.0) 2.0))))

C

x = abs(x);
float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return (t_0 / s) / powf((t_0 + 1.0f), 2.0f);
}

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
    real(4) :: t_0
    t_0 = exp((-abs(x) / s))
    code = (t_0 / s) / ((t_0 + 1.0e0) ** 2.0e0)
end function

Julia

x = abs(x)
function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(Float32(t_0 / s) / (Float32(t_0 + Float32(1.0)) ^ Float32(2.0)))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = (t_0 / s) / ((t_0 + single(1.0)) ^ single(2.0));
end

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 99.4%

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

   1. associate-/r* 99.4%

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

   2. associate-*r/ 99.4%

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

   3. mul-1-neg 99.4%

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

   4. +-commutative 99.4%

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

   5. associate-*r/ 99.4%

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

   6. mul-1-neg 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 5: 73.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 20000000000:\\ \;\;\;\;4 \cdot \frac{\frac{1}{\frac{1}{s} \cdot \frac{1}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{{x}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 1.000000031374395e-22)
   (/ 0.25 s)
   (if (<= x 20000000000.0)
     (* 4.0 (/ (/ 1.0 (* (/ 1.0 s) (/ 1.0 s))) s))
     (/ s (pow x 2.0)))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 1.000000031374395e-22f) {
		tmp = 0.25f / s;
	} else if (x <= 20000000000.0f) {
		tmp = 4.0f * ((1.0f / ((1.0f / s) * (1.0f / s))) / s);
	} else {
		tmp = s / powf(x, 2.0f);
	}
	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 <= 1.000000031374395e-22) then
        tmp = 0.25e0 / s
    else if (x <= 20000000000.0e0) then
        tmp = 4.0e0 * ((1.0e0 / ((1.0e0 / s) * (1.0e0 / s))) / s)
    else
        tmp = s / (x ** 2.0e0)
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.000000031374395e-22))
		tmp = Float32(Float32(0.25) / s);
	elseif (x <= Float32(20000000000.0))
		tmp = Float32(Float32(4.0) * Float32(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / s) * Float32(Float32(1.0) / s))) / s));
	else
		tmp = Float32(s / (x ^ Float32(2.0)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.000000031374395e-22))
		tmp = single(0.25) / s;
	elseif (x <= single(20000000000.0))
		tmp = single(4.0) * ((single(1.0) / ((single(1.0) / s) * (single(1.0) / s))) / s);
	else
		tmp = s / (x ^ single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < 1.00000003e-22

   1. Initial program 99.0%

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

      1. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in s around inf 34.6%

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

   if 1.00000003e-22 < x < 2e10

   1. Initial program 99.6%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   5. Applied egg-rr 90.4%

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

   6. Taylor expanded in x around 0 22.1%

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

      1. add-sqr-sqrt 19.3%

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

      2. sqrt-unprod 20.6%

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

      3. sqr-neg 20.6%

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

      4. sqrt-unprod 0.7%

         \[\leadsto e^{\color{blue}{\sqrt{\log s + 2 \cdot \log 2} \cdot \sqrt{\log s + 2 \cdot \log 2}}} \]

      5. add-sqr-sqrt 12.9%

         \[\leadsto e^{\color{blue}{\log s + 2 \cdot \log 2}} \]

      6. exp-sum 12.9%

         \[\leadsto \color{blue}{e^{\log s} \cdot e^{2 \cdot \log 2}} \]

      7. add-exp-log 12.9%

         \[\leadsto \color{blue}{s} \cdot e^{2 \cdot \log 2} \]

      8. *-commutative 12.9%

         \[\leadsto s \cdot e^{\color{blue}{\log 2 \cdot 2}} \]

      9. exp-to-pow 12.9%

         \[\leadsto s \cdot \color{blue}{{2}^{2}} \]

      10. metadata-eval 12.9%

          \[\leadsto s \cdot \color{blue}{4} \]

      11. +-lft-identity 12.9%

          \[\leadsto \color{blue}{0 + s \cdot 4} \]

      12. mul0-rgt 12.9%

          \[\leadsto \color{blue}{\left|x\right| \cdot 0} + s \cdot 4 \]

