Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 14.0s

Alternatives: 11

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

   2. fabs-neg 99.0%

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

   3. distribute-lft-in 99.0%

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

   4. *-rgt-identity 99.0%

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

   5. fabs-neg 99.0%

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

   6. distribute-rgt-in 99.1%

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

   7. cancel-sign-sub 99.1%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 75.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.25999999046325684:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{x}{-s}}}{s}}{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 0.25999999046325684)
   (/ (exp (- (/ x s) (* 2.0 (log1p (exp (/ x s)))))) s)
   (/ (/ (exp (/ x (- s))) s) 4.0)))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 0.25999999046325684f) {
		tmp = expf(((x / s) - (2.0f * log1pf(expf((x / s)))))) / s;
	} else {
		tmp = (expf((x / -s)) / s) / 4.0f;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.25999999046325684))
		tmp = Float32(exp(Float32(Float32(x / s) - Float32(Float32(2.0) * log1p(exp(Float32(x / s)))))) / s);
	else
		tmp = Float32(Float32(exp(Float32(x / Float32(-s))) / s) / Float32(4.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.25999999

   1. Initial program 97.9%

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

      1. *-commutative 97.9%

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

   3. Simplified 97.9%

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

   6. Applied egg-rr 91.8%

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

      1. exp-diff 77.1%

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

      2. add-exp-log 81.2%

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

      3. *-commutative 81.2%

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

      4. associate-/r* 81.0%

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

      5. +-commutative 81.0%

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

   8. Applied egg-rr 81.0%

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

      1. add-exp-log 81.0%

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

      2. log-div 80.9%

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

      3. add-log-exp 95.9%

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

      4. +-commutative 95.9%

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

      5. log-pow 97.2%

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

      6. +-commutative 97.2%

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

      7. log1p-define 97.4%

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

   10. Applied egg-rr 97.4%

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

   if 0.25999999 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

      2. mul-1-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. mul-1-neg 100.0%

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

      6. distribute-neg-frac2 100.0%

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

   7. Simplified 100.0%

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

      1. distribute-frac-neg2 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 3.1%

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

      8. div-inv 3.1%

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

      9. exp-prod 3.1%

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

      10. add-sqr-sqrt 1.6%

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

      11. fabs-sqr 1.6%

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

      12. add-sqr-sqrt 50.1%

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

      13. add-sqr-sqrt -0.0%

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

      14. sqrt-unprod 53.0%

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

      15. sqr-neg 53.0%

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

      16. sqrt-unprod 53.0%

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

      17. add-sqr-sqrt 53.0%

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

      18. exp-prod 53.0%

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

      19. div-inv 53.0%

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

   9. Applied egg-rr 53.0%

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

       1. rec-exp 53.0%

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

       2. distribute-frac-neg 53.0%

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

   11. Simplified 53.0%

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

   12. Taylor expanded in s around inf 53.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.25999999046325684:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{x}{-s}}}{s}}{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x (- s))))) (/ (/ t_0 s) (pow (+ 1.0 t_0) 2.0))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in x around 0 99.0%

