Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 10.7s

Alternatives: 10

Speedup: 0.9×

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Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% 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: 63.3% 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 99.7%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{s \cdot {\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}}} \]
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 48.7%

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

   4. fabs-sqr 48.7%

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

   5. rem-square-sqrt 62.7%

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

   6. exp-prod 62.6%

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

   7. neg-mul-1 62.6%

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

   8. distribute-neg-frac2 62.6%

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

7. Simplified 63.5%

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

8. Final simplification 63.5%

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

Alternative 3: 60.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((x / -s));
	return (t_0 / (1.0f + t_0)) / (s + (s / (1.0f + (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((x / -s))
    code = (t_0 / (1.0e0 + t_0)) / (s + (s / (1.0e0 + (x / s))))
end function

Julia

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

MATLAB

function tmp = code(x, s)
	t_0 = exp((x / -s));
	tmp = (t_0 / (single(1.0) + t_0)) / (s + (s / (single(1.0) + (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. Taylor expanded in x around 0 99.8%

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

   1. associate-/r* 99.8%

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

7. Simplified 64.2%

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

8. Taylor expanded in x around 0 62.6%

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

9. Final simplification 62.6%

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

Alternative 4: 59.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{x \cdot \frac{-1}{s}}}{2}}{s + \frac{s}{e^{\frac{x}{s}}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (exp((x * (single(-1.0) / s))) / single(2.0)) / (s + (s / exp((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. Taylor expanded in x around 0 99.8%

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

   1. associate-/r* 99.8%

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

7. Simplified 64.2%

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

8. Taylor expanded in x around 0 60.8%

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

   1. distribute-frac-neg2 60.8%

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

   2. div-inv 60.8%

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

   3. distribute-lft-neg-in 60.8%

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

10. Applied egg-rr 60.8%

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

11. Final simplification 60.8%

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

Alternative 5: 59.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{\frac{x}{-s}}}{2}}{s + \frac{s}{e^{\frac{x}{s}}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (exp((x / -s)) / single(2.0)) / (s + (s / exp((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. Taylor expanded in x around 0 99.8%

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

   1. associate-/r* 99.8%

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

7. Simplified 64.2%

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

8. Taylor expanded in x around 0 60.8%

   \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{\color{blue}{2}}}{s + \frac{s}{e^{\frac{x}{s}}}} \]
9. Add Preprocessing

Alternative 6: 60.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (single(0.5) / s) / (single(1.0) + exp((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%

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

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. Step-by-step derivation

   1. add-sqr-sqrt 99.7%

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

   2. times-frac 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-lft-in 99.7%

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

   5. *-rgt-identity 99.7%

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

   6. fma-undefine 99.8%

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

6. Applied egg-rr 61.8%

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

   1. associate-*l/ 61.9%

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

   2. /-rgt-identity 61.9%

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

   3. times-frac 61.8%

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

   4. rem-square-sqrt 61.9%

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

   5. *-lft-identity 61.9%

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

   6. +-commutative 61.9%

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

8. Simplified 61.9%

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

9. Taylor expanded in x around 0 62.4%

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

Alternative 7: 59.8% 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. *-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 99.7%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{s \cdot {\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}}} \]
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 48.7%

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

   4. fabs-sqr 48.7%

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

   5. rem-square-sqrt 62.7%

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

   6. exp-prod 62.6%

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

   7. neg-mul-1 62.6%

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

   8. distribute-neg-frac2 62.6%

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

7. Simplified 63.5%

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

8. Taylor expanded in x around 0 61.6%

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

Alternative 8: 53.8% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{2 + \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{x}{s} \cdot -0.25}{s + \left(s + x \cdot \left(\frac{x}{s} \cdot 0.5 + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= s 9.999999960041972e-13)
   (/ (/ 0.5 s) (+ 2.0 (/ x s)))
   (/ (+ 0.5 (* (/ x s) -0.25)) (+ s (+ s (* x (+ (* (/ x s) 0.5) -1.0)))))))

C

float code(float x, float s) {
	float tmp;
	if (s <= 9.999999960041972e-13f) {
		tmp = (0.5f / s) / (2.0f + (x / s));
	} else {
		tmp = (0.5f + ((x / s) * -0.25f)) / (s + (s + (x * (((x / s) * 0.5f) + -1.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (s <= 9.999999960041972e-13) then
        tmp = (0.5e0 / s) / (2.0e0 + (x / s))
    else
        tmp = (0.5e0 + ((x / s) * (-0.25e0))) / (s + (s + (x * (((x / s) * 0.5e0) + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (s <= Float32(9.999999960041972e-13))
		tmp = Float32(Float32(Float32(0.5) / s) / Float32(Float32(2.0) + Float32(x / s)));
	else
		tmp = Float32(Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(-0.25))) / Float32(s + Float32(s + Float32(x * Float32(Float32(Float32(x / s) * Float32(0.5)) + Float32(-1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (s <= single(9.999999960041972e-13))
		tmp = (single(0.5) / s) / (single(2.0) + (x / s));
	else
		tmp = (single(0.5) + ((x / s) * single(-0.25))) / (s + (s + (x * (((x / s) * single(0.5)) + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 9.99999996e-13

