Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 11.5s

Alternatives: 9

Speedup: 0.9×

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: 99.6% 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: 99.6% 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 99.8%

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

   1. *-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. fabs-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. *-rgt-identity 99.8%

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

   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: 63.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 2.0000000063421537 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{e^{\log x}}{-s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{s}{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 2.0000000063421537e-28)
     (/ (exp (/ (exp (log x)) (- s))) (* s 4.0))
     (/ 1.0 (/ s (exp (- (/ x s) (* 2.0 (log1p (exp (/ x s)))))))))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 2.0000000063421537e-28f) {
		tmp = expf((expf(logf(x)) / -s)) / (s * 4.0f);
	} else {
		tmp = 1.0f / (s / expf(((x / s) - (2.0f * log1pf(expf((x / s)))))));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(2.0000000063421537e-28))
		tmp = Float32(exp(Float32(exp(log(x)) / Float32(-s))) / Float32(s * Float32(4.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(s / exp(Float32(Float32(x / s) - Float32(Float32(2.0) * log1p(exp(Float32(x / s))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 2.00000001e-28

   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%

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

      2. +-commutative 99.9%

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

      3. fabs-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. *-rgt-identity 99.9%

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

      6. fma-define 99.9%

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

      7. fabs-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in s around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. add-sqr-sqrt 55.7%

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

      2. fabs-sqr 55.7%

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

      3. add-sqr-sqrt 57.1%

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

      4. add-exp-log 55.7%

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

   9. Applied egg-rr 55.7%

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

   if 2.00000001e-28 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

   3. Simplified 99.2%

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

      1. associate-/r* 99.2%

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

      2. +-commutative 99.2%

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

      3. distribute-lft-in 99.3%

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

      4. *-rgt-identity 99.3%

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

      5. fma-undefine 99.4%

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

      6. associate-/r* 99.3%

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

      7. div-inv 99.2%

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

      8. *-commutative 99.2%

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

   6. Applied egg-rr 97.9%

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

      1. associate-*l/ 98.1%

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

      2. *-un-lft-identity 98.1%

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

      3. *-commutative 98.1%

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

      4. +-commutative 98.1%

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

      5. associate-/r* 97.9%

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

      6. +-commutative 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. clear-num 98.0%

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

      2. inv-pow 98.0%

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

      3. add-exp-log 98.0%

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

      4. log-div 97.9%

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

      5. add-log-exp 97.9%

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

      6. log-pow 99.1%

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

      7. +-commutative 99.1%

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

      8. log1p-define 99.3%

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

   10. Applied egg-rr 99.3%

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

       1. unpow-1 99.3%

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

   12. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{s}{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)} \leq 2.0000000063421537 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{e^{\log x}}{-s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{s}{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 2.0000000063421537 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{e^{\log x}}{-s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 2.0000000063421537e-28)
     (/ (exp (/ (exp (log x)) (- s))) (* s 4.0))
     (/ (exp (+ (/ x s) (* (log1p (exp (/ x s))) -2.0))) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 2.0000000063421537e-28f) {
		tmp = expf((expf(logf(x)) / -s)) / (s * 4.0f);
	} else {
		tmp = expf(((x / s) + (log1pf(expf((x / s))) * -2.0f))) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(2.0000000063421537e-28))
		tmp = Float32(exp(Float32(exp(log(x)) / Float32(-s))) / Float32(s * Float32(4.0)));
	else
		tmp = Float32(exp(Float32(Float32(x / s) + Float32(log1p(exp(Float32(x / s))) * Float32(-2.0)))) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 2.00000001e-28

   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%

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

      2. +-commutative 99.9%

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

      3. fabs-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. *-rgt-identity 99.9%

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

      6. fma-define 99.9%

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

      7. fabs-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in s around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. add-sqr-sqrt 55.7%

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

      2. fabs-sqr 55.7%

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

      3. add-sqr-sqrt 57.1%

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

      4. add-exp-log 55.7%

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

   9. Applied egg-rr 55.7%

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

   if 2.00000001e-28 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

   3. Simplified 99.2%

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

      1. associate-/r* 99.2%

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

      2. +-commutative 99.2%

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

      3. distribute-lft-in 99.3%

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

      4. *-rgt-identity 99.3%

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

      5. fma-undefine 99.4%

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

      6. associate-/r* 99.3%

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

      7. div-inv 99.2%

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

      8. *-commutative 99.2%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in s around 0 98.1%

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

      1. +-commutative 98.1%

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

      2. *-commutative 98.1%

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

      3. exp-to-pow 98.0%

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

      4. +-commutative 98.0%

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

      5. log1p-undefine 98.0%

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

      6. *-commutative 98.0%

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

      7. rem-exp-log 90.9%

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

      8. exp-sum 91.6%

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

      9. exp-diff 91.1%

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

      10. associate--r+ 91.1%

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

      11. exp-diff 92.1%

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

   9. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)} \leq 2.0000000063421537 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{e^{\log x}}{-s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.8% 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 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. associate-/r* 99.8%