      13. flip-+ 51.9%

          \[\leadsto \color{blue}{\frac{\left(\left|x\right| \cdot 0\right) \cdot \left(\left|x\right| \cdot 0\right) - \left(s \cdot 4\right) \cdot \left(s \cdot 4\right)}{\left|x\right| \cdot 0 - s \cdot 4}} \]

   8. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{\frac{0}{-s \cdot 4} - \frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]
   9. Step-by-step derivation

      1. div0 51.9%

         \[\leadsto \color{blue}{0} - \frac{{s}^{2} \cdot 16}{-s \cdot 4} \]

      2. neg-sub0 51.9%

         \[\leadsto \color{blue}{-\frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]

      3. neg-mul-1 51.9%

         \[\leadsto \color{blue}{-1 \cdot \frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]

      4. metadata-eval 51.9%

         \[\leadsto \color{blue}{\frac{1}{-1}} \cdot \frac{{s}^{2} \cdot 16}{-s \cdot 4} \]

      5. times-frac 51.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \left({s}^{2} \cdot 16\right)}{-1 \cdot \left(-s \cdot 4\right)}} \]

      6. neg-mul-1 51.9%

         \[\leadsto \frac{1 \cdot \left({s}^{2} \cdot 16\right)}{\color{blue}{-\left(-s \cdot 4\right)}} \]

      7. remove-double-neg 51.9%

         \[\leadsto \frac{1 \cdot \left({s}^{2} \cdot 16\right)}{\color{blue}{s \cdot 4}} \]

      8. *-lft-identity 51.9%

         \[\leadsto \frac{\color{blue}{{s}^{2} \cdot 16}}{s \cdot 4} \]

      9. *-commutative 51.9%

         \[\leadsto \frac{\color{blue}{16 \cdot {s}^{2}}}{s \cdot 4} \]

      10. *-commutative 51.9%

          \[\leadsto \frac{16 \cdot {s}^{2}}{\color{blue}{4 \cdot s}} \]

      11. times-frac 51.9%

          \[\leadsto \color{blue}{\frac{16}{4} \cdot \frac{{s}^{2}}{s}} \]

      12. metadata-eval 51.9%

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

   10. Simplified 51.9%

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

       1. pow1 51.9%

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

       2. metadata-eval 51.9%

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

       3. pow-div 51.9%

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

       4. pow1 51.9%

          \[\leadsto 4 \cdot \frac{{\left(\frac{{s}^{2}}{\color{blue}{s}}\right)}^{2}}{s} \]

       5. metadata-eval 51.9%

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

       6. pow-prod-up 51.9%

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

       7. pow1 51.9%

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

       8. clear-num 51.9%

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

       9. pow1 51.9%

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

       10. pow-div 51.9%

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

       11. metadata-eval 51.9%

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

       12. inv-pow 51.9%

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

       13. pow1 51.9%

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

       14. clear-num 51.9%

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

       15. pow1 51.9%

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

       16. pow-div 51.9%

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

       17. metadata-eval 51.9%

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

       18. inv-pow 51.9%

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

       19. frac-times 61.4%

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

       20. metadata-eval 61.4%

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

   12. Applied egg-rr 61.4%

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

   if 2e10 < 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%

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

   4. Taylor expanded in s around -inf 30.3%

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

      1. +-commutative 30.3%

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

      2. mul-1-neg 30.3%

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

      3. distribute-lft1-in 95.7%

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

      4. metadata-eval 95.7%

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

      5. associate-*r/ 95.7%

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

      6. mul-1-neg 95.7%

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

      7. remove-double-neg 95.7%

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

      8. associate-+r+ 95.7%

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

   6. Simplified 95.7%

      \[\leadsto \frac{1}{\color{blue}{\left(\left|x\right| \cdot 0 + s \cdot 4\right) + \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 94.1%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 20000000000:\\ \;\;\;\;4 \cdot \frac{\frac{1}{\frac{1}{s} \cdot \frac{1}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{{x}^{2}}\\ \end{array} \]

Alternative 6: 95.0% accurate, 5.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ (/ (exp (/ (- x) s)) s) 4.0))

C

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

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 = (exp((-x / s)) / s) / 4.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 99.4%

