Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 22.7s

Alternatives: 12

Speedup: 2.0×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{{\left(e^{-0.25}\right)}^{\left(8 \cdot \frac{\frac{x\_m}{s}}{2}\right)}}{s}}{{\left(e^{\frac{x\_m}{-s}} + 1\right)}^{2}} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/
  (/ (pow (exp -0.25) (* 8.0 (/ (/ x_m s) 2.0))) s)
  (pow (+ (exp (/ x_m (- s))) 1.0) 2.0)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (powf(expf(-0.25f), (8.0f * ((x_m / s) / 2.0f))) / s) / powf((expf((x_m / -s)) + 1.0f), 2.0f);
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = ((exp((-0.25e0)) ** (8.0e0 * ((x_m / s) / 2.0e0))) / s) / ((exp((x_m / -s)) + 1.0e0) ** 2.0e0)
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32((exp(Float32(-0.25)) ^ Float32(Float32(8.0) * Float32(Float32(x_m / s) / Float32(2.0)))) / s) / (Float32(exp(Float32(x_m / Float32(-s))) + Float32(1.0)) ^ Float32(2.0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = ((exp(single(-0.25)) ^ (single(8.0) * ((x_m / s) / single(2.0)))) / s) / ((exp((x_m / -s)) + single(1.0)) ^ single(2.0));
end

Derivation

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. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

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. Add Preprocessing

5. Taylor expanded in x around 0 99.4%

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

   1. associate-/r* 99.5%

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

   2. mul-1-neg 99.5%

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

   3. distribute-neg-frac2 99.5%

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

   4. +-commutative 99.5%

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

   5. mul-1-neg 99.5%

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

   6. distribute-neg-frac2 99.5%

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

7. Simplified 99.5%

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

   1. distribute-frac-neg2 99.5%

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

   2. rec-exp 99.5%

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

   3. remove-double-neg 99.5%

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

   4. add-sqr-sqrt 99.5%

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

   5. sqrt-unprod 95.7%

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

   6. sqr-neg 95.7%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 91.2%

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

   9. frac-2neg 91.2%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 96.0%

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

   12. sqr-neg 96.0%

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

   13. sqrt-unprod 99.5%

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

   14. add-sqr-sqrt 99.5%

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

   15. add-sqr-sqrt 53.2%

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

   16. fabs-sqr 53.2%

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

   17. add-sqr-sqrt 95.9%

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

9. Applied egg-rr 95.9%

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

    1. rec-exp 95.9%

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

11. Simplified 95.9%

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

    1. distribute-frac-neg2 99.5%

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

    2. rec-exp 99.5%

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

    3. remove-double-neg 99.5%

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

    4. add-sqr-sqrt 99.5%

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

    5. sqrt-unprod 95.7%

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

    6. sqr-neg 95.7%

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

    7. sqrt-unprod -0.0%

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

    8. add-sqr-sqrt 91.2%

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

    9. frac-2neg 91.2%

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

    10. add-sqr-sqrt -0.0%

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

    11. sqrt-unprod 96.0%

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

    12. sqr-neg 96.0%

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

    13. sqrt-unprod 99.5%

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

    14. add-sqr-sqrt 99.5%

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

    15. add-sqr-sqrt 53.2%

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

    16. fabs-sqr 53.2%

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

    17. add-sqr-sqrt 95.9%

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

13. Applied egg-rr 63.0%

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

    1. rec-exp 95.9%

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

15. Simplified 63.3%

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

    1. neg-mul-1 63.3%

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

    2. exp-prod 63.3%

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

17. Applied egg-rr 63.3%

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

    1. add-sqr-sqrt 63.3%

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

    2. unpow-prod-down 63.3%

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

19. Applied egg-rr 63.3%

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

    1. pow-sqr 63.3%

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

21. Simplified 63.3%

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

    1. add-sqr-sqrt 63.3%

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

    2. unpow-prod-down 63.2%

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

    3. pow1/2 63.2%

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

    4. pow-exp 63.2%

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

    5. metadata-eval 63.2%

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

    6. associate-*r/ 63.2%

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

    7. pow1/2 63.2%

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

    8. pow-exp 63.2%

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

    9. metadata-eval 63.2%

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

    10. associate-*r/ 63.2%

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

23. Applied egg-rr 63.2%

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

    1. pow-sqr 63.2%

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

    2. associate-/l* 63.2%

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

25. Simplified 63.2%

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

    1. sqr-pow 63.2%

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

    2. pow1/2 63.2%

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

    3. pow-exp 63.2%

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

    4. metadata-eval 63.2%

       \[\leadsto \frac{\frac{{\left(e^{\color{blue}{-0.25}}\right)}^{\left(\frac{2 \cdot \left(2 \cdot \frac{x}{s}\right)}{2}\right)} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{2 \cdot \left(2 \cdot \frac{x}{s}\right)}{2}\right)}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

    5. associate-*r* 63.2%

       \[\leadsto \frac{\frac{{\left(e^{-0.25}\right)}^{\left(\frac{\color{blue}{\left(2 \cdot 2\right) \cdot \frac{x}{s}}}{2}\right)} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{2 \cdot \left(2 \cdot \frac{x}{s}\right)}{2}\right)}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

    6. metadata-eval 63.2%

       \[\leadsto \frac{\frac{{\left(e^{-0.25}\right)}^{\left(\frac{\color{blue}{4} \cdot \frac{x}{s}}{2}\right)} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{2 \cdot \left(2 \cdot \frac{x}{s}\right)}{2}\right)}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

    7. pow1/2 63.2%

       \[\leadsto \frac{\frac{{\left(e^{-0.25}\right)}^{\left(\frac{4 \cdot \frac{x}{s}}{2}\right)} \cdot {\color{blue}{\left({\left(e^{-0.5}\right)}^{0.5}\right)}}^{\left(\frac{2 \cdot \left(2 \cdot \frac{x}{s}\right)}{2}\right)}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

    8. pow-exp 63.2%

       \[\leadsto \frac{\frac{{\left(e^{-0.25}\right)}^{\left(\frac{4 \cdot \frac{x}{s}}{2}\right)} \cdot {\color{blue}{\left(e^{-0.5 \cdot 0.5}\right)}}^{\left(\frac{2 \cdot \left(2 \cdot \frac{x}{s}\right)}{2}\right)}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

    9. metadata-eval 63.2%

       \[\leadsto \frac{\frac{{\left(e^{-0.25}\right)}^{\left(\frac{4 \cdot \frac{x}{s}}{2}\right)} \cdot {\left(e^{\color{blue}{-0.25}}\right)}^{\left(\frac{2 \cdot \left(2 \cdot \frac{x}{s}\right)}{2}\right)}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

