Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 11.3s

Alternatives: 10

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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. distribute-lft-in 99.8%

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

   3. *-rgt-identity 99.8%

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

   4. fabs-neg 99.8%

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

   5. +-commutative 99.8%

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

   6. fma-define 99.8%

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

   7. fabs-neg 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 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(Float32(-abs(x)) / 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 99.7%

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

   1. *-commutative 99.7%

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

   2. fabs-neg 99.7%

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

   3. distribute-lft-in 99.8%

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

   4. *-rgt-identity 99.8%

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

   5. fabs-neg 99.8%

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

   6. distribute-rgt-in 99.8%

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

   7. cancel-sign-sub 99.8%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 73.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;\left|x\right| \leq 0.0020000000949949026:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(t\_0\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{-s}}}{s \cdot {\left(1 + t\_0\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= (fabs x) 0.0020000000949949026)
     (/ (exp (- (/ x s) (* 2.0 (log1p t_0)))) s)
     (/ (exp (/ x (- s))) (* s (pow (+ 1.0 t_0) 2.0))))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (fabsf(x) <= 0.0020000000949949026f) {
		tmp = expf(((x / s) - (2.0f * log1pf(t_0)))) / s;
	} else {
		tmp = expf((x / -s)) / (s * powf((1.0f + t_0), 2.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.0020000000949949026))
		tmp = Float32(exp(Float32(Float32(x / s) - Float32(Float32(2.0) * log1p(t_0)))) / s);
	else
		tmp = Float32(exp(Float32(x / Float32(-s))) / Float32(s * (Float32(Float32(1.0) + t_0) ^ Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.00200000009

   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. *-commutative 99.5%

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

   3. Simplified 99.5%

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

      1. associate-/r* 99.5%

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

      2. +-commutative 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. *-rgt-identity 99.5%

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

      5. fma-undefine 99.5%

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

      6. associate-/r* 99.5%

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

      7. div-inv 99.5%

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

   6. Applied egg-rr 77.8%

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

      1. associate-*r/ 77.8%

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

      2. *-rgt-identity 77.8%

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

      3. associate-/r* 77.8%

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

      4. +-commutative 77.8%

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

   8. Simplified 77.8%

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

      1. +-commutative 77.8%

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

      2. associate-/r* 77.8%

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

      3. add-exp-log 74.5%

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

      4. log-div 74.2%

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

      5. add-log-exp 95.0%

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

      6. +-commutative 95.0%

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

      7. log-prod 94.2%

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

      8. +-commutative 94.2%

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

      9. log-pow 94.9%

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

      10. +-commutative 94.9%

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

      11. log1p-define 94.8%

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

   10. Applied egg-rr 94.8%

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

   11. Taylor expanded in x around inf 94.9%

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

       1. +-commutative 94.9%

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

       2. associate--r+ 95.1%

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

       3. exp-diff 95.1%

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

       4. log1p-define 95.1%

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

       5. rem-exp-log 99.5%

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

   13. Simplified 99.5%

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

   if 0.00200000009 < (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

   6. Applied egg-rr 59.3%

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

      1. unpow-1 59.3%

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

      2. *-commutative 59.3%

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

      3. associate-/r* 59.3%

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

      4. rec-exp 59.3%

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

      5. distribute-neg-frac2 59.3%

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

      6. +-commutative 59.3%

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

   8. Simplified 59.3%

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

Alternative 4: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{{\left(e^{x}\right)}^{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 10.0)
   (/ (exp (- (/ x s) (* 2.0 (log1p (exp (/ x s)))))) s)
   (/ s (pow (exp x) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 10.0f) {
		tmp = expf(((x / s) - (2.0f * log1pf(expf((x / s)))))) / s;
	} else {
		tmp = s / powf(expf(x), s);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(10.0))
		tmp = Float32(exp(Float32(Float32(x / s) - Float32(Float32(2.0) * log1p(exp(Float32(x / s)))))) / s);
	else
		tmp = Float32(s / (exp(x) ^ s));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 10