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

   1. associate-/r* 99.4%

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

   2. associate-*r/ 99.4%

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

   3. mul-1-neg 99.4%

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

   4. +-commutative 99.4%

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

   5. associate-*r/ 99.4%

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

   6. mul-1-neg 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in s around inf 95.1%

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

   1. distribute-frac-neg 95.1%

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

   2. exp-neg 95.1%

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

   3. add-sqr-sqrt 46.9%

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

   4. fabs-sqr 46.9%

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

   5. add-sqr-sqrt 58.3%

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

9. Applied egg-rr 58.3%

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

    1. rec-exp 58.4%

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

    2. distribute-neg-frac 58.4%

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

11. Simplified 58.4%

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

12. Final simplification 58.4%

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

Alternative 7: 62.6% accurate, 41.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;s \leq 3.99999987306209 \cdot 10^{-20}:\\ \;\;\;\;4 \cdot \frac{\frac{1}{\frac{1}{s} \cdot \frac{1}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= s 3.99999987306209e-20)
   (* 4.0 (/ (/ 1.0 (* (/ 1.0 s) (/ 1.0 s))) s))
   (/ 0.25 s)))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (s <= 3.99999987306209e-20f) {
		tmp = 4.0f * ((1.0f / ((1.0f / s) * (1.0f / s))) / s);
	} else {
		tmp = 0.25f / s;
	}
	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 (s <= 3.99999987306209e-20) then
        tmp = 4.0e0 * ((1.0e0 / ((1.0e0 / s) * (1.0e0 / s))) / s)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (s <= Float32(3.99999987306209e-20))
		tmp = Float32(Float32(4.0) * Float32(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / s) * Float32(Float32(1.0) / s))) / s));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (s <= single(3.99999987306209e-20))
		tmp = single(4.0) * ((single(1.0) / ((single(1.0) / s) * (single(1.0) / s))) / s);
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 3.99999987e-20

   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* 99.2%

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

   3. Simplified 99.2%

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

   5. Applied egg-rr 70.1%

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

   6. Taylor expanded in x around 0 9.6%

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

      1. add-sqr-sqrt 9.4%

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

      2. sqrt-unprod 9.6%

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

      3. sqr-neg 9.6%

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

      4. sqrt-unprod -0.0%

         \[\leadsto e^{\color{blue}{\sqrt{\log s + 2 \cdot \log 2} \cdot \sqrt{\log s + 2 \cdot \log 2}}} \]

      5. add-sqr-sqrt 12.9%

         \[\leadsto e^{\color{blue}{\log s + 2 \cdot \log 2}} \]

      6. exp-sum 12.9%

         \[\leadsto \color{blue}{e^{\log s} \cdot e^{2 \cdot \log 2}} \]

      7. add-exp-log 12.9%

         \[\leadsto \color{blue}{s} \cdot e^{2 \cdot \log 2} \]

      8. *-commutative 12.9%

         \[\leadsto s \cdot e^{\color{blue}{\log 2 \cdot 2}} \]

      9. exp-to-pow 12.9%

         \[\leadsto s \cdot \color{blue}{{2}^{2}} \]

      10. metadata-eval 12.9%

          \[\leadsto s \cdot \color{blue}{4} \]

      11. +-lft-identity 12.9%

          \[\leadsto \color{blue}{0 + s \cdot 4} \]

      12. mul0-rgt 12.9%

          \[\leadsto \color{blue}{\left|x\right| \cdot 0} + s \cdot 4 \]

      13. flip-+ 73.4%

          \[\leadsto \color{blue}{\frac{\left(\left|x\right| \cdot 0\right) \cdot \left(\left|x\right| \cdot 0\right) - \left(s \cdot 4\right) \cdot \left(s \cdot 4\right)}{\left|x\right| \cdot 0 - s \cdot 4}} \]

   8. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{0}{-s \cdot 4} - \frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]
   9. Step-by-step derivation

      1. div0 76.3%

         \[\leadsto \color{blue}{0} - \frac{{s}^{2} \cdot 16}{-s \cdot 4} \]

      2. neg-sub0 76.3%

         \[\leadsto \color{blue}{-\frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]