    10. associate-*r* 63.2%

        \[\leadsto \frac{\frac{{\left(e^{-0.25}\right)}^{\left(\frac{4 \cdot \frac{x}{s}}{2}\right)} \cdot {\left(e^{-0.25}\right)}^{\left(\frac{\color{blue}{\left(2 \cdot 2\right) \cdot \frac{x}{s}}}{2}\right)}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

    11. metadata-eval 63.2%

        \[\leadsto \frac{\frac{{\left(e^{-0.25}\right)}^{\left(\frac{4 \cdot \frac{x}{s}}{2}\right)} \cdot {\left(e^{-0.25}\right)}^{\left(\frac{\color{blue}{4} \cdot \frac{x}{s}}{2}\right)}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

27. Applied egg-rr 63.2%

    \[\leadsto \frac{\frac{\color{blue}{{\left(e^{-0.25}\right)}^{\left(\frac{4 \cdot \frac{x}{s}}{2}\right)} \cdot {\left(e^{-0.25}\right)}^{\left(\frac{4 \cdot \frac{x}{s}}{2}\right)}}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]
28. Step-by-step derivation

    1. pow-sqr 63.2%

       \[\leadsto \frac{\frac{\color{blue}{{\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{4 \cdot \frac{x}{s}}{2}\right)}}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

    2. associate-/l* 63.2%

       \[\leadsto \frac{\frac{{\left(e^{-0.25}\right)}^{\left(2 \cdot \color{blue}{\left(4 \cdot \frac{\frac{x}{s}}{2}\right)}\right)}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

    3. associate-*r* 63.2%

       \[\leadsto \frac{\frac{{\left(e^{-0.25}\right)}^{\color{blue}{\left(\left(2 \cdot 4\right) \cdot \frac{\frac{x}{s}}{2}\right)}}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

    4. metadata-eval 63.2%

       \[\leadsto \frac{\frac{{\left(e^{-0.25}\right)}^{\left(\color{blue}{8} \cdot \frac{\frac{x}{s}}{2}\right)}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

29. Simplified 63.2%

    \[\leadsto \frac{\frac{\color{blue}{{\left(e^{-0.25}\right)}^{\left(8 \cdot \frac{\frac{x}{s}}{2}\right)}}}{s}}{{\left(e^{-\frac{x}{s}} + 1\right)}^{2}} \]

30. Final simplification 63.2%

    \[\leadsto \frac{\frac{{\left(e^{-0.25}\right)}^{\left(8 \cdot \frac{\frac{x}{s}}{2}\right)}}{s}}{{\left(e^{\frac{x}{-s}} + 1\right)}^{2}} \]
31. Add Preprocessing

Alternative 2: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{x\_m}{s}}\\ \mathbf{if}\;\left|x\_m\right| \leq 0.00011600000289035961:\\ \;\;\;\;\frac{1}{s \cdot e^{2 \cdot \mathsf{log1p}\left(t\_0\right) - \frac{x\_m}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m s))))
   (if (<= (fabs x_m) 0.00011600000289035961)
     (/ 1.0 (* s (exp (- (* 2.0 (log1p t_0)) (/ x_m s)))))
     (/ (/ 0.5 s) (+ 1.0 t_0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((x_m / s));
	float tmp;
	if (fabsf(x_m) <= 0.00011600000289035961f) {
		tmp = 1.0f / (s * expf(((2.0f * log1pf(t_0)) - (x_m / s))));
	} else {
		tmp = (0.5f / s) / (1.0f + t_0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(x_m / s))
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(0.00011600000289035961))
		tmp = Float32(Float32(1.0) / Float32(s * exp(Float32(Float32(Float32(2.0) * log1p(t_0)) - Float32(x_m / s)))));
	else
		tmp = Float32(Float32(Float32(0.5) / s) / Float32(Float32(1.0) + t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1.16000003e-4

   1. Initial program 99.2%

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

      1. fabs-neg 99.2%

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

      2. distribute-frac-neg 99.2%

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

      3. distribute-frac-neg2 99.2%

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

      4. fabs-neg 99.2%

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

      5. *-commutative 99.2%

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

      6. fabs-neg 99.2%

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

      7. +-commutative 99.2%

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

      8. fabs-neg 99.2%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. associate-/r* 99.2%

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

      2. mul-1-neg 99.2%

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

      3. distribute-neg-frac2 99.2%

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

      4. +-commutative 99.2%

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

      5. mul-1-neg 99.2%

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

      6. distribute-neg-frac2 99.2%

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

   7. Simplified 99.2%

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

   9. Applied egg-rr 94.6%

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

   10. Taylor expanded in x around -inf 94.9%

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

       1. exp-neg 94.9%

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

       2. exp-sum 95.3%

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

       3. rem-exp-log 98.9%

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

       4. neg-mul-1 98.9%

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

       5. distribute-frac-neg2 98.9%

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

       6. +-commutative 98.9%

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

       7. log1p-define 99.2%

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

       8. distribute-frac-neg2 99.2%

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

       9. unsub-neg 99.2%

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

   12. Simplified 99.2%

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

   if 1.16000003e-4 < (fabs.f32 x)

   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. Step-by-step derivation

      1. fabs-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-frac-neg2 99.8%

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

      4. fabs-neg 99.8%

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

      5. *-commutative 99.8%

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

      6. fabs-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. fabs-neg 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 48.4%

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

      1. associate-*l/ 48.4%

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

      2. *-lft-identity 48.4%

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

      3. +-commutative 48.4%

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

   8. Simplified 48.4%

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

   9. Taylor expanded in x around 0 53.2%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{s}}}{1 + e^{\frac{x}{s}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{x\_m}{s}}\\ \mathbf{if}\;\left|x\_m\right| \leq 0.00011600000289035961:\\ \;\;\;\;\frac{e^{\frac{x\_m}{s} + \mathsf{log1p}\left(t\_0\right) \cdot -2}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m s))))
   (if (<= (fabs x_m) 0.00011600000289035961)
     (/ (exp (+ (/ x_m s) (* (log1p t_0) -2.0))) s)
     (/ (/ 0.5 s) (+ 1.0 t_0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((x_m / s));
	float tmp;
	if (fabsf(x_m) <= 0.00011600000289035961f) {
		tmp = expf(((x_m / s) + (log1pf(t_0) * -2.0f))) / s;
	} else {
		tmp = (0.5f / s) / (1.0f + t_0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(x_m / s))
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(0.00011600000289035961))
		tmp = Float32(exp(Float32(Float32(x_m / s) + Float32(log1p(t_0) * Float32(-2.0)))) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / s) / Float32(Float32(1.0) + t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1.16000003e-4