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

   1. associate-/r* 98.9%

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

   2. mul-1-neg 98.9%

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

   3. distribute-neg-frac2 98.9%

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

   4. +-commutative 98.9%

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

   5. mul-1-neg 98.9%

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

   6. distribute-neg-frac2 98.9%

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

7. Simplified 98.9%

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

   1. distribute-frac-neg2 98.9%

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

   2. exp-neg 98.8%

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

   3. add-sqr-sqrt 98.7%

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

   4. sqrt-unprod 92.5%

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

   5. sqr-neg 92.5%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 21.8%

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

   8. div-inv 21.8%

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

   9. exp-prod 22.2%

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

   10. add-sqr-sqrt 12.7%

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

   11. fabs-sqr 12.7%

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

   12. add-sqr-sqrt 52.8%

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

   13. add-sqr-sqrt 19.8%

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

   14. sqrt-unprod 52.5%

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

   15. sqr-neg 52.5%

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

   16. sqrt-unprod 52.5%

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

   17. add-sqr-sqrt 52.5%

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

   18. exp-prod 60.7%

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

   19. div-inv 60.7%

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

9. Applied egg-rr 60.7%

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

    1. rec-exp 60.8%

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

    2. distribute-frac-neg 60.8%

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

11. Simplified 60.8%

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

    1. distribute-frac-neg2 98.9%

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

    2. exp-neg 98.8%

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

    3. add-sqr-sqrt 98.7%

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

    4. sqrt-unprod 92.5%

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

    5. sqr-neg 92.5%

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

    6. sqrt-unprod -0.0%

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

    7. add-sqr-sqrt 21.8%

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

    8. div-inv 21.8%

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

    9. exp-prod 22.2%

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

    10. add-sqr-sqrt 12.7%

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

    11. fabs-sqr 12.7%

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

    12. add-sqr-sqrt 52.8%

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

    13. add-sqr-sqrt 19.8%

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

    14. sqrt-unprod 52.5%

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

    15. sqr-neg 52.5%

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

    16. sqrt-unprod 52.5%

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

    17. add-sqr-sqrt 52.5%

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

    18. exp-prod 60.7%

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

    19. div-inv 60.7%

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

13. Applied egg-rr 61.7%

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

    1. rec-exp 60.8%

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

    2. distribute-frac-neg 60.8%

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

15. Simplified 61.8%

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

16. Final simplification 61.8%

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

Alternative 4: 60.7% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ (/ (exp (/ x (- s))) s) (pow (- 2.0 (/ x s)) 2.0)))

C

float code(float x, float s) {
	return (expf((x / -s)) / s) / powf((2.0f - (x / s)), 2.0f);
}

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(exp(Float32(x / Float32(-s))) / s) / (Float32(Float32(2.0) - Float32(x / s)) ^ Float32(2.0)))
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in x around 0 99.0%

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

   1. associate-/r* 98.9%

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

   2. mul-1-neg 98.9%

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

   3. distribute-neg-frac2 98.9%

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

   4. +-commutative 98.9%

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

   5. mul-1-neg 98.9%

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

   6. distribute-neg-frac2 98.9%

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

7. Simplified 98.9%

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

   1. distribute-frac-neg2 98.9%

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

   2. exp-neg 98.8%

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

   3. add-sqr-sqrt 98.7%

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

   4. sqrt-unprod 92.5%

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

   5. sqr-neg 92.5%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 21.8%

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

   8. div-inv 21.8%

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

   9. exp-prod 22.2%

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

   10. add-sqr-sqrt 12.7%

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

   11. fabs-sqr 12.7%

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

   12. add-sqr-sqrt 52.8%

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

   13. add-sqr-sqrt 19.8%

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

   14. sqrt-unprod 52.5%

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

   15. sqr-neg 52.5%

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

   16. sqrt-unprod 52.5%

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

   17. add-sqr-sqrt 52.5%

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

   18. exp-prod 60.7%

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

   19. div-inv 60.7%

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

9. Applied egg-rr 60.7%

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

    1. rec-exp 60.8%

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

    2. distribute-frac-neg 60.8%

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

11. Simplified 60.8%

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

    1. distribute-frac-neg2 98.9%

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

    2. exp-neg 98.8%

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

    3. add-sqr-sqrt 98.7%

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

    4. sqrt-unprod 92.5%

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

    5. sqr-neg 92.5%

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

    6. sqrt-unprod -0.0%

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

    7. add-sqr-sqrt 21.8%

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

    8. div-inv 21.8%

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

    9. exp-prod 22.2%

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

    10. add-sqr-sqrt 12.7%

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

    11. fabs-sqr 12.7%

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

    12. add-sqr-sqrt 52.8%

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

    13. add-sqr-sqrt 19.8%

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

    14. sqrt-unprod 52.5%

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

    15. sqr-neg 52.5%

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

    16. sqrt-unprod 52.5%

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

    17. add-sqr-sqrt 52.5%

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

    18. exp-prod 60.7%

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

    19. div-inv 60.7%

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

13. Applied egg-rr 61.7%

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

    1. rec-exp 60.8%

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

    2. distribute-frac-neg 60.8%

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

15. Simplified 61.8%

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

16. Taylor expanded in x around 0 56.7%

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

    1. mul-1-neg 56.7%

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

    2. unsub-neg 56.7%

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

18. Simplified 56.7%

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

19. Final simplification 56.7%

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

Alternative 5: 60.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{x}{s}}} \cdot \frac{0.5}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (* (/ 1.0 (+ 1.0 (exp (/ x s)))) (/ 0.5 s)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

