Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 12.8s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% 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 99.9%

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

   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. Final simplification 99.9%

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

Alternative 2: 99.6% 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 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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. fabs-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. fabs-neg 99.8%

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

   5. distribute-lft-in 99.8%

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

   6. *-rgt-identity 99.8%

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

   7. +-commutative 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 63.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.8%

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

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 50.7%

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

   4. fabs-sqr 50.7%

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

   5. rem-square-sqrt 63.1%

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

   6. exp-prod 63.1%

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

   7. neg-mul-1 63.1%

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

   8. distribute-neg-frac2 63.1%

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

   9. +-commutative 63.1%

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

   10. exp-prod 63.1%

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

   11. rem-square-sqrt 50.7%

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

   12. fabs-sqr 50.7%

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

   13. rem-square-sqrt 64.2%

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

   14. exp-prod 64.2%

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

   15. neg-mul-1 64.2%

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

   16. distribute-neg-frac2 64.2%

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

7. Simplified 64.2%

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

8. Final simplification 64.2%

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

Alternative 4: 63.5% accurate, 2.0× speedup

Math

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

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(t_0 / Float32(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%

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

   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 99.9%

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

   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 x around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. distribute-lft-in 99.8%

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

   4. +-commutative 99.8%

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

   5. mul-1-neg 99.8%

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

   6. rec-exp 99.8%

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

   7. +-commutative 99.8%

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

   8. associate-*r* 99.8%

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

   9. rec-exp 99.8%

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

   10. mul-1-neg 99.8%

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

   11. unpow2 99.8%

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

7. Simplified 97.6%

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

8. Taylor expanded in x around 0 97.6%

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

   1. mul-1-neg 97.6%

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

   2. rem-square-sqrt 50.7%

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

   3. fabs-sqr 50.7%

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

   4. rem-square-sqrt 64.2%

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

10. Simplified 64.2%

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

11. Final simplification 64.2%

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

Alternative 5: 95.7% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ (exp (/ (- (fabs x)) s)) (+ (* s 4.0) (* x (- (* 3.0 (/ x s)) 4.0)))))

C

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

Fortran

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

Julia

function code(x, s)
	return Float32(exp(Float32(Float32(-abs(x)) / s)) / Float32(Float32(s * Float32(4.0)) + Float32(x * Float32(Float32(Float32(3.0) * Float32(x / s)) - Float32(4.0)))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 99.9%

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

   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 x around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. distribute-lft-in 99.8%

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

   4. +-commutative 99.8%

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

   5. mul-1-neg 99.8%

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

   6. rec-exp 99.8%

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

   7. +-commutative 99.8%

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

   8. associate-*r* 99.8%

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

   9. rec-exp 99.8%

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

   10. mul-1-neg 99.8%

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

   11. unpow2 99.8%

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

7. Simplified 97.6%

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

8. Taylor expanded in x around 0 97.2%

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

9. Final simplification 97.2%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4 + x \cdot \left(3 \cdot \frac{x}{s} - 4\right)} \]
10. Add Preprocessing

Alternative 6: 61.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{x}{-s}}}{s \cdot 4 + x \cdot \left(\frac{x}{s} - 4\right)} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	return expf((x / -s)) / ((s * 4.0f) + (x * ((x / 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) + (x * ((x / s) - 4.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 99.9%

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

   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 x around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. distribute-lft-in 99.8%

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

   4. +-commutative 99.8%

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

   5. mul-1-neg 99.8%

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

   6. rec-exp 99.8%

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

   7. +-commutative 99.8%

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

   8. associate-*r* 99.8%

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

   9. rec-exp 99.8%

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

   10. mul-1-neg 99.8%

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

   11. unpow2 99.8%

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

7. Simplified 97.6%

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

8. Taylor expanded in x around 0 97.6%

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

   1. mul-1-neg 97.6%

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

   2. rem-square-sqrt 50.7%

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

   3. fabs-sqr 50.7%

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

   4. rem-square-sqrt 64.2%

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

10. Simplified 64.2%

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

11. Taylor expanded in x around 0 62.7%

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

    1. neg-mul-1 62.7%

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

    2. unsub-neg 62.7%

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

13. Simplified 62.7%

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

14. Taylor expanded in x around 0 62.9%

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

15. Final simplification 62.9%

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

Alternative 7: 60.9% accurate, 5.4× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return (expf((x / -s)) / s) / (4.0f + ((x / 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 + ((x / s) * (-4.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.8%

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

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 50.7%

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

   4. fabs-sqr 50.7%

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

   5. rem-square-sqrt 63.1%

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

   6. exp-prod 63.1%

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

   7. neg-mul-1 63.1%

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

   8. distribute-neg-frac2 63.1%

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

   9. +-commutative 63.1%

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

   10. exp-prod 63.1%

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

   11. rem-square-sqrt 50.7%

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

   12. fabs-sqr 50.7%

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

   13. rem-square-sqrt 64.2%

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

   14. exp-prod 64.2%

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

   15. neg-mul-1 64.2%

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

   16. distribute-neg-frac2 64.2%

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

7. Simplified 64.2%

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

8. Taylor expanded in x around 0 63.0%

   \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{s}}{\color{blue}{4 + -4 \cdot \frac{x}{s}}} \]
9. Step-by-step derivation