   1. Initial program 99.2%

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

      1. fabs-neg 99.2%

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

      2. distribute-frac-neg 99.2%

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

      3. distribute-frac-neg2 99.2%

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

      4. fabs-neg 99.2%

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

      5. *-commutative 99.2%

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

      6. fabs-neg 99.2%

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

      7. +-commutative 99.2%

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

      8. fabs-neg 99.2%

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

   3. Simplified 99.1%

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

   6. Applied egg-rr 70.6%

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

      1. associate-*r/ 70.6%

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

      2. *-rgt-identity 70.6%

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

      3. rem-exp-log 67.0%

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

      4. exp-to-pow 66.9%

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

      5. +-commutative 66.9%

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

      6. log1p-undefine 66.9%

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

      7. *-commutative 66.9%

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

      8. exp-sum 68.1%

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

      9. +-commutative 68.1%

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

      10. exp-diff 94.8%

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

      11. associate--r+ 94.8%

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

      12. exp-diff 95.2%

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

   8. Simplified 99.2%

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

   if 1.16000003e-4 < (fabs.f32 x)

   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. Step-by-step derivation

      1. fabs-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-frac-neg2 99.8%

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

      4. fabs-neg 99.8%

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

      5. *-commutative 99.8%

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

      6. fabs-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. fabs-neg 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 48.4%

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

      1. associate-*l/ 48.4%

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

      2. *-lft-identity 48.4%

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

      3. +-commutative 48.4%

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

   8. Simplified 48.4%

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

   9. Taylor expanded in x around 0 53.2%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{s}}}{1 + e^{\frac{x}{s}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.00011600000289035961:\\ \;\;\;\;\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{1 + e^{\frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{x\_m}{-s}}\\ \frac{\frac{t\_0}{s}}{{\left(t\_0 + 1\right)}^{2}} \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m (- s))))) (/ (/ t_0 s) (pow (+ t_0 1.0) 2.0))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

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. Add Preprocessing

5. Taylor expanded in x around 0 99.4%

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

   1. associate-/r* 99.5%

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

   2. mul-1-neg 99.5%

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

   3. distribute-neg-frac2 99.5%

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

   4. +-commutative 99.5%

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

   5. mul-1-neg 99.5%

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

   6. distribute-neg-frac2 99.5%

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

7. Simplified 99.5%

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

   1. distribute-frac-neg2 99.5%

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

   2. rec-exp 99.5%

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

   3. remove-double-neg 99.5%

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

   4. add-sqr-sqrt 99.5%

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

   5. sqrt-unprod 95.7%

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

   6. sqr-neg 95.7%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 91.2%

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

   9. frac-2neg 91.2%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 96.0%

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

   12. sqr-neg 96.0%

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

   13. sqrt-unprod 99.5%

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

   14. add-sqr-sqrt 99.5%

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

   15. add-sqr-sqrt 53.2%

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

   16. fabs-sqr 53.2%

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

   17. add-sqr-sqrt 95.9%

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

9. Applied egg-rr 95.9%

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

    1. rec-exp 95.9%

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

11. Simplified 95.9%

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

    1. distribute-frac-neg2 99.5%

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

    2. rec-exp 99.5%

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

    3. remove-double-neg 99.5%

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

    4. add-sqr-sqrt 99.5%

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

    5. sqrt-unprod 95.7%

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

    6. sqr-neg 95.7%

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

    7. sqrt-unprod -0.0%

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

    8. add-sqr-sqrt 91.2%

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

    9. frac-2neg 91.2%

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

    10. add-sqr-sqrt -0.0%

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

    11. sqrt-unprod 96.0%

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

    12. sqr-neg 96.0%

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

    13. sqrt-unprod 99.5%

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

    14. add-sqr-sqrt 99.5%

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

    15. add-sqr-sqrt 53.2%

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

    16. fabs-sqr 53.2%

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

    17. add-sqr-sqrt 95.9%

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

13. Applied egg-rr 63.0%

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

    1. rec-exp 95.9%

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

15. Simplified 63.3%

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

16. Final simplification 63.3%

    \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{s}}{{\left(e^{\frac{x}{-s}} + 1\right)}^{2}} \]
17. Add Preprocessing

Alternative 5: 94.8% accurate, 5.7× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ (/ 0.5 s) (+ 1.0 (exp (/ x_m s)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (0.5f / s) / (1.0f + expf((x_m / s)));
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = (0.5e0 / s) / (1.0e0 + exp((x_m / s)))
end function

Julia

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

MATLAB

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

Derivation

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. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

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. Add Preprocessing

6. Applied egg-rr 60.1%

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

   1. associate-*l/ 60.1%

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

   2. *-lft-identity 60.1%

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

   3. +-commutative 60.1%

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

8. Simplified 60.1%

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

9. Taylor expanded in x around 0 58.2%

   \[\leadsto \frac{\color{blue}{\frac{0.5}{s}}}{1 + e^{\frac{x}{s}}} \]
10. Add Preprocessing

Alternative 6: 94.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\frac{x\_m}{-s}} \cdot 0.25}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ (* (exp (/ x_m (- s))) 0.25) s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (expf((x_m / -s)) * 0.25f) / s;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = (exp((x_m / -s)) * 0.25e0) / s
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(exp(Float32(x_m / Float32(-s))) * Float32(0.25)) / s)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (exp((x_m / -s)) * single(0.25)) / s;
end

Derivation

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. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

