Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 11.6s

Alternatives: 11

Speedup: N/A×

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, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{x\_m}{s}}\\ \frac{e^{-\mathsf{log1p}\left(t\_0\right)}}{s + \frac{s}{t\_0}} \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m s)))) (/ (exp (- (log1p t_0))) (+ s (/ s t_0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((x_m / s));
	return expf(-log1pf(t_0)) / (s + (s / t_0));
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(x_m / s))
	return Float32(exp(Float32(-log1p(t_0))) / Float32(s + Float32(s / t_0)))
end

Derivation

1. Initial program 99.2%

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

   1. *-commutative 99.2%

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

   2. fabs-neg 99.2%

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

   3. +-commutative 99.2%

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

   4. fabs-neg 99.2%

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

   5. distribute-lft-in 99.2%

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

   6. *-rgt-identity 99.2%

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

   7. +-commutative 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 99.2%

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

   1. associate-/r* 99.3%

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

7. Simplified 64.5%

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

   1. distribute-frac-neg2 64.5%

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

   2. rec-exp 64.4%

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

9. Applied egg-rr 64.4%

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

10. Taylor expanded in x around inf 64.5%

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

    1. distribute-lft-in 25.4%

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

    2. *-rgt-identity 25.4%

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

    3. mul-1-neg 25.4%

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

    4. rec-exp 25.4%

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

    5. rgt-mult-inverse 99.3%

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

    6. +-commutative 99.3%

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

12. Simplified 99.3%

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

    1. add-exp-log 99.3%

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

    2. log-rec 99.3%

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

    3. log1p-define 99.3%

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

14. Applied egg-rr 99.3%

    \[\leadsto \frac{\color{blue}{e^{-\mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}}{s + \frac{s}{e^{\frac{x}{s}}}} \]
15. Add Preprocessing

Alternative 2: 99.5% accurate, 2.9× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m s)))) (/ (/ 1.0 (+ t_0 1.0)) (+ s (/ s t_0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((x_m / s));
	return (1.0f / (t_0 + 1.0f)) / (s + (s / t_0));
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. *-commutative 99.2%

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

   2. fabs-neg 99.2%

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

   3. +-commutative 99.2%

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

   4. fabs-neg 99.2%

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

   5. distribute-lft-in 99.2%

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

   6. *-rgt-identity 99.2%

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

   7. +-commutative 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 99.2%

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

   1. associate-/r* 99.3%

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

7. Simplified 64.5%

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

   1. distribute-frac-neg2 64.5%

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

   2. rec-exp 64.4%

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

9. Applied egg-rr 64.4%

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

10. Taylor expanded in x around inf 64.5%

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

    1. distribute-lft-in 25.4%

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

    2. *-rgt-identity 25.4%

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

    3. mul-1-neg 25.4%

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

    4. rec-exp 25.4%

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

    5. rgt-mult-inverse 99.3%

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

    6. +-commutative 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 3: 99.5% accurate, 2.9× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m s)))) (/ 1.0 (* (+ s (/ s t_0)) (+ t_0 1.0)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. *-commutative 99.2%

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

   2. fabs-neg 99.2%

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

   3. +-commutative 99.2%

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

   4. fabs-neg 99.2%

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

   5. distribute-lft-in 99.2%

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

   6. *-rgt-identity 99.2%

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

   7. +-commutative 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 99.2%

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

   1. associate-/r* 99.3%

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

7. Simplified 64.5%

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

   1. distribute-frac-neg2 64.5%

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

   2. rec-exp 64.4%

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

9. Applied egg-rr 64.4%

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

10. Taylor expanded in x around inf 64.5%

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

    1. associate-*r* 64.4%

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

    2. distribute-lft-in 25.4%

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

    3. *-rgt-identity 25.4%

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

    4. mul-1-neg 25.4%

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

    5. rec-exp 25.4%

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

    6. rgt-mult-inverse 99.3%

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

    7. +-commutative 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 4: 96.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{1}{e^{\frac{x\_m}{s}} + 1}}{s + \frac{s}{\frac{x\_m}{s} + 1}} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (/ 1.0 (+ (exp (/ x_m s)) 1.0)) (+ s (/ s (+ (/ x_m s) 1.0)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. *-commutative 99.2%