3. Simplified 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. times-frac 98.9%

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

6. Applied egg-rr 64.5%

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

7. Taylor expanded in x around 0 56.8%

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

8. Final simplification 56.8%

   \[\leadsto \frac{1}{1 + e^{\frac{x}{s}}} \cdot \frac{0.5}{s} \]
9. Add Preprocessing

Alternative 6: 60.0% accurate, 5.7× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ (/ (exp (/ x (- s))) s) 4.0))

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (exp((x / -s)) / s) / 4.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in x around 0 99.0%

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

   1. associate-/r* 98.9%

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

   2. mul-1-neg 98.9%

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

   3. distribute-neg-frac2 98.9%

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

   4. +-commutative 98.9%

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

   5. mul-1-neg 98.9%

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

   6. distribute-neg-frac2 98.9%

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

7. Simplified 98.9%

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

   1. distribute-frac-neg2 98.9%

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

   2. exp-neg 98.8%

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

   3. add-sqr-sqrt 98.7%

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

   4. sqrt-unprod 92.5%

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

   5. sqr-neg 92.5%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 21.8%

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

   8. div-inv 21.8%

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

   9. exp-prod 22.2%

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

   10. add-sqr-sqrt 12.7%

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

   11. fabs-sqr 12.7%

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

   12. add-sqr-sqrt 52.8%

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

   13. add-sqr-sqrt 19.8%

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

   14. sqrt-unprod 52.5%

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

   15. sqr-neg 52.5%

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

   16. sqrt-unprod 52.5%

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

   17. add-sqr-sqrt 52.5%

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

   18. exp-prod 60.7%

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

   19. div-inv 60.7%

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

9. Applied egg-rr 60.7%

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

    1. rec-exp 60.8%

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

    2. distribute-frac-neg 60.8%

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

11. Simplified 60.8%

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

12. Taylor expanded in s around inf 55.9%

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

13. Final simplification 55.9%

    \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{s}}{4} \]
14. Add Preprocessing

Alternative 7: 28.8% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.3333333333333333 + 0.3333333333333333 \cdot \frac{s}{x}\right) - \frac{s}{x} \cdot 0.4444444444444444}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 3.9999998989515007e-5)
   (/ 0.25 s)
   (/
    (-
     (+ 0.3333333333333333 (* 0.3333333333333333 (/ s x)))
     (* (/ s x) 0.4444444444444444))
    x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 3.9999998989515007e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = ((0.3333333333333333f + (0.3333333333333333f * (s / x))) - ((s / x) * 0.4444444444444444f)) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 3.9999998989515007e-5) then
        tmp = 0.25e0 / s
    else
        tmp = ((0.3333333333333333e0 + (0.3333333333333333e0 * (s / x))) - ((s / x) * 0.4444444444444444e0)) / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.9999998989515007e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(Float32(0.3333333333333333) + Float32(Float32(0.3333333333333333) * Float32(s / x))) - Float32(Float32(s / x) * Float32(0.4444444444444444))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.9999998989515007e-5))
		tmp = single(0.25) / s;
	else
		tmp = ((single(0.3333333333333333) + (single(0.3333333333333333) * (s / x))) - ((s / x) * single(0.4444444444444444))) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999e-5

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in s around inf 33.5%

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

   if 3.9999999e-5 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 100.0%

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

      3. div-inv 100.0%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 3.1%

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

      6. sqr-neg 3.1%

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

      7. sqrt-unprod 3.1%

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

      8. add-sqr-sqrt 3.1%

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

      9. add-sqr-sqrt 3.1%

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

      10. fabs-sqr 3.1%

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

      11. add-sqr-sqrt 3.1%

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

      12. *-commutative 3.1%

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

      13. associate-*l* 3.1%

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

   6. Applied egg-rr -0.0%

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

      1. associate-*r/ -0.0%

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

      2. *-rgt-identity -0.0%

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

      3. +-commutative -0.0%

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

   8. Simplified -0.0%

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

   9. Taylor expanded in x around 0 0.9%

      \[\leadsto \frac{e^{\frac{x}{s}}}{\color{blue}{4 \cdot s + x \cdot \left(4 + 3 \cdot \frac{x}{s}\right)}} \]