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. Add Preprocessing

5. Taylor expanded in x around 0 99.4%

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

   1. associate-/r* 99.5%

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

   2. mul-1-neg 99.5%

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

   3. distribute-neg-frac2 99.5%

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

   4. +-commutative 99.5%

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

   5. mul-1-neg 99.5%

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

   6. distribute-neg-frac2 99.5%

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

7. Simplified 99.5%

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

   1. distribute-frac-neg2 99.5%

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

   2. rec-exp 99.5%

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

   3. remove-double-neg 99.5%

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

   4. add-sqr-sqrt 99.5%

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

   5. sqrt-unprod 95.7%

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

   6. sqr-neg 95.7%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 91.2%

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

   9. frac-2neg 91.2%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 96.0%

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

   12. sqr-neg 96.0%

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

   13. sqrt-unprod 99.5%

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

   14. add-sqr-sqrt 99.5%

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

   15. add-sqr-sqrt 53.2%

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

   16. fabs-sqr 53.2%

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

   17. add-sqr-sqrt 95.9%

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

9. Applied egg-rr 95.9%

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

    1. rec-exp 95.9%

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

11. Simplified 95.9%

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

    1. distribute-frac-neg2 99.5%

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

    2. rec-exp 99.5%

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

    3. remove-double-neg 99.5%

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

    4. add-sqr-sqrt 99.5%

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

    5. sqrt-unprod 95.7%

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

    6. sqr-neg 95.7%

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

    7. sqrt-unprod -0.0%

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

    8. add-sqr-sqrt 91.2%

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

    9. frac-2neg 91.2%

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

    10. add-sqr-sqrt -0.0%

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

    11. sqrt-unprod 96.0%

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

    12. sqr-neg 96.0%

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

    13. sqrt-unprod 99.5%

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

    14. add-sqr-sqrt 99.5%

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

    15. add-sqr-sqrt 53.2%

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

    16. fabs-sqr 53.2%

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

    17. add-sqr-sqrt 95.9%

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

13. Applied egg-rr 63.0%

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

    1. rec-exp 95.9%

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

15. Simplified 63.3%

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

16. Taylor expanded in x around 0 57.3%

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

17. Taylor expanded in x around inf 57.3%

    \[\leadsto \color{blue}{0.25 \cdot \frac{e^{-\frac{x}{s}}}{s}} \]
18. Step-by-step derivation

    1. associate-*r/ 57.3%

       \[\leadsto \color{blue}{\frac{0.25 \cdot e^{-\frac{x}{s}}}{s}} \]

    2. distribute-neg-frac2 57.3%

       \[\leadsto \frac{0.25 \cdot e^{\color{blue}{\frac{x}{-s}}}}{s} \]

19. Simplified 57.3%

    \[\leadsto \color{blue}{\frac{0.25 \cdot e^{\frac{x}{-s}}}{s}} \]

20. Final simplification 57.3%

    \[\leadsto \frac{e^{\frac{x}{-s}} \cdot 0.25}{s} \]
21. Add Preprocessing

Alternative 7: 88.7% accurate, 28.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{-0.25 \cdot x\_m + s \cdot 0.25}{s} - -0.25 \cdot \frac{x\_m}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{s}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 5.000000229068525e-19)
   (/ (- (/ (+ (* -0.25 x_m) (* s 0.25)) s) (* -0.25 (/ x_m s))) s)
   (/ (/ 0.0 s) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 5.000000229068525e-19f) {
		tmp = ((((-0.25f * x_m) + (s * 0.25f)) / s) - (-0.25f * (x_m / s))) / s;
	} else {
		tmp = (0.0f / s) / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 5.000000229068525e-19) then
        tmp = (((((-0.25e0) * x_m) + (s * 0.25e0)) / s) - ((-0.25e0) * (x_m / s))) / s
    else
        tmp = (0.0e0 / s) / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(5.000000229068525e-19))
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * x_m) + Float32(s * Float32(0.25))) / s) - Float32(Float32(-0.25) * Float32(x_m / s))) / s);
	else
		tmp = Float32(Float32(Float32(0.0) / s) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(5.000000229068525e-19))
		tmp = ((((single(-0.25) * x_m) + (s * single(0.25))) / s) - (single(-0.25) * (x_m / s))) / s;
	else
		tmp = (single(0.0) / s) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.00000023e-19

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. fabs-neg 99.4%

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

      5. *-commutative 99.4%

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

      6. fabs-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. fabs-neg 99.4%

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

   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. Add Preprocessing

   6. Applied egg-rr 91.6%

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

      1. associate-*r/ 91.7%

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

      2. times-frac 91.1%

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

      3. associate-*r/ 91.2%

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

      4. *-rgt-identity 91.2%

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

      5. +-commutative 91.2%

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

   8. Simplified 91.2%

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

      1. *-un-lft-identity 91.2%

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

      2. exp-prod 91.1%

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

   10. Applied egg-rr 91.1%

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

       1. exp-1-e 91.1%

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

   12. Simplified 91.1%

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

   13. Taylor expanded in s around -inf 31.5%

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

       1. mul-1-neg 31.5%

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

       2. log-E 62.1%

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

       3. *-commutative 62.1%

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

       4. associate-*r/ 62.2%

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

       5. *-commutative 62.2%

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

   15. Simplified 62.2%

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

   16. Taylor expanded in s around 0 63.4%

       \[\leadsto -\frac{-0.25 \cdot \frac{1 \cdot x}{s} - \color{blue}{\frac{-0.25 \cdot x + 0.25 \cdot s}{s}}}{s} \]

   if 5.00000023e-19 < x

   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. fabs-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. distribute-frac-neg2 99.6%

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

      4. fabs-neg 99.6%

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

      5. *-commutative 99.6%

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

      6. fabs-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. fabs-neg 99.6%

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

   3. Simplified 99.5%

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

   6. Applied egg-rr 11.5%

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

      1. associate-*l/ 11.5%

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

      2. *-lft-identity 11.5%

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

      3. +-commutative 11.5%

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

   8. Simplified 11.5%

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

   9. Taylor expanded in s around inf 38.8%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s}}}{1 + e^{\frac{x}{s}}} \]
   10. Step-by-step derivation

       1. div-inv 38.8%

          \[\leadsto \color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

       2. associate--l+ 38.7%

          \[\leadsto \frac{\color{blue}{0.5 + \left(0.5 \cdot \frac{x}{s} - 0.25 \cdot \frac{x}{s}\right)}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       3. associate-*r/ 38.7%

          \[\leadsto \frac{0.5 + \left(\color{blue}{\frac{0.5 \cdot x}{s}} - 0.25 \cdot \frac{x}{s}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       4. associate-*r/ 38.7%

          \[\leadsto \frac{0.5 + \left(\frac{0.5 \cdot x}{s} - \color{blue}{\frac{0.25 \cdot x}{s}}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       5. sub-div 38.7%

          \[\leadsto \frac{0.5 + \color{blue}{\frac{0.5 \cdot x - 0.25 \cdot x}{s}}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

   11. Applied egg-rr 38.7%

       \[\leadsto \color{blue}{\frac{0.5 + \frac{0.5 \cdot x - 0.25 \cdot x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

   12. Taylor expanded in s around inf 63.5%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) - 0.125 \cdot \frac{x}{s}}{s}} \]
   13. Step-by-step derivation

       1. cancel-sign-sub-inv 63.5%

          \[\leadsto \frac{\color{blue}{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}}{s} \]