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

   2. fabs-neg 99.2%

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

   3. +-commutative 99.2%

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

   4. fabs-neg 99.2%

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

   5. distribute-lft-in 99.2%

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

   6. *-rgt-identity 99.2%

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

   7. +-commutative 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 99.2%

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

   1. associate-/r* 99.3%

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

7. Simplified 64.5%

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

   1. distribute-frac-neg2 64.5%

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

   2. rec-exp 64.4%

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

9. Applied egg-rr 64.4%

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

10. Taylor expanded in x around inf 64.5%

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

    1. distribute-lft-in 25.4%

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

    2. *-rgt-identity 25.4%

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

    3. mul-1-neg 25.4%

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

    4. rec-exp 25.4%

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

    5. rgt-mult-inverse 99.3%

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

    6. +-commutative 99.3%

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

12. Simplified 99.3%

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

13. Taylor expanded in x around 0 62.5%

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

14. Final simplification 62.5%

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

Alternative 5: 96.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\left(e^{\frac{x\_m}{s}} + 1\right) \cdot \left(s + \frac{s}{\frac{x\_m}{s} + 1}\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ 1.0 (* (+ (exp (/ x_m s)) 1.0) (+ s (/ s (+ (/ x_m s) 1.0))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. *-commutative 99.2%

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

   2. fabs-neg 99.2%

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

   3. +-commutative 99.2%

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

   4. fabs-neg 99.2%

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

   5. distribute-lft-in 99.2%

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

   6. *-rgt-identity 99.2%

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

   7. +-commutative 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 99.2%

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

   1. associate-/r* 99.3%

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

7. Simplified 64.5%

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

   1. distribute-frac-neg2 64.5%

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

   2. rec-exp 64.4%

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

9. Applied egg-rr 64.4%

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

10. Taylor expanded in x around inf 64.5%

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

    1. associate-*r* 64.4%

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

    2. distribute-lft-in 25.4%

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

    3. *-rgt-identity 25.4%

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

    4. mul-1-neg 25.4%

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

    5. rec-exp 25.4%

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

    6. rgt-mult-inverse 99.3%

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

    7. +-commutative 99.3%

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

12. Simplified 99.3%

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

13. Taylor expanded in x around 0 62.5%

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

14. Final simplification 62.5%

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

Alternative 6: 95.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{1}{e^{\frac{x\_m}{s}} + 1}}{s + s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (/ 1.0 (+ (exp (/ x_m s)) 1.0)) (+ s s)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. *-commutative 99.2%