   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. *-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r* 99.6%

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

      2. +-commutative 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. *-rgt-identity 99.6%

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

      5. fma-undefine 99.7%

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

      6. associate-/r* 99.7%

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

      7. div-inv 99.7%

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

   6. Applied egg-rr 81.0%

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

      1. associate-*r/ 81.0%

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

      2. *-rgt-identity 81.0%

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

      3. associate-/r* 81.0%

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

      4. +-commutative 81.0%

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

   8. Simplified 81.0%

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

      1. +-commutative 81.0%

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

      2. associate-/r* 81.0%

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

      3. add-exp-log 78.8%

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

      4. log-div 78.6%

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

      5. add-log-exp 96.7%

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

      6. +-commutative 96.7%

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

      7. log-prod 96.2%

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

      8. +-commutative 96.2%

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

      9. log-pow 96.6%

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

      10. +-commutative 96.6%

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

      11. log1p-define 96.5%

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

   10. Applied egg-rr 96.5%

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

   11. Taylor expanded in x around inf 96.6%

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

       1. +-commutative 96.6%

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

       2. associate--r+ 96.8%

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

       3. exp-diff 96.8%

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

       4. log1p-define 96.8%

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

       5. rem-exp-log 99.7%

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

   13. Simplified 99.7%

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

   if 10 < 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. associate-/r* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. fma-undefine 100.0%

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

      6. associate-/r* 100.0%

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

      7. div-inv 98.5%

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

   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. associate-/r* -0.0%

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

      4. +-commutative -0.0%

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

   8. Simplified -0.0%

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

      1. +-commutative -0.0%

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

      2. associate-/r* -0.0%

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

      3. add-exp-log -0.0%

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

      4. log-div -0.0%

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

      5. add-log-exp 48.5%

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

      6. +-commutative 48.5%

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

      7. log-prod 48.5%

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

      8. +-commutative 48.5%

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

      9. log-pow 48.5%

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

      10. +-commutative 48.5%

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

      11. log1p-define 48.5%

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

   10. Applied egg-rr 48.5%

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

   12. Applied egg-rr 98.9%

       \[\leadsto \color{blue}{\frac{s}{{\left(e^{x}\right)}^{s}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 62.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 100:\\ \;\;\;\;\left(0.5 + x \cdot \frac{0.25}{s}\right) \cdot \frac{\frac{1}{s}}{1 + e^{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{{\left(e^{x}\right)}^{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 100.0)
   (* (+ 0.5 (* x (/ 0.25 s))) (/ (/ 1.0 s) (+ 1.0 (exp (/ x s)))))
   (/ s (pow (exp x) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 100.0f) {
		tmp = (0.5f + (x * (0.25f / s))) * ((1.0f / s) / (1.0f + expf((x / s))));
	} else {
		tmp = s / powf(expf(x), s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 100.0e0) then
        tmp = (0.5e0 + (x * (0.25e0 / s))) * ((1.0e0 / s) / (1.0e0 + exp((x / s))))
    else
        tmp = s / (exp(x) ** s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(100.0))
		tmp = Float32(Float32(Float32(0.5) + Float32(x * Float32(Float32(0.25) / s))) * Float32(Float32(Float32(1.0) / s) / Float32(Float32(1.0) + exp(Float32(x / s)))));
	else
		tmp = Float32(s / (exp(x) ^ s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(100.0))
		tmp = (single(0.5) + (x * (single(0.25) / s))) * ((single(1.0) / s) / (single(1.0) + exp((x / s))));
	else
		tmp = s / (exp(x) ^ s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 100

   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. *-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r* 99.6%

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

      2. +-commutative 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. *-rgt-identity 99.6%

