Logistic distribution

Percentage Accurate: 99.4% → 99.3%

Time: 10.2s

Alternatives: 10

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.4% 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, 2.0× speedup

Math

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

FPCore

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

C

x = abs(x);
float code(float x, float s) {
	return (1.0f / s) / (expf((-x / s)) + (expf((fabsf(x) / s)) + 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
    code = (1.0e0 / s) / (exp((-x / s)) + (exp((abs(x) / s)) + 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. neg-mul-1 99.9%

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

   3. times-frac 99.9%

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

   4. metadata-eval 99.9%

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

   5. metadata-eval 99.9%

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

   6. times-frac 99.9%

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

   7. neg-mul-1 99.9%

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

   8. *-un-lft-identity 99.9%

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

   9. distribute-frac-neg 99.9%

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

   10. rec-exp 99.9%

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

   11. add-sqr-sqrt 99.9%

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

   12. sqrt-unprod 98.1%

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

   13. sqr-neg 98.1%

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

   14. sqrt-unprod -0.0%

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

   15. add-sqr-sqrt 95.6%

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

5. Applied egg-rr 97.3%

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

   1. rec-exp 97.3%

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

   2. distribute-neg-frac 97.3%

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

7. Simplified 97.3%

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

8. Final simplification 97.3%

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

Alternative 2: 96.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{s}}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 + e^{\frac{x}{s}}\right)\right)} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ (/ 1.0 s) (expm1 (log1p (+ 3.0 (exp (/ x s)))))))

C

x = abs(x);
float code(float x, float s) {
	return (1.0f / s) / expm1f(log1pf((3.0f + expf((x / s)))));
}

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / expm1(log1p(Float32(Float32(3.0) + exp(Float32(x / s))))))
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. neg-mul-1 99.9%

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

   3. times-frac 99.9%

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

   4. metadata-eval 99.9%

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

   5. metadata-eval 99.9%

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

   6. times-frac 99.9%

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

   7. neg-mul-1 99.9%

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

   8. *-un-lft-identity 99.9%

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

   9. distribute-frac-neg 99.9%

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

   10. rec-exp 99.9%

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

   11. add-sqr-sqrt 99.9%

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

   12. sqrt-unprod 98.1%

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

   13. sqr-neg 98.1%

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

   14. sqrt-unprod -0.0%

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

   15. add-sqr-sqrt 95.6%

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

5. Applied egg-rr 97.3%

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

   1. rec-exp 97.3%

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

   2. distribute-neg-frac 97.3%

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

7. Simplified 97.3%

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

8. Taylor expanded in x around 0 96.7%

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

   1. associate-/r* 96.7%

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

10. Simplified 96.7%

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

    1. expm1-log1p-u 96.7%

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

    2. add-sqr-sqrt 49.4%

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

    3. fabs-sqr 49.4%

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

    4. add-sqr-sqrt 60.8%

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

12. Applied egg-rr 60.8%

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

13. Final simplification 60.8%

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

Alternative 3: 80.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(3 + \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 1.9999999996399175e-23)
   (/ 0.25 s)
   (/ 1.0 (* s (+ 3.0 (+ 1.0 (+ (/ x s) (* 0.5 (/ (* x x) (* s s))))))))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 1.9999999996399175e-23f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (s * (3.0f + (1.0f + ((x / s) + (0.5f * ((x * x) / (s * 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 (abs(x) <= 1.9999999996399175e-23) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (s * (3.0e0 + (1.0e0 + ((x / s) + (0.5e0 * ((x * x) / (s * s)))))))
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(3.0) + Float32(Float32(1.0) + Float32(Float32(x / s) + Float32(Float32(0.5) * Float32(Float32(x * x) / Float32(s * s))))))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(1.9999999996399175e-23))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (s * (single(3.0) + (single(1.0) + ((x / s) + (single(0.5) * ((x * x) / (s * s)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 2e-23

   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. Taylor expanded in s around inf 76.1%

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

   if 2e-23 < (fabs.f32 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. neg-mul-1 99.9%

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

      3. times-frac 99.9%

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

      4. metadata-eval 99.9%

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

      5. metadata-eval 99.9%

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

      6. times-frac 99.9%

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

      7. neg-mul-1 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. distribute-frac-neg 99.9%

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

      10. rec-exp 99.9%

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

      11. add-sqr-sqrt 99.9%

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

      12. sqrt-unprod 99.8%

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

      13. sqr-neg 99.8%

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

      14. sqrt-unprod -0.0%

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

      15. add-sqr-sqrt 97.2%

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

   5. Applied egg-rr 98.5%

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

      1. rec-exp 98.5%

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

      2. distribute-neg-frac 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around 0 98.0%

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

      1. distribute-lft-in 98.0%

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

      2. add-sqr-sqrt 49.3%

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

      3. fabs-sqr 49.3%

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

      4. add-sqr-sqrt 56.4%

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

   10. Applied egg-rr 56.4%

       \[\leadsto \frac{1}{\color{blue}{s \cdot 3 + s \cdot e^{\frac{x}{s}}}} \]
   11. Step-by-step derivation