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

   2. fabs-neg 99.2%

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

   3. +-commutative 99.2%

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

   4. fabs-neg 99.2%

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

   5. distribute-lft-in 99.2%

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

   6. *-rgt-identity 99.2%

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

   7. +-commutative 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 99.2%

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

   1. associate-/r* 99.3%

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

7. Simplified 64.5%

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

   1. distribute-frac-neg2 64.5%

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

   2. rec-exp 64.4%

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

9. Applied egg-rr 64.4%

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

10. Taylor expanded in x around inf 64.5%

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

    1. distribute-lft-in 25.4%

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

    2. *-rgt-identity 25.4%

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

    3. mul-1-neg 25.4%

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

    4. rec-exp 25.4%

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

    5. rgt-mult-inverse 99.3%

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

    6. +-commutative 99.3%

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

12. Simplified 99.3%

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

13. Taylor expanded in s around inf 61.2%

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

14. Final simplification 61.2%

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

Alternative 7: 90.8% accurate, 25.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.500000013088254 \cdot 10^{-9}:\\ \;\;\;\;\frac{x\_m \cdot \left(\frac{0.125}{s} + \frac{0.25}{x\_m}\right) + -0.25 \cdot \left(x\_m \cdot \frac{0.5}{s}\right)}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 1.500000013088254e-9)
   (/ (+ (* x_m (+ (/ 0.125 s) (/ 0.25 x_m))) (* -0.25 (* x_m (/ 0.5 s)))) s)
   0.0))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 1.500000013088254e-9f) {
		tmp = ((x_m * ((0.125f / s) + (0.25f / x_m))) + (-0.25f * (x_m * (0.5f / s)))) / s;
	} else {
		tmp = 0.0f;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 1.500000013088254e-9) then
        tmp = ((x_m * ((0.125e0 / s) + (0.25e0 / x_m))) + ((-0.25e0) * (x_m * (0.5e0 / s)))) / s
    else
        tmp = 0.0e0
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(1.500000013088254e-9))
		tmp = Float32(Float32(Float32(x_m * Float32(Float32(Float32(0.125) / s) + Float32(Float32(0.25) / x_m))) + Float32(Float32(-0.25) * Float32(x_m * Float32(Float32(0.5) / s)))) / s);
	else
		tmp = Float32(0.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(1.500000013088254e-9))
		tmp = ((x_m * ((single(0.125) / s) + (single(0.25) / x_m))) + (single(-0.25) * (x_m * (single(0.5) / s)))) / s;
	else
		tmp = single(0.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.50000001e-9

   1. Initial program 98.9%

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

      1. fabs-neg 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-frac-neg2 98.9%

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

      4. fabs-neg 98.9%

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

      5. *-commutative 98.9%

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

      6. fabs-neg 98.9%

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

      7. +-commutative 98.9%

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

      8. fabs-neg 98.9%

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

   3. Simplified 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in s around inf 70.3%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot e^{-\log 2} + 0.5 \cdot \frac{e^{-\log 2} \cdot \left(x - 0.5 \cdot x\right)}{s}\right) - 0.25 \cdot \frac{x \cdot e^{-\log 2}}{s}}{s}} \]
   10. Step-by-step derivation

   11. Simplified 55.3%

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

   12. Taylor expanded in x around inf 69.6%

       \[\leadsto \frac{\color{blue}{x \cdot \left(0.125 \cdot \frac{1}{s} + 0.25 \cdot \frac{1}{x}\right)} + -0.25 \cdot \left(x \cdot \frac{0.5}{s}\right)}{s} \]
   13. Step-by-step derivation

       1. associate-*r/ 69.6%

          \[\leadsto \frac{x \cdot \left(\color{blue}{\frac{0.125 \cdot 1}{s}} + 0.25 \cdot \frac{1}{x}\right) + -0.25 \cdot \left(x \cdot \frac{0.5}{s}\right)}{s} \]

       2. metadata-eval 69.6%

          \[\leadsto \frac{x \cdot \left(\frac{\color{blue}{0.125}}{s} + 0.25 \cdot \frac{1}{x}\right) + -0.25 \cdot \left(x \cdot \frac{0.5}{s}\right)}{s} \]

       3. associate-*r/ 69.6%

          \[\leadsto \frac{x \cdot \left(\frac{0.125}{s} + \color{blue}{\frac{0.25 \cdot 1}{x}}\right) + -0.25 \cdot \left(x \cdot \frac{0.5}{s}\right)}{s} \]

       4. metadata-eval 69.6%

          \[\leadsto \frac{x \cdot \left(\frac{0.125}{s} + \frac{\color{blue}{0.25}}{x}\right) + -0.25 \cdot \left(x \cdot \frac{0.5}{s}\right)}{s} \]

   14. Simplified 69.6%

       \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{0.125}{s} + \frac{0.25}{x}\right)} + -0.25 \cdot \left(x \cdot \frac{0.5}{s}\right)}{s} \]

   if 1.50000001e-9 < x

   1. Initial program 99.9%

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

      1. fabs-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. fabs-neg 99.9%

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

      5. *-commutative 99.9%

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

      6. fabs-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. fabs-neg 99.9%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 64.7%

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

      1. associate-*r/ 64.7%

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

      2. *-rgt-identity 64.7%

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

   8. Simplified 64.7%

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

   9. Taylor expanded in s around inf 58.5%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot e^{-\log 2} + 0.5 \cdot \frac{e^{-\log 2} \cdot \left(x - 0.5 \cdot x\right)}{s}\right) - 0.25 \cdot \frac{x \cdot e^{-\log 2}}{s}}{s}} \]
   10. Step-by-step derivation