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

      5. fma-undefine 99.7%

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

      6. associate-/r* 99.7%

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

      7. div-inv 99.7%

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

   6. Applied egg-rr 80.6%

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

      1. associate-*r/ 80.6%

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

      2. *-rgt-identity 80.6%

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

      3. associate-/r* 80.6%

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

      4. +-commutative 80.6%

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

   8. Simplified 80.6%

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

   10. Applied egg-rr 99.1%

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

       1. add-cube-cbrt 99.0%

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

       2. fma-neg 99.0%

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

       3. pow2 99.0%

          \[\leadsto e^{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\frac{x}{s}}\right)}^{2}}, \sqrt[3]{\frac{x}{s}}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)} \cdot \frac{\frac{1}{s}}{e^{\frac{x}{s}} + 1} \]

   12. Applied egg-rr 99.0%

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\frac{x}{s}}\right)}^{2}, \sqrt[3]{\frac{x}{s}}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}} \cdot \frac{\frac{1}{s}}{e^{\frac{x}{s}} + 1} \]

   13. Taylor expanded in x around 0 57.4%

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

       1. exp-neg 57.4%

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

       2. rem-exp-log 57.4%

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

       3. metadata-eval 57.4%

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

       4. associate-/l* 57.4%

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

       5. associate-*r* 57.9%

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

       6. *-commutative 57.9%

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

       7. associate-*r* 57.9%

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

       8. associate-*r/ 57.9%

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

       9. exp-neg 57.9%

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

       10. rem-exp-log 57.9%

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

       11. metadata-eval 57.9%

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

       12. metadata-eval 57.9%

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

   15. Simplified 57.9%

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

   if 100 < 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. associate-/r* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. fma-undefine 100.0%

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

      6. associate-/r* 100.0%

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

      7. div-inv 98.5%

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

   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. associate-/r* -0.0%

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

      4. +-commutative -0.0%

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

   8. Simplified -0.0%

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

      1. +-commutative -0.0%

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

      2. associate-/r* -0.0%

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

      3. add-exp-log -0.0%

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

      4. log-div -0.0%

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

      5. add-log-exp 49.3%

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

      6. +-commutative 49.3%

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

      7. log-prod 49.3%

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

      8. +-commutative 49.3%

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

      9. log-pow 49.3%

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

      10. +-commutative 49.3%

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

      11. log1p-define 49.3%

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

   10. Applied egg-rr 49.3%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 100:\\ \;\;\;\;\left(0.5 + x \cdot \frac{0.25}{s}\right) \cdot \frac{\frac{1}{s}}{1 + e^{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{{\left(e^{x}\right)}^{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{s}}{1 + e^{\frac{x}{s}}}\\ \mathbf{if}\;x \leq 0.0020000000949949026:\\ \;\;\;\;\left(0.5 + x \cdot \frac{0.25}{s}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (/ 1.0 s) (+ 1.0 (exp (/ x s))))))
   (if (<= x 0.0020000000949949026)
     (* (+ 0.5 (* x (/ 0.25 s))) t_0)
     (* t_0 1.5))))

C

float code(float x, float s) {
	float t_0 = (1.0f / s) / (1.0f + expf((x / s)));
	float tmp;
	if (x <= 0.0020000000949949026f) {
		tmp = (0.5f + (x * (0.25f / s))) * t_0;
	} else {
		tmp = t_0 * 1.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (1.0e0 / s) / (1.0e0 + exp((x / s)))
    if (x <= 0.0020000000949949026e0) then
        tmp = (0.5e0 + (x * (0.25e0 / s))) * t_0
    else
        tmp = t_0 * 1.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(Float32(1.0) / s) / Float32(Float32(1.0) + exp(Float32(x / s))))
	tmp = Float32(0.0)
	if (x <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(Float32(0.5) + Float32(x * Float32(Float32(0.25) / s))) * t_0);
	else
		tmp = Float32(t_0 * Float32(1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = (single(1.0) / s) / (single(1.0) + exp((x / s)));
	tmp = single(0.0);
	if (x <= single(0.0020000000949949026))
		tmp = (single(0.5) + (x * (single(0.25) / s))) * t_0;
	else
		tmp = t_0 * single(1.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00200000009

   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. *-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r* 99.6%