      3. neg-mul-1 76.3%

         \[\leadsto \color{blue}{-1 \cdot \frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]

      4. metadata-eval 76.3%

         \[\leadsto \color{blue}{\frac{1}{-1}} \cdot \frac{{s}^{2} \cdot 16}{-s \cdot 4} \]

      5. times-frac 76.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \left({s}^{2} \cdot 16\right)}{-1 \cdot \left(-s \cdot 4\right)}} \]

      6. neg-mul-1 76.3%

         \[\leadsto \frac{1 \cdot \left({s}^{2} \cdot 16\right)}{\color{blue}{-\left(-s \cdot 4\right)}} \]

      7. remove-double-neg 76.3%

         \[\leadsto \frac{1 \cdot \left({s}^{2} \cdot 16\right)}{\color{blue}{s \cdot 4}} \]

      8. *-lft-identity 76.3%

         \[\leadsto \frac{\color{blue}{{s}^{2} \cdot 16}}{s \cdot 4} \]

      9. *-commutative 76.3%

         \[\leadsto \frac{\color{blue}{16 \cdot {s}^{2}}}{s \cdot 4} \]

      10. *-commutative 76.3%

          \[\leadsto \frac{16 \cdot {s}^{2}}{\color{blue}{4 \cdot s}} \]

      11. times-frac 76.3%

          \[\leadsto \color{blue}{\frac{16}{4} \cdot \frac{{s}^{2}}{s}} \]

      12. metadata-eval 76.3%

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

   10. Simplified 76.3%

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

       1. pow1 76.3%

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

       2. metadata-eval 76.3%

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

       3. pow-div 76.3%

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

       4. pow1 76.3%

          \[\leadsto 4 \cdot \frac{{\left(\frac{{s}^{2}}{\color{blue}{s}}\right)}^{2}}{s} \]

       5. metadata-eval 76.3%

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

       6. pow-prod-up 76.3%

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

       7. pow1 76.3%

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

       8. clear-num 76.3%

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

       9. pow1 76.3%

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

       10. pow-div 76.3%

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

       11. metadata-eval 76.3%

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

       12. inv-pow 76.3%

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

       13. pow1 76.3%

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

       14. clear-num 76.3%

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

       15. pow1 76.3%

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

       16. pow-div 76.3%

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

       17. metadata-eval 76.3%

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

       18. inv-pow 76.3%

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

       19. frac-times 91.6%

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

       20. metadata-eval 91.6%

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

   12. Applied egg-rr 91.6%

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

   if 3.99999987e-20 < s

   1. Initial program 99.5%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 40.5%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 3.99999987306209 \cdot 10^{-20}:\\ \;\;\;\;4 \cdot \frac{\frac{1}{\frac{1}{s} \cdot \frac{1}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 8: 58.5% accurate, 67.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;s \leq 2.5000000784359874 \cdot 10^{-23}:\\ \;\;\;\;4 \cdot \frac{s \cdot s}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= s 2.5000000784359874e-23) (* 4.0 (/ (* s s) s)) (/ 0.25 s)))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (s <= 2.5000000784359874e-23f) {
		tmp = 4.0f * ((s * s) / s);
	} else {
		tmp = 0.25f / s;
	}
	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 (s <= 2.5000000784359874e-23) then
        tmp = 4.0e0 * ((s * s) / s)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (s <= Float32(2.5000000784359874e-23))
		tmp = Float32(Float32(4.0) * Float32(Float32(s * s) / s));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (s <= single(2.5000000784359874e-23))
		tmp = single(4.0) * ((s * s) / s);
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 2.50000008e-23

   1. Initial program 98.9%

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

      1. associate-*l* 99.0%

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

   3. Simplified 99.0%

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

   5. Applied egg-rr 68.2%

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

   6. Taylor expanded in x around 0 8.3%

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

      1. add-sqr-sqrt 8.0%

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

      2. sqrt-unprod 8.3%

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

      3. sqr-neg 8.3%

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

      4. sqrt-unprod -0.0%

         \[\leadsto e^{\color{blue}{\sqrt{\log s + 2 \cdot \log 2} \cdot \sqrt{\log s + 2 \cdot \log 2}}} \]