       2. distribute-rgt-out-- 63.5%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{\color{blue}{x \cdot \left(0.5 - 0.25\right)}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       3. metadata-eval 63.5%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{x \cdot \color{blue}{0.25}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       4. associate-/l* 48.3%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \color{blue}{\left(x \cdot \frac{0.25}{s}\right)}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       5. metadata-eval 48.3%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{-0.125} \cdot \frac{x}{s}}{s} \]

       6. *-commutative 48.3%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{\frac{x}{s} \cdot -0.125}}{s} \]

   14. Simplified 48.3%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \frac{x}{s} \cdot -0.125}{s}} \]

   15. Taylor expanded in s around 0 88.2%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + 0.125 \cdot x}{s}}}{s} \]
   16. Step-by-step derivation

       1. distribute-rgt-out 88.2%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(-0.125 + 0.125\right)}}{s}}{s} \]

       2. metadata-eval 88.2%

          \[\leadsto \frac{\frac{x \cdot \color{blue}{0}}{s}}{s} \]

       3. mul0-rgt 88.2%

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

   17. Simplified 88.2%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{-0.25 \cdot x + s \cdot 0.25}{s} - -0.25 \cdot \frac{x}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{s}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.7% accurate, 31.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5.499999890083859 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(0.25 + \frac{-0.25 \cdot x\_m}{s}\right) - -0.25 \cdot \frac{x\_m}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{s}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 5.499999890083859e-19)
   (/ (- (+ 0.25 (/ (* -0.25 x_m) s)) (* -0.25 (/ x_m s))) s)
   (/ (/ 0.0 s) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 5.499999890083859e-19f) {
		tmp = ((0.25f + ((-0.25f * x_m) / s)) - (-0.25f * (x_m / s))) / s;
	} else {
		tmp = (0.0f / s) / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 5.499999890083859e-19) then
        tmp = ((0.25e0 + (((-0.25e0) * x_m) / s)) - ((-0.25e0) * (x_m / s))) / s
    else
        tmp = (0.0e0 / s) / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(5.499999890083859e-19))
		tmp = Float32(Float32(Float32(Float32(0.25) + Float32(Float32(Float32(-0.25) * x_m) / s)) - Float32(Float32(-0.25) * Float32(x_m / s))) / s);
	else
		tmp = Float32(Float32(Float32(0.0) / s) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(5.499999890083859e-19))
		tmp = ((single(0.25) + ((single(-0.25) * x_m) / s)) - (single(-0.25) * (x_m / s))) / s;
	else
		tmp = (single(0.0) / s) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.49999989e-19

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. fabs-neg 99.4%

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

      5. *-commutative 99.4%

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

      6. fabs-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. fabs-neg 99.4%

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

   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. Add Preprocessing

   6. Applied egg-rr 91.7%

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

      1. associate-*r/ 91.7%

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

      2. times-frac 91.1%

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

      3. associate-*r/ 91.2%

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

      4. *-rgt-identity 91.2%

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

      5. +-commutative 91.2%

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

   8. Simplified 91.2%

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

      1. *-un-lft-identity 91.2%

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

      2. exp-prod 91.2%

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

   10. Applied egg-rr 91.2%

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

       1. exp-1-e 91.2%

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

   12. Simplified 91.2%

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

   13. Taylor expanded in s around -inf 31.9%

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

       1. mul-1-neg 31.9%

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

       2. log-E 62.3%

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

       3. *-commutative 62.3%

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

       4. associate-*r/ 62.4%

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

       5. *-commutative 62.4%

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

   15. Simplified 62.4%

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

   if 5.49999989e-19 < x

   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. fabs-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. distribute-frac-neg2 99.6%

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

      4. fabs-neg 99.6%

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

      5. *-commutative 99.6%

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

      6. fabs-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. fabs-neg 99.6%

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

   3. Simplified 99.6%

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

   6. Applied egg-rr 10.7%

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

      1. associate-*l/ 10.6%

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

      2. *-lft-identity 10.6%

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

      3. +-commutative 10.6%

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

   8. Simplified 10.6%

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

   9. Taylor expanded in s around inf 38.2%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s}}}{1 + e^{\frac{x}{s}}} \]
   10. Step-by-step derivation

       1. div-inv 38.2%

          \[\leadsto \color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

       2. associate--l+ 38.1%

          \[\leadsto \frac{\color{blue}{0.5 + \left(0.5 \cdot \frac{x}{s} - 0.25 \cdot \frac{x}{s}\right)}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       3. associate-*r/ 38.1%

          \[\leadsto \frac{0.5 + \left(\color{blue}{\frac{0.5 \cdot x}{s}} - 0.25 \cdot \frac{x}{s}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       4. associate-*r/ 38.1%

          \[\leadsto \frac{0.5 + \left(\frac{0.5 \cdot x}{s} - \color{blue}{\frac{0.25 \cdot x}{s}}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       5. sub-div 38.1%

          \[\leadsto \frac{0.5 + \color{blue}{\frac{0.5 \cdot x - 0.25 \cdot x}{s}}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

   11. Applied egg-rr 38.1%

       \[\leadsto \color{blue}{\frac{0.5 + \frac{0.5 \cdot x - 0.25 \cdot x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

   12. Taylor expanded in s around inf 63.1%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) - 0.125 \cdot \frac{x}{s}}{s}} \]
   13. Step-by-step derivation

       1. cancel-sign-sub-inv 63.1%

          \[\leadsto \frac{\color{blue}{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}}{s} \]

       2. distribute-rgt-out-- 63.1%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{\color{blue}{x \cdot \left(0.5 - 0.25\right)}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       3. metadata-eval 63.1%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{x \cdot \color{blue}{0.25}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       4. associate-/l* 47.9%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \color{blue}{\left(x \cdot \frac{0.25}{s}\right)}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       5. metadata-eval 47.9%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{-0.125} \cdot \frac{x}{s}}{s} \]

       6. *-commutative 47.9%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{\frac{x}{s} \cdot -0.125}}{s} \]

   14. Simplified 47.9%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \frac{x}{s} \cdot -0.125}{s}} \]

   15. Taylor expanded in s around 0 89.0%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + 0.125 \cdot x}{s}}}{s} \]
   16. Step-by-step derivation

       1. distribute-rgt-out 89.0%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(-0.125 + 0.125\right)}}{s}}{s} \]

       2. metadata-eval 89.0%

          \[\leadsto \frac{\frac{x \cdot \color{blue}{0}}{s}}{s} \]