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

      2. +-commutative 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. *-rgt-identity 99.6%

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

      5. fma-undefine 99.7%

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

      6. associate-/r* 99.7%

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

      7. div-inv 99.6%

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

   6. Applied egg-rr 84.6%

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

      1. associate-*r/ 84.6%

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

      2. *-rgt-identity 84.6%

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

      3. associate-/r* 84.6%

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

      4. +-commutative 84.6%

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

   8. Simplified 84.6%

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

   10. Applied egg-rr 99.6%

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

       1. add-cube-cbrt 99.5%

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

       2. fma-neg 99.5%

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

       3. pow2 99.5%

          \[\leadsto e^{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\frac{x}{s}}\right)}^{2}}, \sqrt[3]{\frac{x}{s}}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)} \cdot \frac{\frac{1}{s}}{e^{\frac{x}{s}} + 1} \]

   12. Applied egg-rr 99.5%

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\frac{x}{s}}\right)}^{2}, \sqrt[3]{\frac{x}{s}}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}} \cdot \frac{\frac{1}{s}}{e^{\frac{x}{s}} + 1} \]

   13. Taylor expanded in x around 0 55.8%

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

       1. exp-neg 55.8%

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

       2. rem-exp-log 55.8%

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

       3. metadata-eval 55.8%

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

       4. associate-/l* 55.8%

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

       5. associate-*r* 55.8%

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

       6. *-commutative 55.8%

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

       7. associate-*r* 55.8%

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

       8. associate-*r/ 55.8%

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

       9. exp-neg 55.8%

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

       10. rem-exp-log 55.8%

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

       11. metadata-eval 55.8%

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

       12. metadata-eval 55.8%

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

   15. Simplified 55.8%

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

   if 0.00200000009 < 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. associate-/r* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. fma-undefine 100.0%

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

      6. associate-/r* 100.0%

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

      7. div-inv 98.7%

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

   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. associate-/r* -0.0%

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

      4. +-commutative -0.0%

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

   8. Simplified -0.0%

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

   10. Applied egg-rr 53.9%

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

   11. Taylor expanded in x around 0 98.7%

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

       1. neg-mul-1 98.7%

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

   13. Simplified 98.7%

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

   15. Applied egg-rr 98.7%

       \[\leadsto \color{blue}{1.5} \cdot \frac{\frac{1}{s}}{e^{\frac{x}{s}} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0020000000949949026:\\ \;\;\;\;\left(0.5 + x \cdot \frac{0.25}{s}\right) \cdot \frac{\frac{1}{s}}{1 + e^{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{1 + e^{\frac{x}{s}}} \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.499999924127398 \cdot 10^{-19}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{1 + e^{\frac{x}{s}}} \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 8.499999924127398e-19)
   (/ 0.25 s)
   (* (/ (/ 1.0 s) (+ 1.0 (exp (/ x s)))) 1.5)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 8.499999924127398e-19f) {
		tmp = 0.25f / s;
	} else {
		tmp = ((1.0f / s) / (1.0f + expf((x / s)))) * 1.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 8.499999924127398e-19) then
        tmp = 0.25e0 / s
    else
        tmp = ((1.0e0 / s) / (1.0e0 + exp((x / s)))) * 1.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(8.499999924127398e-19))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(Float32(1.0) / s) / Float32(Float32(1.0) + exp(Float32(x / s)))) * Float32(1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(8.499999924127398e-19))
		tmp = single(0.25) / s;
	else
		tmp = ((single(1.0) / s) / (single(1.0) + exp((x / s)))) * single(1.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 8.49999992e-19

   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. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in s around inf 43.3%

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

   if 8.49999992e-19 < x

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/r* 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-rgt-identity 99.9%