   10. Taylor expanded in x around 0 22.3%

       \[\leadsto \frac{\color{blue}{1 + \frac{x}{s}}}{4 \cdot s + x \cdot \left(4 + 3 \cdot \frac{x}{s}\right)} \]

   11. Taylor expanded in x around inf 11.0%

       \[\leadsto \color{blue}{\frac{\left(0.3333333333333333 + 0.3333333333333333 \cdot \frac{s}{x}\right) - 0.4444444444444444 \cdot \frac{s}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.3333333333333333 + 0.3333333333333333 \cdot \frac{s}{x}\right) - \frac{s}{x} \cdot 0.4444444444444444}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.9% accurate, 32.6× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \frac{x}{s}}{s \cdot 4 + x \cdot \left(4 + \frac{x}{s} \cdot 3\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (+ 1.0 (/ x s)) (+ (* s 4.0) (* x (+ 4.0 (* (/ x s) 3.0))))))

C

float code(float x, float s) {
	return (1.0f + (x / s)) / ((s * 4.0f) + (x * (4.0f + ((x / s) * 3.0f))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 + (x / s)) / ((s * 4.0e0) + (x * (4.0e0 + ((x / s) * 3.0e0))))
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) + Float32(x / s)) / Float32(Float32(s * Float32(4.0)) + Float32(x * Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(3.0))))))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) + (x / s)) / ((s * single(4.0)) + (x * (single(4.0) + ((x / s) * single(3.0)))));
end

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

3. Simplified 99.0%

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

   1. div-inv 99.0%

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

   2. div-inv 99.0%

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

   3. div-inv 99.0%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 22.0%

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

   6. sqr-neg 22.0%

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

   7. sqrt-unprod 21.8%

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

   8. add-sqr-sqrt 21.8%

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

   9. add-sqr-sqrt 12.5%

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

   10. fabs-sqr 12.5%

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

   11. add-sqr-sqrt 60.0%

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

   12. *-commutative 60.0%

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

   13. associate-*l* 60.0%

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

6. Applied egg-rr 63.8%

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

   1. associate-*r/ 63.8%

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

   2. *-rgt-identity 63.8%

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

   3. +-commutative 63.8%

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

8. Simplified 63.8%

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

9. Taylor expanded in x around 0 59.7%

   \[\leadsto \frac{e^{\frac{x}{s}}}{\color{blue}{4 \cdot s + x \cdot \left(4 + 3 \cdot \frac{x}{s}\right)}} \]

10. Taylor expanded in x around 0 36.1%

    \[\leadsto \frac{\color{blue}{1 + \frac{x}{s}}}{4 \cdot s + x \cdot \left(4 + 3 \cdot \frac{x}{s}\right)} \]

11. Final simplification 36.1%

    \[\leadsto \frac{1 + \frac{x}{s}}{s \cdot 4 + x \cdot \left(4 + \frac{x}{s} \cdot 3\right)} \]
12. Add Preprocessing

Alternative 9: 28.8% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 + \frac{s \cdot -0.1111111111111111}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 3.9999998989515007e-5)
   (/ 0.25 s)
   (/ (+ 0.3333333333333333 (/ (* s -0.1111111111111111) x)) x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 3.9999998989515007e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.3333333333333333f + ((s * -0.1111111111111111f) / x)) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 3.9999998989515007e-5) then
        tmp = 0.25e0 / s
    else
        tmp = (0.3333333333333333e0 + ((s * (-0.1111111111111111e0)) / x)) / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.9999998989515007e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.3333333333333333) + Float32(Float32(s * Float32(-0.1111111111111111)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.9999998989515007e-5))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.3333333333333333) + ((s * single(-0.1111111111111111)) / x)) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999e-5

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in s around inf 33.5%

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

   if 3.9999999e-5 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 100.0%

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

      3. div-inv 100.0%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 3.1%

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

      6. sqr-neg 3.1%

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

      7. sqrt-unprod 3.1%

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

      8. add-sqr-sqrt 3.1%

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

      9. add-sqr-sqrt 3.1%

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

      10. fabs-sqr 3.1%

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

      11. add-sqr-sqrt 3.1%

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

      12. *-commutative 3.1%

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

      13. associate-*l* 3.1%

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

   6. Applied egg-rr -0.0%

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

      1. associate-*r/ -0.0%

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

      2. *-rgt-identity -0.0%

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

      3. +-commutative -0.0%

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

   8. Simplified -0.0%

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

   9. Taylor expanded in x around 0 0.9%

      \[\leadsto \frac{e^{\frac{x}{s}}}{\color{blue}{4 \cdot s + x \cdot \left(4 + 3 \cdot \frac{x}{s}\right)}} \]