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

   2. exp-prod 99.8%

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

   3. rem-square-sqrt 54.9%

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

   4. fabs-sqr 54.9%

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

   5. rem-square-sqrt 66.3%

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

   6. exp-prod 66.3%

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

   7. neg-mul-1 66.3%

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

   8. distribute-neg-frac2 66.3%

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

   9. +-commutative 66.3%

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

   10. exp-prod 66.3%

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

   11. rem-square-sqrt 54.9%

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

   12. fabs-sqr 54.9%

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

   13. rem-square-sqrt 67.3%

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

   14. exp-prod 67.4%

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

   15. neg-mul-1 67.4%

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

   16. distribute-neg-frac2 67.4%

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

7. Simplified 67.4%

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

8. Final simplification 67.4%

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

Alternative 5: 47.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. fabs-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. *-rgt-identity 99.8%

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

   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. Taylor expanded in s around inf 95.7%

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

   1. *-commutative 95.7%

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

7. Simplified 95.7%

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

   1. add-sqr-sqrt 52.8%

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

   2. fabs-sqr 52.8%

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

   3. add-sqr-sqrt 64.2%

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

   4. add-exp-log 52.8%

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

9. Applied egg-rr 52.8%

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

10. Final simplification 52.8%

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

Alternative 6: 59.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot \frac{-1}{s}}}{s \cdot 4} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (exp (* x (/ -1.0 s))) (* s 4.0)))

C

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

Fortran

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

Julia

function code(x, s)
	return Float32(exp(Float32(x * Float32(Float32(-1.0) / s))) / Float32(s * Float32(4.0)))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. fabs-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. *-rgt-identity 99.8%

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

   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. Taylor expanded in s around inf 95.7%

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

   1. *-commutative 95.7%

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

7. Simplified 95.7%

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

   1. frac-2neg 95.7%

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

   2. div-inv 95.7%

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

   3. remove-double-neg 95.7%

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

   4. add-sqr-sqrt 52.8%

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

   5. fabs-sqr 52.8%

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

   6. add-sqr-sqrt 64.2%

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

9. Applied egg-rr 64.2%

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

10. Taylor expanded in s around 0 64.2%

    \[\leadsto \frac{e^{x \cdot \color{blue}{\frac{-1}{s}}}}{s \cdot 4} \]
11. Add Preprocessing

Alternative 7: 59.6% accurate, 5.7× speedup

Math

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

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(exp(Float32(x / Float32(-s))) / Float32(s * Float32(4.0)))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. fabs-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. *-rgt-identity 99.8%

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

   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. Taylor expanded in s around inf 95.7%

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

   1. *-commutative 95.7%

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

7. Simplified 95.7%

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

   1. add-sqr-sqrt 95.7%

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

   2. sqrt-unprod 93.8%

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

   3. sqr-abs 93.8%

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

   4. sqr-neg 93.8%

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

   5. add-sqr-sqrt 51.8%

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

   6. fabs-sqr 51.8%

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

   7. add-sqr-sqrt 60.7%

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

   8. add-sqr-sqrt 51.8%

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

   9. fabs-sqr 51.8%

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

   10. add-sqr-sqrt 93.8%

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

   11. sqrt-unprod -0.0%

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

   12. add-sqr-sqrt 24.3%

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

   13. neg-sub0 24.3%

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

   14. add-sqr-sqrt 12.9%

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

   15. fabs-sqr 12.9%

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

   16. add-sqr-sqrt 55.8%

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

   17. sub-neg 55.8%

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

   18. add-sqr-sqrt 12.9%

       \[\leadsto \frac{e^{\frac{-\left(0 + \left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right)}{s}}}{s \cdot 4} \]

   19. fabs-sqr 12.9%

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

   20. add-sqr-sqrt 24.3%

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

   21. add-sqr-sqrt -0.0%

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

   22. sqrt-unprod 93.8%

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

9. Applied egg-rr 64.2%

   \[\leadsto \frac{e^{\frac{-\color{blue}{\left(0 + x\right)}}{s}}}{s \cdot 4} \]
10. Step-by-step derivation

    1. +-lft-identity 64.2%

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

11. Simplified 64.2%

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

12. Final simplification 64.2%

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

Alternative 8: 59.6% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. fabs-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. *-rgt-identity 99.8%

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

   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. Taylor expanded in s around inf 95.7%

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

   1. *-commutative 95.7%

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

7. Simplified 95.7%

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

   1. frac-2neg 95.7%

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

   2. div-inv 95.7%

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

   3. remove-double-neg 95.7%

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

   4. add-sqr-sqrt 52.8%

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

   5. fabs-sqr 52.8%

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

   6. add-sqr-sqrt 64.2%

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

9. Applied egg-rr 64.2%

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

10. Taylor expanded in x around inf 64.2%

    \[\leadsto \color{blue}{0.25 \cdot \frac{e^{-1 \cdot \frac{x}{s}}}{s}} \]
11. Step-by-step derivation

    1. associate-*r/ 64.2%

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

    2. mul-1-neg 64.2%

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

    3. rec-exp 64.1%

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

    4. associate-*r/ 64.1%

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

    5. metadata-eval 64.1%

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

12. Simplified 64.1%

    \[\leadsto \color{blue}{\frac{\frac{0.25}{e^{\frac{x}{s}}}}{s}} \]
13. Add Preprocessing

Alternative 9: 27.0% 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 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in s around inf 26.9%

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

Reproduce

herbie shell --seed 2024191 
(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))))))