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

      5. fma-undefine 99.9%

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

      6. associate-/r* 99.9%

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

      7. div-inv 98.9%

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

   6. Applied egg-rr 8.5%

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

      1. associate-*r/ 8.5%

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

      2. *-rgt-identity 8.5%

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

      3. associate-/r* 8.5%

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

      4. +-commutative 8.5%

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

   8. Simplified 8.5%

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

   10. Applied egg-rr 65.8%

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

   11. Taylor expanded in x around 0 95.5%

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

       1. neg-mul-1 95.5%

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

   13. Simplified 95.5%

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

   15. Applied egg-rr 92.6%

       \[\leadsto \color{blue}{1.5} \cdot \frac{\frac{1}{s}}{e^{\frac{x}{s}} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.499999924127398 \cdot 10^{-19}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{1 + e^{\frac{x}{s}}} \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0020000000949949026:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{s \cdot \left(-x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.0020000000949949026) (/ 0.25 s) (exp (* s (- x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.0020000000949949026f) {
		tmp = 0.25f / s;
	} else {
		tmp = expf((s * -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 <= 0.0020000000949949026e0) then
        tmp = 0.25e0 / s
    else
        tmp = exp((s * -x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = exp(Float32(s * Float32(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.0020000000949949026))
		tmp = single(0.25) / s;
	else
		tmp = exp((s * -x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00200000009

   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. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in s around inf 41.2%

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

   if 0.00200000009 < 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. associate-/r* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. fma-undefine 100.0%

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

      6. associate-/r* 100.0%

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

      7. div-inv 98.7%

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

   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. associate-/r* -0.0%

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

      4. +-commutative -0.0%

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

   8. Simplified -0.0%

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

      1. +-commutative -0.0%

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

      2. associate-/r* -0.0%

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

      3. add-exp-log -0.0%

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

      4. log-div -0.0%

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

      5. add-log-exp 53.9%

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

      6. +-commutative 53.9%

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

      7. log-prod 53.9%

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

      8. +-commutative 53.9%

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

      9. log-pow 53.9%

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

      10. +-commutative 53.9%

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

      11. log1p-define 53.9%

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

   10. Applied egg-rr 53.9%

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

   12. Applied egg-rr 88.9%

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

       1. associate-/r* 88.9%

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

       2. *-inverses 88.9%

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

       3. exp-prod 45.9%

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

       4. rec-exp 45.9%

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

       5. distribute-rgt-neg-in 45.9%

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

   14. Simplified 45.9%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0020000000949949026:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{s \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 28.5% accurate, 77.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0020000000949949026:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.0020000000949949026) (/ 0.25 s) (/ s -2.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.0020000000949949026f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / -2.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.0020000000949949026e0) then
        tmp = 0.25e0 / s
    else
        tmp = s / (-2.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(-2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.0020000000949949026))
		tmp = single(0.25) / s;
	else
		tmp = s / single(-2.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00200000009

   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. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in s around inf 41.2%

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

   if 0.00200000009 < 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. associate-/r* 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. fma-undefine 100.0%

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

      6. associate-/r* 100.0%

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

      7. div-inv 98.7%

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

   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. associate-/r* -0.0%

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

      4. +-commutative -0.0%

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

   8. Simplified -0.0%

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

      1. +-commutative -0.0%

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

      2. associate-/r* -0.0%

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

      3. add-exp-log -0.0%

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

      4. log-div -0.0%

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

      5. add-log-exp 53.9%

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

      6. +-commutative 53.9%

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

      7. log-prod 53.9%

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

      8. +-commutative 53.9%

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

      9. log-pow 53.9%

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

      10. +-commutative 53.9%

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

      11. log1p-define 53.9%

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

   10. Applied egg-rr 53.9%

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

   12. Applied egg-rr 1.4%

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

       1. associate-/l/ 1.4%

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

       2. associate-/r* 1.4%

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

       3. associate-/l* 1.4%

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

       4. *-inverses 11.1%

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

       5. *-rgt-identity 11.1%

          \[\leadsto \frac{\color{blue}{s}}{-2} \]

   14. Simplified 11.1%

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

Alternative 10: 26.7% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 30.3%

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

Reproduce

herbie shell --seed 2024111 
(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))))))