   11. Simplified 48.7%

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

   12. Taylor expanded in s around 0 97.9%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + 0.25 \cdot \left(x + -0.5 \cdot x\right)}{s}}}{s} \]

   13. Taylor expanded in x around 0 97.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.1% accurate, 28.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;\frac{-0.25 \cdot \left(x\_m \cdot \frac{0.5}{s}\right) + \left(0.25 + \frac{x\_m \cdot 0.125}{s}\right)}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 4.999999980020986e-13)
   (/ (+ (* -0.25 (* x_m (/ 0.5 s))) (+ 0.25 (/ (* x_m 0.125) s))) s)
   0.0))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 4.999999980020986e-13f) {
		tmp = ((-0.25f * (x_m * (0.5f / s))) + (0.25f + ((x_m * 0.125f) / s))) / s;
	} else {
		tmp = 0.0f;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 4.999999980020986e-13) then
        tmp = (((-0.25e0) * (x_m * (0.5e0 / s))) + (0.25e0 + ((x_m * 0.125e0) / s))) / s
    else
        tmp = 0.0e0
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(4.999999980020986e-13))
		tmp = Float32(Float32(Float32(Float32(-0.25) * Float32(x_m * Float32(Float32(0.5) / s))) + Float32(Float32(0.25) + Float32(Float32(x_m * Float32(0.125)) / s))) / s);
	else
		tmp = Float32(0.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(4.999999980020986e-13))
		tmp = ((single(-0.25) * (x_m * (single(0.5) / s))) + (single(0.25) + ((x_m * single(0.125)) / s))) / s;
	else
		tmp = single(0.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999998e-13

   1. Initial program 99.0%

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

      1. fabs-neg 99.0%

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

      2. distribute-frac-neg 99.0%

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

      3. distribute-frac-neg2 99.0%

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

      4. fabs-neg 99.0%

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

      5. *-commutative 99.0%

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

      6. fabs-neg 99.0%

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

      7. +-commutative 99.0%

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

      8. fabs-neg 99.0%

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

   3. Simplified 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. associate-*r/ 99.0%

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

      2. *-rgt-identity 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in s around inf 70.5%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot e^{-\log 2} + 0.5 \cdot \frac{e^{-\log 2} \cdot \left(x - 0.5 \cdot x\right)}{s}\right) - 0.25 \cdot \frac{x \cdot e^{-\log 2}}{s}}{s}} \]
   10. Step-by-step derivation

   11. Simplified 55.7%

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

   12. Taylor expanded in x around 0 55.1%

       \[\leadsto \frac{\left(0.25 + \color{blue}{0.125 \cdot \frac{x}{s}}\right) + -0.25 \cdot \left(x \cdot \frac{0.5}{s}\right)}{s} \]
   13. Step-by-step derivation

       1. associate-*r/ 55.7%

          \[\leadsto \frac{\left(0.25 + \color{blue}{\frac{0.125 \cdot x}{s}}\right) + -0.25 \cdot \left(x \cdot \frac{0.5}{s}\right)}{s} \]

       2. *-commutative 55.7%

          \[\leadsto \frac{\left(0.25 + \frac{\color{blue}{x \cdot 0.125}}{s}\right) + -0.25 \cdot \left(x \cdot \frac{0.5}{s}\right)}{s} \]

   14. Simplified 55.7%

       \[\leadsto \frac{\left(0.25 + \color{blue}{\frac{x \cdot 0.125}{s}}\right) + -0.25 \cdot \left(x \cdot \frac{0.5}{s}\right)}{s} \]

   if 4.99999998e-13 < x

   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. fabs-neg 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. distribute-frac-neg2 99.7%

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

      4. fabs-neg 99.7%

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

      5. *-commutative 99.7%

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

      6. fabs-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. fabs-neg 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 66.3%

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

      1. associate-*r/ 66.4%

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

      2. *-rgt-identity 66.4%

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

   8. Simplified 66.4%

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

   9. Taylor expanded in s around inf 58.8%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot e^{-\log 2} + 0.5 \cdot \frac{e^{-\log 2} \cdot \left(x - 0.5 \cdot x\right)}{s}\right) - 0.25 \cdot \frac{x \cdot e^{-\log 2}}{s}}{s}} \]
   10. Step-by-step derivation