   10. Taylor expanded in x around 0 22.3%

       \[\leadsto \frac{\color{blue}{1 + \frac{x}{s}}}{4 \cdot s + x \cdot \left(4 + 3 \cdot \frac{x}{s}\right)} \]

   11. Taylor expanded in x around inf 11.0%

       \[\leadsto \color{blue}{\frac{\left(0.3333333333333333 + 0.3333333333333333 \cdot \frac{s}{x}\right) - 0.4444444444444444 \cdot \frac{s}{x}}{x}} \]
   12. Step-by-step derivation

       1. associate--l+ 11.0%

          \[\leadsto \frac{\color{blue}{0.3333333333333333 + \left(0.3333333333333333 \cdot \frac{s}{x} - 0.4444444444444444 \cdot \frac{s}{x}\right)}}{x} \]

       2. associate-*r/ 11.0%

          \[\leadsto \frac{0.3333333333333333 + \left(\color{blue}{\frac{0.3333333333333333 \cdot s}{x}} - 0.4444444444444444 \cdot \frac{s}{x}\right)}{x} \]

       3. associate-*r/ 11.0%

          \[\leadsto \frac{0.3333333333333333 + \left(\frac{0.3333333333333333 \cdot s}{x} - \color{blue}{\frac{0.4444444444444444 \cdot s}{x}}\right)}{x} \]

       4. div-sub 11.0%

          \[\leadsto \frac{0.3333333333333333 + \color{blue}{\frac{0.3333333333333333 \cdot s - 0.4444444444444444 \cdot s}{x}}}{x} \]

       5. distribute-rgt-out-- 11.0%

          \[\leadsto \frac{0.3333333333333333 + \frac{\color{blue}{s \cdot \left(0.3333333333333333 - 0.4444444444444444\right)}}{x}}{x} \]

       6. metadata-eval 11.0%

          \[\leadsto \frac{0.3333333333333333 + \frac{s \cdot \color{blue}{-0.1111111111111111}}{x}}{x} \]

   13. Simplified 11.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333 + \frac{s \cdot -0.1111111111111111}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 + \frac{s \cdot -0.1111111111111111}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 28.8% accurate, 77.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 3.9999998989515007e-5) (/ 0.25 s) (/ 0.3333333333333333 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 3.9999998989515007e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.3333333333333333f / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 3.9999998989515007e-5) then
        tmp = 0.25e0 / s
    else
        tmp = 0.3333333333333333e0 / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.9999998989515007e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.3333333333333333) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.9999998989515007e-5))
		tmp = single(0.25) / s;
	else
		tmp = single(0.3333333333333333) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999e-5

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in s around inf 33.5%

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

   if 3.9999999e-5 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 100.0%

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

      3. div-inv 100.0%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 3.1%

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

      6. sqr-neg 3.1%

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

      7. sqrt-unprod 3.1%

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

      8. add-sqr-sqrt 3.1%

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

      9. add-sqr-sqrt 3.1%

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

      10. fabs-sqr 3.1%

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

      11. add-sqr-sqrt 3.1%

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

      12. *-commutative 3.1%

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

      13. associate-*l* 3.1%

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

   6. Applied egg-rr -0.0%

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

      1. associate-*r/ -0.0%

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

      2. *-rgt-identity -0.0%

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

      3. +-commutative -0.0%

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

   8. Simplified -0.0%

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

   9. Taylor expanded in x around 0 0.9%

      \[\leadsto \frac{e^{\frac{x}{s}}}{\color{blue}{4 \cdot s + x \cdot \left(4 + 3 \cdot \frac{x}{s}\right)}} \]

   10. Taylor expanded in x around 0 22.3%

       \[\leadsto \frac{\color{blue}{1 + \frac{x}{s}}}{4 \cdot s + x \cdot \left(4 + 3 \cdot \frac{x}{s}\right)} \]

   11. Taylor expanded in x around inf 11.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.1% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 25.1%

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

6. Final simplification 25.1%

   \[\leadsto \frac{0.25}{s} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(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))))))