       3. mul0-rgt 89.0%

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

   17. Simplified 89.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.499999890083859 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(0.25 + \frac{-0.25 \cdot x}{s}\right) - -0.25 \cdot \frac{x}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{s}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.5% accurate, 31.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5.499999890083859 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(0.25 + \frac{x\_m}{s} \cdot 0.125\right) + \frac{x\_m}{s} \cdot -0.125}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{s}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 5.499999890083859e-19)
   (/ (+ (+ 0.25 (* (/ x_m s) 0.125)) (* (/ x_m s) -0.125)) s)
   (/ (/ 0.0 s) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 5.499999890083859e-19f) {
		tmp = ((0.25f + ((x_m / s) * 0.125f)) + ((x_m / s) * -0.125f)) / s;
	} else {
		tmp = (0.0f / s) / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 5.499999890083859e-19) then
        tmp = ((0.25e0 + ((x_m / s) * 0.125e0)) + ((x_m / s) * (-0.125e0))) / s
    else
        tmp = (0.0e0 / s) / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(5.499999890083859e-19))
		tmp = Float32(Float32(Float32(Float32(0.25) + Float32(Float32(x_m / s) * Float32(0.125))) + Float32(Float32(x_m / s) * Float32(-0.125))) / s);
	else
		tmp = Float32(Float32(Float32(0.0) / s) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(5.499999890083859e-19))
		tmp = ((single(0.25) + ((x_m / s) * single(0.125))) + ((x_m / s) * single(-0.125))) / s;
	else
		tmp = (single(0.0) / s) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.49999989e-19

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. fabs-neg 99.4%

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

      5. *-commutative 99.4%

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

      6. fabs-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. fabs-neg 99.4%

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

   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. Add Preprocessing

   6. Applied egg-rr 92.3%

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

      1. associate-*l/ 92.3%

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

      2. *-lft-identity 92.3%

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

      3. +-commutative 92.3%

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

   8. Simplified 92.3%

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

   9. Taylor expanded in s around inf 34.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s}}}{1 + e^{\frac{x}{s}}} \]
   10. Step-by-step derivation

       1. div-inv 34.3%

          \[\leadsto \color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

       2. associate--l+ 34.4%

          \[\leadsto \frac{\color{blue}{0.5 + \left(0.5 \cdot \frac{x}{s} - 0.25 \cdot \frac{x}{s}\right)}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       3. associate-*r/ 34.4%

          \[\leadsto \frac{0.5 + \left(\color{blue}{\frac{0.5 \cdot x}{s}} - 0.25 \cdot \frac{x}{s}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       4. associate-*r/ 34.4%

          \[\leadsto \frac{0.5 + \left(\frac{0.5 \cdot x}{s} - \color{blue}{\frac{0.25 \cdot x}{s}}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       5. sub-div 35.0%

          \[\leadsto \frac{0.5 + \color{blue}{\frac{0.5 \cdot x - 0.25 \cdot x}{s}}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

   11. Applied egg-rr 35.0%

       \[\leadsto \color{blue}{\frac{0.5 + \frac{0.5 \cdot x - 0.25 \cdot x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

   12. Taylor expanded in s around inf 62.4%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) - 0.125 \cdot \frac{x}{s}}{s}} \]
   13. Step-by-step derivation

       1. cancel-sign-sub-inv 62.4%

          \[\leadsto \frac{\color{blue}{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}}{s} \]

       2. distribute-rgt-out-- 62.4%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{\color{blue}{x \cdot \left(0.5 - 0.25\right)}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       3. metadata-eval 62.4%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{x \cdot \color{blue}{0.25}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       4. associate-/l* 55.5%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \color{blue}{\left(x \cdot \frac{0.25}{s}\right)}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       5. metadata-eval 55.5%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{-0.125} \cdot \frac{x}{s}}{s} \]

       6. *-commutative 55.5%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{\frac{x}{s} \cdot -0.125}}{s} \]

   14. Simplified 55.5%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \frac{x}{s} \cdot -0.125}{s}} \]

   15. Taylor expanded in x around 0 62.3%

       \[\leadsto \frac{\color{blue}{\left(0.25 + 0.125 \cdot \frac{x}{s}\right)} + \frac{x}{s} \cdot -0.125}{s} \]
   16. Step-by-step derivation

       1. *-commutative 62.3%

          \[\leadsto \frac{\left(0.25 + \color{blue}{\frac{x}{s} \cdot 0.125}\right) + \frac{x}{s} \cdot -0.125}{s} \]

   17. Simplified 62.3%

       \[\leadsto \frac{\color{blue}{\left(0.25 + \frac{x}{s} \cdot 0.125\right)} + \frac{x}{s} \cdot -0.125}{s} \]

   if 5.49999989e-19 < x

   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. fabs-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. distribute-frac-neg2 99.6%

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

      4. fabs-neg 99.6%

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

      5. *-commutative 99.6%

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

      6. fabs-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. fabs-neg 99.6%

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

   3. Simplified 99.6%

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

   6. Applied egg-rr 10.7%

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

      1. associate-*l/ 10.6%

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

      2. *-lft-identity 10.6%

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

      3. +-commutative 10.6%

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

   8. Simplified 10.6%

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

   9. Taylor expanded in s around inf 38.2%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s}}}{1 + e^{\frac{x}{s}}} \]
   10. Step-by-step derivation

       1. div-inv 38.2%

          \[\leadsto \color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

       2. associate--l+ 38.1%

          \[\leadsto \frac{\color{blue}{0.5 + \left(0.5 \cdot \frac{x}{s} - 0.25 \cdot \frac{x}{s}\right)}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       3. associate-*r/ 38.1%

          \[\leadsto \frac{0.5 + \left(\color{blue}{\frac{0.5 \cdot x}{s}} - 0.25 \cdot \frac{x}{s}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       4. associate-*r/ 38.1%

          \[\leadsto \frac{0.5 + \left(\frac{0.5 \cdot x}{s} - \color{blue}{\frac{0.25 \cdot x}{s}}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       5. sub-div 38.1%

          \[\leadsto \frac{0.5 + \color{blue}{\frac{0.5 \cdot x - 0.25 \cdot x}{s}}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

   11. Applied egg-rr 38.1%

       \[\leadsto \color{blue}{\frac{0.5 + \frac{0.5 \cdot x - 0.25 \cdot x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

   12. Taylor expanded in s around inf 63.1%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) - 0.125 \cdot \frac{x}{s}}{s}} \]
   13. Step-by-step derivation

       1. cancel-sign-sub-inv 63.1%

          \[\leadsto \frac{\color{blue}{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}}{s} \]