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 99.7%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. *-rgt-identity 99.7%

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

      6. fma-undefine 99.7%

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

   6. Applied egg-rr 51.7%

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

      1. associate-*l/ 51.7%

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

      2. /-rgt-identity 51.7%

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

      3. times-frac 51.7%

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

      4. rem-square-sqrt 51.7%

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

      5. *-lft-identity 51.7%

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

      6. +-commutative 51.7%

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

   8. Simplified 51.7%

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

   9. Taylor expanded in s around -inf 24.2%

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

       1. associate-*r/ 24.2%

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

       2. sub-neg 24.2%

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

       3. mul-1-neg 24.2%

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

       4. distribute-rgt-out-- 24.2%

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

       5. metadata-eval 24.2%

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

       6. *-commutative 24.2%

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

       7. associate-*r/ 24.2%

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

       8. distribute-lft-neg-in 24.2%

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

       9. metadata-eval 24.2%

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

       10. metadata-eval 24.2%

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

       11. distribute-lft-in 24.2%

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

       12. neg-mul-1 24.2%

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

       13. distribute-lft-neg-in 24.2%

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

       14. metadata-eval 24.2%

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

       15. metadata-eval 24.2%

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

   11. Simplified 24.2%

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

   12. Taylor expanded in x around 0 15.2%

       \[\leadsto \frac{\frac{0.25 \cdot \frac{x}{s} + 0.5}{s}}{\color{blue}{2 + \frac{x}{s}}} \]

   13. Taylor expanded in x around 0 49.2%

       \[\leadsto \frac{\frac{\color{blue}{0.5}}{s}}{2 + \frac{x}{s}} \]

   if 9.99999996e-13 < s

   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. fabs-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fabs-neg 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. *-rgt-identity 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. associate-/r* 99.9%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in x around 0 80.8%

      \[\leadsto \frac{\color{blue}{0.5 + -0.25 \cdot \frac{x}{s}}}{s + \frac{s}{e^{\frac{x}{s}}}} \]
   9. Step-by-step derivation

      1. *-commutative 80.8%

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

   10. Simplified 80.8%

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

   11. Taylor expanded in x around 0 73.2%

       \[\leadsto \frac{0.5 + \frac{x}{s} \cdot -0.25}{s + \color{blue}{\left(s + x \cdot \left(0.5 \cdot \frac{x}{s} - 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{2 + \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{x}{s} \cdot -0.25}{s + \left(s + x \cdot \left(\frac{x}{s} \cdot 0.5 + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.7% accurate, 68.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{s}}{2 + \frac{x}{s}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 0.5 s) (+ 2.0 (/ x s))))

C

float code(float x, float s) {
	return (0.5f / s) / (2.0f + (x / s));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (0.5e0 / s) / (2.0e0 + (x / s))
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(0.5) / s) / Float32(Float32(2.0) + Float32(x / s)))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.5) / s) / (single(2.0) + (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%

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

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. Step-by-step derivation

   1. add-sqr-sqrt 99.7%

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

   2. times-frac 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-lft-in 99.7%

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

   5. *-rgt-identity 99.7%

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

   6. fma-undefine 99.8%

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

6. Applied egg-rr 61.8%

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

   1. associate-*l/ 61.9%

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

   2. /-rgt-identity 61.9%

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

   3. times-frac 61.8%

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

   4. rem-square-sqrt 61.9%

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

   5. *-lft-identity 61.9%

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

   6. +-commutative 61.9%

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

8. Simplified 61.9%

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

9. Taylor expanded in s around -inf 35.7%

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

    1. associate-*r/ 35.7%

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

    2. sub-neg 35.7%

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

    3. mul-1-neg 35.7%

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

    4. distribute-rgt-out-- 35.7%

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

    5. metadata-eval 35.7%

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

    6. *-commutative 35.7%

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

    7. associate-*r/ 35.7%

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

    8. distribute-lft-neg-in 35.7%

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

    9. metadata-eval 35.7%

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

    10. metadata-eval 35.7%

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

    11. distribute-lft-in 35.7%

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

    12. neg-mul-1 35.7%

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

    13. distribute-lft-neg-in 35.7%

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

    14. metadata-eval 35.7%

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

    15. metadata-eval 35.7%

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

11. Simplified 35.7%

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

12. Taylor expanded in x around 0 26.6%

    \[\leadsto \frac{\frac{0.25 \cdot \frac{x}{s} + 0.5}{s}}{\color{blue}{2 + \frac{x}{s}}} \]

13. Taylor expanded in x around 0 51.2%

    \[\leadsto \frac{\frac{\color{blue}{0.5}}{s}}{2 + \frac{x}{s}} \]
14. Add Preprocessing

Alternative 10: 26.7% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 27.9%

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

Reproduce

herbie shell --seed 2024155 
(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))))))