      5. add-sqr-sqrt 13.8%

         \[\leadsto e^{\color{blue}{\log s + 2 \cdot \log 2}} \]

      6. exp-sum 13.8%

         \[\leadsto \color{blue}{e^{\log s} \cdot e^{2 \cdot \log 2}} \]

      7. add-exp-log 13.8%

         \[\leadsto \color{blue}{s} \cdot e^{2 \cdot \log 2} \]

      8. *-commutative 13.8%

         \[\leadsto s \cdot e^{\color{blue}{\log 2 \cdot 2}} \]

      9. exp-to-pow 13.8%

         \[\leadsto s \cdot \color{blue}{{2}^{2}} \]

      10. metadata-eval 13.8%

          \[\leadsto s \cdot \color{blue}{4} \]

      11. +-lft-identity 13.8%

          \[\leadsto \color{blue}{0 + s \cdot 4} \]

      12. mul0-rgt 13.8%

          \[\leadsto \color{blue}{\left|x\right| \cdot 0} + s \cdot 4 \]

      13. flip-+ 88.8%

          \[\leadsto \color{blue}{\frac{\left(\left|x\right| \cdot 0\right) \cdot \left(\left|x\right| \cdot 0\right) - \left(s \cdot 4\right) \cdot \left(s \cdot 4\right)}{\left|x\right| \cdot 0 - s \cdot 4}} \]

   8. Applied egg-rr 92.4%

      \[\leadsto \color{blue}{\frac{0}{-s \cdot 4} - \frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]
   9. Step-by-step derivation

      1. div0 92.4%

         \[\leadsto \color{blue}{0} - \frac{{s}^{2} \cdot 16}{-s \cdot 4} \]

      2. neg-sub0 92.4%

         \[\leadsto \color{blue}{-\frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]

      3. neg-mul-1 92.4%

         \[\leadsto \color{blue}{-1 \cdot \frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]

      4. metadata-eval 92.4%

         \[\leadsto \color{blue}{\frac{1}{-1}} \cdot \frac{{s}^{2} \cdot 16}{-s \cdot 4} \]

      5. times-frac 92.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \left({s}^{2} \cdot 16\right)}{-1 \cdot \left(-s \cdot 4\right)}} \]

      6. neg-mul-1 92.4%

         \[\leadsto \frac{1 \cdot \left({s}^{2} \cdot 16\right)}{\color{blue}{-\left(-s \cdot 4\right)}} \]

      7. remove-double-neg 92.4%

         \[\leadsto \frac{1 \cdot \left({s}^{2} \cdot 16\right)}{\color{blue}{s \cdot 4}} \]

      8. *-lft-identity 92.4%

         \[\leadsto \frac{\color{blue}{{s}^{2} \cdot 16}}{s \cdot 4} \]

      9. *-commutative 92.4%

         \[\leadsto \frac{\color{blue}{16 \cdot {s}^{2}}}{s \cdot 4} \]

      10. *-commutative 92.4%

          \[\leadsto \frac{16 \cdot {s}^{2}}{\color{blue}{4 \cdot s}} \]

      11. times-frac 92.4%

          \[\leadsto \color{blue}{\frac{16}{4} \cdot \frac{{s}^{2}}{s}} \]

      12. metadata-eval 92.4%

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

   10. Simplified 92.4%

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

       1. unpow2 92.4%

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

   12. Applied egg-rr 92.4%

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

   if 2.50000008e-23 < s

   1. Initial program 99.5%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 36.8%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.5000000784359874 \cdot 10^{-23}:\\ \;\;\;\;4 \cdot \frac{s \cdot s}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 9: 30.4% accurate, 121.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot 4\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 5.000000058430487e-8) (/ 0.25 s) (* s 4.0)))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 5.000000058430487e-8f) {
		tmp = 0.25f / s;
	} else {
		tmp = s * 4.0f;
	}
	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 <= 5.000000058430487e-8) then
        tmp = 0.25e0 / s
    else
        tmp = s * 4.0e0
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(5.000000058430487e-8))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s * Float32(4.0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.000000058430487e-8))
		tmp = single(0.25) / s;
	else
		tmp = s * single(4.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.00000006e-8