       1. distribute-lft-in 56.4%

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

   12. Simplified 56.4%

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

   13. Taylor expanded in x around 0 67.0%

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

       1. unpow2 67.0%

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

       2. unpow2 67.0%

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

   15. Simplified 67.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(3 + \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)\right)}\\ \end{array} \]

Alternative 4: 96.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. neg-mul-1 99.9%

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

   3. times-frac 99.9%

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

   4. metadata-eval 99.9%

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

   5. metadata-eval 99.9%

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

   6. times-frac 99.9%

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

   7. neg-mul-1 99.9%

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

   8. *-un-lft-identity 99.9%

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

   9. distribute-frac-neg 99.9%

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

   10. rec-exp 99.9%

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

   11. add-sqr-sqrt 99.9%

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

   12. sqrt-unprod 98.1%

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

   13. sqr-neg 98.1%

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

   14. sqrt-unprod -0.0%

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

   15. add-sqr-sqrt 95.6%

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

5. Applied egg-rr 97.3%

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

   1. rec-exp 97.3%

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

   2. distribute-neg-frac 97.3%

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

7. Simplified 97.3%

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

8. Taylor expanded in x around 0 96.7%

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

   1. distribute-lft-in 96.7%

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

   2. add-sqr-sqrt 49.4%

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

   3. fabs-sqr 49.4%

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

   4. add-sqr-sqrt 60.8%

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

10. Applied egg-rr 60.8%

    \[\leadsto \frac{1}{\color{blue}{s \cdot 3 + s \cdot e^{\frac{x}{s}}}} \]
11. Step-by-step derivation

    1. distribute-lft-in 60.8%

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

12. Simplified 60.8%

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

13. Final simplification 60.8%

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

Alternative 5: 80.3% accurate, 29.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999996399175e-23)
   (/ 0.25 s)
   (/ (/ 1.0 s) (+ 4.0 (+ (/ x s) (* 0.5 (/ (* x x) (* s s))))))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999996399175e-23f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / s) / (4.0f + ((x / s) + (0.5f * ((x * x) / (s * 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 (x <= 1.9999999996399175e-23) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) + (0.5e0 * ((x * x) / (s * s)))))
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) + Float32(Float32(0.5) * Float32(Float32(x * x) / Float32(s * s))))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999996399175e-23))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) + (single(0.5) * ((x * x) / (s * s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2e-23

   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. Taylor expanded in s around inf 34.0%

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

   if 2e-23 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. neg-mul-1 99.9%

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

      3. times-frac 99.9%

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

      4. metadata-eval 99.9%

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

      5. metadata-eval 99.9%

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

      6. times-frac 99.9%

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

      7. neg-mul-1 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. distribute-frac-neg 99.9%

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

      10. rec-exp 99.9%

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

      11. add-sqr-sqrt 99.9%

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

      12. sqrt-unprod 99.6%

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

      13. sqr-neg 99.6%

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

      14. sqrt-unprod -0.0%

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

      15. add-sqr-sqrt 97.4%

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

   5. Applied egg-rr 99.9%

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

      1. rec-exp 99.9%

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

      2. distribute-neg-frac 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 98.2%

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

      1. associate-/r* 98.2%

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

   10. Simplified 98.2%

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

       1. expm1-log1p-u 98.3%

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

       2. add-sqr-sqrt 98.2%

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

       3. fabs-sqr 98.2%

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

       4. add-sqr-sqrt 98.3%

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

   12. Applied egg-rr 98.3%

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

   13. Taylor expanded in x around 0 79.2%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{4 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   14. Step-by-step derivation

       1. *-commutative 79.2%

          \[\leadsto \frac{\frac{1}{s}}{4 + \left(\frac{x}{s} + \color{blue}{\frac{{x}^{2}}{{s}^{2}} \cdot 0.5}\right)} \]

       2. unpow2 79.2%

          \[\leadsto \frac{\frac{1}{s}}{4 + \left(\frac{x}{s} + \frac{\color{blue}{x \cdot x}}{{s}^{2}} \cdot 0.5\right)} \]

       3. unpow2 79.2%

          \[\leadsto \frac{\frac{1}{s}}{4 + \left(\frac{x}{s} + \frac{x \cdot x}{\color{blue}{s \cdot s}} \cdot 0.5\right)} \]

   15. Simplified 79.2%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{4 + \left(\frac{x}{s} + \frac{x \cdot x}{s \cdot s} \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \]

Alternative 6: 51.3% accurate, 56.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{s}}{4 + 2 \cdot \frac{x}{s}} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ (/ 1.0 s) (+ 4.0 (* 2.0 (/ x s)))))

C

x = abs(x);
float code(float x, float s) {
	return (1.0f / s) / (4.0f + (2.0f * (x / s)));
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / (4.0e0 + (2.0e0 * (x / s)))
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(2.0) * Float32(x / s))))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = (single(1.0) / s) / (single(4.0) + (single(2.0) * (x / s)));
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