   1. *-commutative 63.0%

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

10. Simplified 63.0%

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

Alternative 8: 60.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.8%

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

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 50.7%

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

   4. fabs-sqr 50.7%

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

   5. rem-square-sqrt 63.1%

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

   6. exp-prod 63.1%

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

   7. neg-mul-1 63.1%

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

   8. distribute-neg-frac2 63.1%

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

   9. +-commutative 63.1%

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

   10. exp-prod 63.1%

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

   11. rem-square-sqrt 50.7%

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

   12. fabs-sqr 50.7%

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

   13. rem-square-sqrt 64.2%

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

   14. exp-prod 64.2%

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

   15. neg-mul-1 64.2%

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

   16. distribute-neg-frac2 64.2%

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

7. Simplified 64.2%

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

8. Taylor expanded in x around 0 62.6%

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

   1. neg-mul-1 62.7%

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

   2. unsub-neg 62.7%

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

10. Simplified 62.6%

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

11. Taylor expanded in x around 0 62.0%

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

Alternative 9: 83.0% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4999999206638063 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s \cdot \frac{s}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.4999999206638063e-22)
   (/ 1.0 (* s (+ 4.0 (/ x (* s (/ s x))))))
   0.0))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.4999999206638063e-22f) {
		tmp = 1.0f / (s * (4.0f + (x / (s * (s / x)))));
	} else {
		tmp = 0.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.4999999206638063e-22) then
        tmp = 1.0e0 / (s * (4.0e0 + (x / (s * (s / x)))))
    else
        tmp = 0.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.4999999206638063e-22))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(x / Float32(s * Float32(s / x))))));
	else
		tmp = Float32(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.4999999206638063e-22))
		tmp = single(1.0) / (s * (single(4.0) + (x / (s * (s / x)))));
	else
		tmp = single(0.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.4999999e-22

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-frac-neg2 99.8%

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

      4. fabs-neg 99.8%

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

      5. *-commutative 99.8%

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

      6. fabs-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. fabs-neg 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   6. Applied egg-rr 96.2%

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

      1. unpow-1 96.2%

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

   8. Simplified 96.2%

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

   9. Taylor expanded in x around 0 63.0%

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

   10. Taylor expanded in s around inf 71.7%

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

       1. +-commutative 71.7%

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

       2. unpow2 71.7%

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

       3. unpow2 71.7%

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

       4. times-frac 72.9%

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

       5. unpow2 72.9%

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

   12. Simplified 72.9%

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

       1. unpow2 72.9%

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

       2. clear-num 72.9%

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

       3. frac-times 75.5%

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

       4. *-un-lft-identity 75.5%

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

   14. Applied egg-rr 75.5%

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

   if 2.4999999e-22 < x

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-frac-neg2 99.8%

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

      4. fabs-neg 99.8%

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

      5. *-commutative 99.8%

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

      6. fabs-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. fabs-neg 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in s around inf 11.8%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 11.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{s}\right)\right)} \]

      2. expm1-undefine 11.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{s}\right)} - 1} \]

   7. Applied egg-rr 11.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{s}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-define 11.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{s}\right)\right)} \]

   9. Simplified 11.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{s}\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 11.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{s}\right)} - 1} \]

       2. log1p-undefine 11.5%

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

       3. rem-exp-log 11.8%

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

   11. Applied egg-rr 11.8%

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

   12. Taylor expanded in s around inf 92.5%

       \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4999999206638063 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s \cdot \frac{s}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.4% accurate, 77.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.000000097707407 \cdot 10^{-26}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 5.000000097707407e-26) (/ 0.25 s) 0.0))

C

float code(float x, float s) {
	float tmp;
	if (x <= 5.000000097707407e-26f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 5.000000097707407e-26) then
        tmp = 0.25e0 / s
    else
        tmp = 0.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(5.000000097707407e-26))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.000000097707407e-26))
		tmp = single(0.25) / s;
	else
		tmp = single(0.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.0000001e-26

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-frac-neg2 99.8%

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

      4. fabs-neg 99.8%

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

      5. *-commutative 99.8%

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

      6. fabs-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. fabs-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around inf 32.4%

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

   if 5.0000001e-26 < x

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-frac-neg2 99.8%

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

      4. fabs-neg 99.8%

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

      5. *-commutative 99.8%

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

      6. fabs-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. fabs-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around inf 12.6%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 12.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{s}\right)\right)} \]

      2. expm1-undefine 12.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{s}\right)} - 1} \]

   7. Applied egg-rr 12.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{s}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-define 12.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{s}\right)\right)} \]

   9. Simplified 12.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{s}\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 12.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{s}\right)} - 1} \]

       2. log1p-undefine 12.1%

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

       3. rem-exp-log 12.6%

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

   11. Applied egg-rr 12.6%

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

   12. Taylor expanded in s around inf 91.1%

       \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.000000097707407 \cdot 10^{-26}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.0% accurate, 620.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.0)

C

float code(float x, float s) {
	return 0.0f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(x, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.0);
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. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in s around inf 23.4%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 22.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{s}\right)\right)} \]

   2. expm1-undefine 22.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{s}\right)} - 1} \]

7. Applied egg-rr 22.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{s}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-define 22.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{s}\right)\right)} \]

9. Simplified 22.1%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{s}\right)\right)} \]
10. Step-by-step derivation

    1. expm1-undefine 22.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{s}\right)} - 1} \]

    2. log1p-undefine 22.0%

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

    3. rem-exp-log 23.3%

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

11. Applied egg-rr 23.3%

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

12. Taylor expanded in s around inf 79.2%

    \[\leadsto \color{blue}{1} - 1 \]

13. Final simplification 79.2%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024151 
(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))))))