   1. Initial program 99.0%

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

      1. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in s around inf 35.5%

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

   if 5.00000006e-8 < 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. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 48.1%

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

   6. Taylor expanded in x around 0 4.9%

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

      1. add-sqr-sqrt 4.7%

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

      2. sqrt-unprod 4.9%

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

      3. sqr-neg 4.9%

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

      4. sqrt-unprod 0.2%

         \[\leadsto e^{\color{blue}{\sqrt{\log s + 2 \cdot \log 2} \cdot \sqrt{\log s + 2 \cdot \log 2}}} \]

      5. add-sqr-sqrt 11.6%

         \[\leadsto e^{\color{blue}{\log s + 2 \cdot \log 2}} \]

      6. exp-sum 11.6%

         \[\leadsto \color{blue}{e^{\log s} \cdot e^{2 \cdot \log 2}} \]

      7. add-exp-log 11.6%

         \[\leadsto \color{blue}{s} \cdot e^{2 \cdot \log 2} \]

      8. *-commutative 11.6%

         \[\leadsto s \cdot e^{\color{blue}{\log 2 \cdot 2}} \]

      9. exp-to-pow 11.6%

         \[\leadsto s \cdot \color{blue}{{2}^{2}} \]

      10. metadata-eval 11.6%

          \[\leadsto s \cdot \color{blue}{4} \]

      11. +-lft-identity 11.6%

          \[\leadsto \color{blue}{0 + s \cdot 4} \]

      12. mul0-rgt 11.6%

          \[\leadsto \color{blue}{\left|x\right| \cdot 0} + s \cdot 4 \]

      13. flip-+ 54.8%

          \[\leadsto \color{blue}{\frac{\left(\left|x\right| \cdot 0\right) \cdot \left(\left|x\right| \cdot 0\right) - \left(s \cdot 4\right) \cdot \left(s \cdot 4\right)}{\left|x\right| \cdot 0 - s \cdot 4}} \]

   8. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{\frac{0}{-s \cdot 4} - \frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]
   9. Step-by-step derivation

      1. div0 55.9%

         \[\leadsto \color{blue}{0} - \frac{{s}^{2} \cdot 16}{-s \cdot 4} \]

      2. neg-sub0 55.9%

         \[\leadsto \color{blue}{-\frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]

      3. neg-mul-1 55.9%

         \[\leadsto \color{blue}{-1 \cdot \frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]

      4. metadata-eval 55.9%

         \[\leadsto \color{blue}{\frac{1}{-1}} \cdot \frac{{s}^{2} \cdot 16}{-s \cdot 4} \]

      5. times-frac 55.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \left({s}^{2} \cdot 16\right)}{-1 \cdot \left(-s \cdot 4\right)}} \]

      6. neg-mul-1 55.9%

         \[\leadsto \frac{1 \cdot \left({s}^{2} \cdot 16\right)}{\color{blue}{-\left(-s \cdot 4\right)}} \]

      7. remove-double-neg 55.9%

         \[\leadsto \frac{1 \cdot \left({s}^{2} \cdot 16\right)}{\color{blue}{s \cdot 4}} \]

      8. *-lft-identity 55.9%

         \[\leadsto \frac{\color{blue}{{s}^{2} \cdot 16}}{s \cdot 4} \]

      9. *-commutative 55.9%

         \[\leadsto \frac{\color{blue}{16 \cdot {s}^{2}}}{s \cdot 4} \]

      10. *-commutative 55.9%

          \[\leadsto \frac{16 \cdot {s}^{2}}{\color{blue}{4 \cdot s}} \]

      11. times-frac 55.9%

          \[\leadsto \color{blue}{\frac{16}{4} \cdot \frac{{s}^{2}}{s}} \]

      12. metadata-eval 55.9%

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

   10. Simplified 55.9%

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

   11. Taylor expanded in s around 0 11.6%

       \[\leadsto 4 \cdot \color{blue}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot 4\\ \end{array} \]

Alternative 10: 10.9% accurate, 206.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ s \cdot 4 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (* s 4.0))