   1. expm1-log1p-u 97.9%

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

   2. expm1-udef 97.9%

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

5. Applied egg-rr 59.2%

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

   1. expm1-def 59.2%

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

   2. expm1-log1p 60.3%

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

   3. associate-/r* 60.3%

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

   4. *-lft-identity 60.3%

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

   5. associate-*l/ 60.3%

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

   6. associate-*r/ 60.3%

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

   7. *-rgt-identity 60.3%

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

   8. +-commutative 60.3%

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

   9. +-commutative 60.3%

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

   10. associate-+l+ 60.3%

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

   11. count-2 60.3%

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

7. Simplified 60.3%

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

8. Taylor expanded in x around 0 51.4%

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

9. Final simplification 51.4%

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

Alternative 7: 51.1% accurate, 68.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{s}}{\frac{x}{s} + 4} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ (/ 1.0 s) (+ (/ x s) 4.0)))

C

x = abs(x);
float code(float x, float s) {
	return (1.0f / s) / ((x / 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 = (1.0e0 / s) / ((x / s) + 4.0e0)
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(Float32(x / s) + Float32(4.0)))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = (single(1.0) / s) / ((x / s) + single(4.0));
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. neg-mul-1 99.9%

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

   3. times-frac 99.9%

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

   4. metadata-eval 99.9%

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

   5. metadata-eval 99.9%

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

   6. times-frac 99.9%

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

   7. neg-mul-1 99.9%

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

   8. *-un-lft-identity 99.9%

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

   9. distribute-frac-neg 99.9%

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

   10. rec-exp 99.9%

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

   11. add-sqr-sqrt 99.9%

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

   12. sqrt-unprod 98.1%

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

   13. sqr-neg 98.1%

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

   14. sqrt-unprod -0.0%

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

   15. add-sqr-sqrt 95.6%

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

5. Applied egg-rr 97.3%

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

   1. rec-exp 97.3%

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

   2. distribute-neg-frac 97.3%

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

7. Simplified 97.3%

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

8. Taylor expanded in x around 0 96.7%

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

   1. associate-/r* 96.7%

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

10. Simplified 96.7%

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

    1. expm1-log1p-u 96.7%

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

    2. add-sqr-sqrt 49.4%

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

    3. fabs-sqr 49.4%

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

    4. add-sqr-sqrt 60.8%

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

12. Applied egg-rr 60.8%

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

13. Taylor expanded in x around 0 51.4%

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

    1. +-commutative 51.4%

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

15. Simplified 51.4%

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

16. Final simplification 51.4%

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

Alternative 8: 29.2% accurate, 88.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{1}{x + s \cdot 4} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 1.0 (+ x (* s 4.0))))

C

x = abs(x);
float code(float x, float s) {
	return 1.0f / (x + (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 = 1.0e0 / (x + (s * 4.0e0))
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(1.0) / Float32(x + Float32(s * Float32(4.0))))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = single(1.0) / (x + (s * single(4.0)));
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. neg-mul-1 99.9%

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

   3. times-frac 99.9%

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

   4. metadata-eval 99.9%

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

   5. metadata-eval 99.9%

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

   6. times-frac 99.9%

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

   7. neg-mul-1 99.9%

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

   8. *-un-lft-identity 99.9%

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

   9. distribute-frac-neg 99.9%

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

   10. rec-exp 99.9%

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

   11. add-sqr-sqrt 99.9%

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

   12. sqrt-unprod 98.1%

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

   13. sqr-neg 98.1%

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

   14. sqrt-unprod -0.0%

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

   15. add-sqr-sqrt 95.6%

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

5. Applied egg-rr 97.3%

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

   1. rec-exp 97.3%

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

   2. distribute-neg-frac 97.3%

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

7. Simplified 97.3%

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

8. Taylor expanded in x around 0 96.7%

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

   1. distribute-lft-in 96.7%

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

   2. add-sqr-sqrt 49.4%

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

   3. fabs-sqr 49.4%

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

   4. add-sqr-sqrt 60.8%

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

10. Applied egg-rr 60.8%

    \[\leadsto \frac{1}{\color{blue}{s \cdot 3 + s \cdot e^{\frac{x}{s}}}} \]
11. Step-by-step derivation

    1. distribute-lft-in 60.8%

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

12. Simplified 60.8%

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

13. Taylor expanded in s around inf 28.7%

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

14. Final simplification 28.7%

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

Alternative 9: 30.0% accurate, 121.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 9.999999974752427e-7) (/ 0.25 s) (/ 0.5 x)))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 9.999999974752427e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.5f / x;
	}
	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 <= 9.999999974752427e-7) then
        tmp = 0.25e0 / s
    else
        tmp = 0.5e0 / x
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.999999974752427e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.5) / x);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.999999974752427e-7))
		tmp = single(0.25) / s;
	else
		tmp = single(0.5) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 9.99999997e-7

   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. Taylor expanded in s around inf 36.0%

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

   if 9.99999997e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-/r* 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-*l/ 100.0%

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

      6. associate-*r/ 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. associate-+l+ 100.0%

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

      11. count-2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 49.2%

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

   9. Taylor expanded in s around 0 10.3%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 10: 26.9% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in s around inf 26.4%

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

3. Final simplification 26.4%

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

Reproduce

herbie shell --seed 2023228 
(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))))))