       2. distribute-rgt-out-- 63.1%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{\color{blue}{x \cdot \left(0.5 - 0.25\right)}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       3. metadata-eval 63.1%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{x \cdot \color{blue}{0.25}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       4. associate-/l* 47.9%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \color{blue}{\left(x \cdot \frac{0.25}{s}\right)}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       5. metadata-eval 47.9%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{-0.125} \cdot \frac{x}{s}}{s} \]

       6. *-commutative 47.9%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{\frac{x}{s} \cdot -0.125}}{s} \]

   14. Simplified 47.9%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \frac{x}{s} \cdot -0.125}{s}} \]

   15. Taylor expanded in s around 0 89.0%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + 0.125 \cdot x}{s}}}{s} \]
   16. Step-by-step derivation

       1. distribute-rgt-out 89.0%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(-0.125 + 0.125\right)}}{s}}{s} \]

       2. metadata-eval 89.0%

          \[\leadsto \frac{\frac{x \cdot \color{blue}{0}}{s}}{s} \]

       3. mul0-rgt 89.0%

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

   17. Simplified 89.0%

       \[\leadsto \frac{\color{blue}{\frac{0}{s}}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 83.0% accurate, 51.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{s \cdot 0.25}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{s}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 4.0000000126843074e-29) (/ (/ (* s 0.25) s) s) (/ (/ 0.0 s) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 4.0000000126843074e-29f) {
		tmp = ((s * 0.25f) / s) / s;
	} else {
		tmp = (0.0f / s) / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 4.0000000126843074e-29) then
        tmp = ((s * 0.25e0) / s) / s
    else
        tmp = (0.0e0 / s) / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(4.0000000126843074e-29))
		tmp = Float32(Float32(Float32(s * Float32(0.25)) / s) / s);
	else
		tmp = Float32(Float32(Float32(0.0) / s) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(4.0000000126843074e-29))
		tmp = ((s * single(0.25)) / s) / s;
	else
		tmp = (single(0.0) / s) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000001e-29

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. fabs-neg 99.4%

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

      5. *-commutative 99.4%

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

      6. fabs-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. fabs-neg 99.4%

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

   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. Add Preprocessing

   6. Applied egg-rr 97.2%

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

      1. associate-*l/ 97.2%

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

      2. *-lft-identity 97.2%

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

      3. +-commutative 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in s around inf 27.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s}}}{1 + e^{\frac{x}{s}}} \]
   10. Step-by-step derivation

       1. div-inv 27.3%

          \[\leadsto \color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

       2. associate--l+ 27.3%

          \[\leadsto \frac{\color{blue}{0.5 + \left(0.5 \cdot \frac{x}{s} - 0.25 \cdot \frac{x}{s}\right)}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       3. associate-*r/ 27.3%

          \[\leadsto \frac{0.5 + \left(\color{blue}{\frac{0.5 \cdot x}{s}} - 0.25 \cdot \frac{x}{s}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       4. associate-*r/ 27.4%

          \[\leadsto \frac{0.5 + \left(\frac{0.5 \cdot x}{s} - \color{blue}{\frac{0.25 \cdot x}{s}}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       5. sub-div 28.0%

          \[\leadsto \frac{0.5 + \color{blue}{\frac{0.5 \cdot x - 0.25 \cdot x}{s}}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

   11. Applied egg-rr 28.0%

       \[\leadsto \color{blue}{\frac{0.5 + \frac{0.5 \cdot x - 0.25 \cdot x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

   12. Taylor expanded in s around inf 59.3%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) - 0.125 \cdot \frac{x}{s}}{s}} \]
   13. Step-by-step derivation

       1. cancel-sign-sub-inv 59.3%

          \[\leadsto \frac{\color{blue}{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}}{s} \]

       2. distribute-rgt-out-- 59.3%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{\color{blue}{x \cdot \left(0.5 - 0.25\right)}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       3. metadata-eval 59.3%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{x \cdot \color{blue}{0.25}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       4. associate-/l* 52.9%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \color{blue}{\left(x \cdot \frac{0.25}{s}\right)}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       5. metadata-eval 52.9%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{-0.125} \cdot \frac{x}{s}}{s} \]

       6. *-commutative 52.9%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{\frac{x}{s} \cdot -0.125}}{s} \]

   14. Simplified 52.9%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \frac{x}{s} \cdot -0.125}{s}} \]

   15. Taylor expanded in s around 0 81.7%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + \left(0.125 \cdot x + 0.25 \cdot s\right)}{s}}}{s} \]
   16. Step-by-step derivation

       1. +-commutative 81.7%

          \[\leadsto \frac{\frac{\color{blue}{\left(0.125 \cdot x + 0.25 \cdot s\right) + -0.125 \cdot x}}{s}}{s} \]

       2. +-commutative 81.7%

          \[\leadsto \frac{\frac{\color{blue}{\left(0.25 \cdot s + 0.125 \cdot x\right)} + -0.125 \cdot x}{s}}{s} \]

       3. associate-+l+ 26.0%

          \[\leadsto \frac{\frac{\color{blue}{0.25 \cdot s + \left(0.125 \cdot x + -0.125 \cdot x\right)}}{s}}{s} \]

       4. *-commutative 26.0%

          \[\leadsto \frac{\frac{\color{blue}{s \cdot 0.25} + \left(0.125 \cdot x + -0.125 \cdot x\right)}{s}}{s} \]

       5. distribute-rgt-out 26.0%

          \[\leadsto \frac{\frac{s \cdot 0.25 + \color{blue}{x \cdot \left(0.125 + -0.125\right)}}{s}}{s} \]

       6. metadata-eval 26.0%

          \[\leadsto \frac{\frac{s \cdot 0.25 + x \cdot \color{blue}{0}}{s}}{s} \]

       7. mul0-rgt 26.0%

          \[\leadsto \frac{\frac{s \cdot 0.25 + \color{blue}{0}}{s}}{s} \]

   17. Simplified 26.0%

       \[\leadsto \frac{\color{blue}{\frac{s \cdot 0.25 + 0}{s}}}{s} \]

   if 4.00000001e-29 < x

   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. fabs-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. distribute-frac-neg2 99.6%

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

      4. fabs-neg 99.6%

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

      5. *-commutative 99.6%

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

      6. fabs-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. fabs-neg 99.6%

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

   3. Simplified 99.5%

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

   6. Applied egg-rr 19.3%

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

      1. associate-*l/ 19.3%

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

      2. *-lft-identity 19.3%

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

      3. +-commutative 19.3%

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

   8. Simplified 19.3%

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

   9. Taylor expanded in s around inf 45.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s}}}{1 + e^{\frac{x}{s}}} \]
   10. Step-by-step derivation