C

x = abs(x);
float code(float x, float s) {
	return s * 4.0f;
}

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 = s * 4.0e0
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(s * Float32(4.0))
end

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.4%

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

3. Simplified 99.4%

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

5. Applied egg-rr 81.2%

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

6. Taylor expanded in x around 0 24.4%

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

   1. add-sqr-sqrt 22.8%

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

   2. sqrt-unprod 23.6%

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

   3. sqr-neg 23.6%

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

   4. sqrt-unprod 0.4%

      \[\leadsto e^{\color{blue}{\sqrt{\log s + 2 \cdot \log 2} \cdot \sqrt{\log s + 2 \cdot \log 2}}} \]

   5. add-sqr-sqrt 11.1%

      \[\leadsto e^{\color{blue}{\log s + 2 \cdot \log 2}} \]

   6. exp-sum 11.1%

      \[\leadsto \color{blue}{e^{\log s} \cdot e^{2 \cdot \log 2}} \]

   7. add-exp-log 11.1%

      \[\leadsto \color{blue}{s} \cdot e^{2 \cdot \log 2} \]

   8. *-commutative 11.1%

      \[\leadsto s \cdot e^{\color{blue}{\log 2 \cdot 2}} \]

   9. exp-to-pow 11.1%

      \[\leadsto s \cdot \color{blue}{{2}^{2}} \]

   10. metadata-eval 11.1%

       \[\leadsto s \cdot \color{blue}{4} \]

   11. +-lft-identity 11.1%

       \[\leadsto \color{blue}{0 + s \cdot 4} \]

   12. mul0-rgt 11.1%

       \[\leadsto \color{blue}{\left|x\right| \cdot 0} + s \cdot 4 \]

   13. flip-+ 40.5%

       \[\leadsto \color{blue}{\frac{\left(\left|x\right| \cdot 0\right) \cdot \left(\left|x\right| \cdot 0\right) - \left(s \cdot 4\right) \cdot \left(s \cdot 4\right)}{\left|x\right| \cdot 0 - s \cdot 4}} \]

8. Applied egg-rr 41.9%

   \[\leadsto \color{blue}{\frac{0}{-s \cdot 4} - \frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]
9. Step-by-step derivation

   1. div0 41.9%

      \[\leadsto \color{blue}{0} - \frac{{s}^{2} \cdot 16}{-s \cdot 4} \]

   2. neg-sub0 41.9%

      \[\leadsto \color{blue}{-\frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]

   3. neg-mul-1 41.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{{s}^{2} \cdot 16}{-s \cdot 4}} \]

   4. metadata-eval 41.9%

      \[\leadsto \color{blue}{\frac{1}{-1}} \cdot \frac{{s}^{2} \cdot 16}{-s \cdot 4} \]

   5. times-frac 41.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \left({s}^{2} \cdot 16\right)}{-1 \cdot \left(-s \cdot 4\right)}} \]

   6. neg-mul-1 41.9%

      \[\leadsto \frac{1 \cdot \left({s}^{2} \cdot 16\right)}{\color{blue}{-\left(-s \cdot 4\right)}} \]

   7. remove-double-neg 41.9%

      \[\leadsto \frac{1 \cdot \left({s}^{2} \cdot 16\right)}{\color{blue}{s \cdot 4}} \]

   8. *-lft-identity 41.9%

      \[\leadsto \frac{\color{blue}{{s}^{2} \cdot 16}}{s \cdot 4} \]

   9. *-commutative 41.9%

      \[\leadsto \frac{\color{blue}{16 \cdot {s}^{2}}}{s \cdot 4} \]

   10. *-commutative 41.9%

       \[\leadsto \frac{16 \cdot {s}^{2}}{\color{blue}{4 \cdot s}} \]

   11. times-frac 41.9%

       \[\leadsto \color{blue}{\frac{16}{4} \cdot \frac{{s}^{2}}{s}} \]

   12. metadata-eval 41.9%

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

10. Simplified 41.9%

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

11. Taylor expanded in s around 0 11.1%

    \[\leadsto 4 \cdot \color{blue}{s} \]

12. Final simplification 11.1%

    \[\leadsto s \cdot 4 \]

Reproduce

herbie shell --seed 2023310 
(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))))))