       1. div-inv 45.2%

          \[\leadsto \color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

       2. associate--l+ 45.2%

          \[\leadsto \frac{\color{blue}{0.5 + \left(0.5 \cdot \frac{x}{s} - 0.25 \cdot \frac{x}{s}\right)}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       3. associate-*r/ 45.2%

          \[\leadsto \frac{0.5 + \left(\color{blue}{\frac{0.5 \cdot x}{s}} - 0.25 \cdot \frac{x}{s}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       4. associate-*r/ 45.2%

          \[\leadsto \frac{0.5 + \left(\frac{0.5 \cdot x}{s} - \color{blue}{\frac{0.25 \cdot x}{s}}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       5. sub-div 45.2%

          \[\leadsto \frac{0.5 + \color{blue}{\frac{0.5 \cdot x - 0.25 \cdot x}{s}}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

   11. Applied egg-rr 45.2%

       \[\leadsto \color{blue}{\frac{0.5 + \frac{0.5 \cdot x - 0.25 \cdot x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

   12. Taylor expanded in s around inf 66.3%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) - 0.125 \cdot \frac{x}{s}}{s}} \]
   13. Step-by-step derivation

       1. cancel-sign-sub-inv 66.3%

          \[\leadsto \frac{\color{blue}{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}}{s} \]

       2. distribute-rgt-out-- 66.3%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{\color{blue}{x \cdot \left(0.5 - 0.25\right)}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       3. metadata-eval 66.3%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{x \cdot \color{blue}{0.25}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       4. associate-/l* 52.1%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \color{blue}{\left(x \cdot \frac{0.25}{s}\right)}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       5. metadata-eval 52.1%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{-0.125} \cdot \frac{x}{s}}{s} \]

       6. *-commutative 52.1%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{\frac{x}{s} \cdot -0.125}}{s} \]

   14. Simplified 52.1%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \frac{x}{s} \cdot -0.125}{s}} \]

   15. Taylor expanded in s around 0 80.8%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + 0.125 \cdot x}{s}}}{s} \]
   16. Step-by-step derivation

       1. distribute-rgt-out 80.8%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(-0.125 + 0.125\right)}}{s}}{s} \]

       2. metadata-eval 80.8%

          \[\leadsto \frac{\frac{x \cdot \color{blue}{0}}{s}}{s} \]

       3. mul0-rgt 80.8%

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

   17. Simplified 80.8%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{s \cdot 0.25}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{s}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.0% accurate, 61.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{s}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 4.0000000126843074e-29) (/ 0.25 s) (/ (/ 0.0 s) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 4.0000000126843074e-29f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.0f / s) / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 4.0000000126843074e-29) then
        tmp = 0.25e0 / s
    else
        tmp = (0.0e0 / s) / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(4.0000000126843074e-29))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.0) / s) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(4.0000000126843074e-29))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.0) / s) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000001e-29

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. fabs-neg 99.4%

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

      5. *-commutative 99.4%

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

      6. fabs-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. fabs-neg 99.4%

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

   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. Add Preprocessing

   5. Taylor expanded in s around inf 26.0%

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

   if 4.00000001e-29 < x

   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. fabs-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. distribute-frac-neg2 99.6%

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

      4. fabs-neg 99.6%

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

      5. *-commutative 99.6%

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

      6. fabs-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. fabs-neg 99.6%

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

   3. Simplified 99.5%

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

   6. Applied egg-rr 19.3%

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

      1. associate-*l/ 19.3%

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

      2. *-lft-identity 19.3%

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

      3. +-commutative 19.3%

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

   8. Simplified 19.3%

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

   9. Taylor expanded in s around inf 45.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s}}}{1 + e^{\frac{x}{s}}} \]
   10. Step-by-step derivation

       1. div-inv 45.2%

          \[\leadsto \color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

       2. associate--l+ 45.2%

          \[\leadsto \frac{\color{blue}{0.5 + \left(0.5 \cdot \frac{x}{s} - 0.25 \cdot \frac{x}{s}\right)}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       3. associate-*r/ 45.2%

          \[\leadsto \frac{0.5 + \left(\color{blue}{\frac{0.5 \cdot x}{s}} - 0.25 \cdot \frac{x}{s}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       4. associate-*r/ 45.2%

          \[\leadsto \frac{0.5 + \left(\frac{0.5 \cdot x}{s} - \color{blue}{\frac{0.25 \cdot x}{s}}\right)}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

       5. sub-div 45.2%

          \[\leadsto \frac{0.5 + \color{blue}{\frac{0.5 \cdot x - 0.25 \cdot x}{s}}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]

   11. Applied egg-rr 45.2%

       \[\leadsto \color{blue}{\frac{0.5 + \frac{0.5 \cdot x - 0.25 \cdot x}{s}}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}}} \]

   12. Taylor expanded in s around inf 66.3%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) - 0.125 \cdot \frac{x}{s}}{s}} \]
   13. Step-by-step derivation

       1. cancel-sign-sub-inv 66.3%

          \[\leadsto \frac{\color{blue}{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}}{s} \]

       2. distribute-rgt-out-- 66.3%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{\color{blue}{x \cdot \left(0.5 - 0.25\right)}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       3. metadata-eval 66.3%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{x \cdot \color{blue}{0.25}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       4. associate-/l* 52.1%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \color{blue}{\left(x \cdot \frac{0.25}{s}\right)}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       5. metadata-eval 52.1%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{-0.125} \cdot \frac{x}{s}}{s} \]

       6. *-commutative 52.1%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \color{blue}{\frac{x}{s} \cdot -0.125}}{s} \]

   14. Simplified 52.1%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \left(x \cdot \frac{0.25}{s}\right)\right) + \frac{x}{s} \cdot -0.125}{s}} \]

   15. Taylor expanded in s around 0 80.8%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + 0.125 \cdot x}{s}}}{s} \]
   16. Step-by-step derivation

       1. distribute-rgt-out 80.8%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(-0.125 + 0.125\right)}}{s}}{s} \]

       2. metadata-eval 80.8%

          \[\leadsto \frac{\frac{x \cdot \color{blue}{0}}{s}}{s} \]

       3. mul0-rgt 80.8%

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

   17. Simplified 80.8%

       \[\leadsto \frac{\color{blue}{\frac{0}{s}}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 27.1% accurate, 206.7× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.25 s))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

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. Add Preprocessing

5. Taylor expanded in s around inf 23.3%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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))))))