   11. Simplified 48.5%

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

   12. Taylor expanded in s around 0 95.0%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + 0.25 \cdot \left(x + -0.5 \cdot x\right)}{s}}}{s} \]

   13. Taylor expanded in x around 0 95.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;\frac{-0.25 \cdot \left(x \cdot \frac{0.5}{s}\right) + \left(0.25 + \frac{x \cdot 0.125}{s}\right)}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.2% accurate, 32.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{x\_m \cdot -0.125 + \left(s \cdot 0.25 + 0.25 \cdot \left(x\_m + x\_m \cdot -0.5\right)\right)}{s}}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (/ (+ (* x_m -0.125) (+ (* s 0.25) (* 0.25 (+ x_m (* x_m -0.5))))) s) s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (((x_m * -0.125f) + ((s * 0.25f) + (0.25f * (x_m + (x_m * -0.5f))))) / s) / s;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = (((x_m * (-0.125e0)) + ((s * 0.25e0) + (0.25e0 * (x_m + (x_m * (-0.5e0)))))) / s) / s
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(Float32(x_m * Float32(-0.125)) + Float32(Float32(s * Float32(0.25)) + Float32(Float32(0.25) * Float32(x_m + Float32(x_m * Float32(-0.5)))))) / s) / s)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (((x_m * single(-0.125)) + ((s * single(0.25)) + (single(0.25) * (x_m + (x_m * single(-0.5)))))) / s) / s;
end

Derivation

1. Initial program 99.2%

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

   1. fabs-neg 99.2%

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

   2. distribute-frac-neg 99.2%

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

   3. distribute-frac-neg2 99.2%

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

   4. fabs-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. fabs-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. fabs-neg 99.2%

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

3. Simplified 99.3%

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

6. Applied egg-rr 87.1%

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

   1. associate-*r/ 87.1%

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

   2. *-rgt-identity 87.1%

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

8. Simplified 87.1%

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

9. Taylor expanded in s around inf 66.2%

   \[\leadsto \color{blue}{\frac{\left(0.5 \cdot e^{-\log 2} + 0.5 \cdot \frac{e^{-\log 2} \cdot \left(x - 0.5 \cdot x\right)}{s}\right) - 0.25 \cdot \frac{x \cdot e^{-\log 2}}{s}}{s}} \]
10. Step-by-step derivation

11. Simplified 53.1%

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

12. Taylor expanded in s around 0 88.9%

    \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + \left(0.25 \cdot s + 0.25 \cdot \left(x + -0.5 \cdot x\right)\right)}{s}}}{s} \]

13. Final simplification 88.9%

    \[\leadsto \frac{\frac{x \cdot -0.125 + \left(s \cdot 0.25 + 0.25 \cdot \left(x + x \cdot -0.5\right)\right)}{s}}{s} \]
14. Add Preprocessing

Alternative 10: 85.6% accurate, 77.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 3.99999992980668e-14) (/ 0.25 s) 0.0))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 3.99999992980668e-14f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.0f;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 3.99999992980668e-14) then
        tmp = 0.25e0 / s
    else
        tmp = 0.0e0
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(3.99999992980668e-14))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(0.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(3.99999992980668e-14))
		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 < 3.99999993e-14

   1. Initial program 98.9%

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

      1. fabs-neg 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-frac-neg2 98.9%

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

      4. fabs-neg 98.9%

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

      5. *-commutative 98.9%

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

      6. fabs-neg 98.9%

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

      7. +-commutative 98.9%

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

      8. fabs-neg 98.9%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in s around inf 34.6%

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

   if 3.99999993e-14 < x

   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. fabs-neg 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. distribute-frac-neg2 99.7%

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

      4. fabs-neg 99.7%

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

      5. *-commutative 99.7%

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

      6. fabs-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. fabs-neg 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 67.0%

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

      1. associate-*r/ 67.1%

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

      2. *-rgt-identity 67.1%

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

   8. Simplified 67.1%

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

   9. Taylor expanded in s around inf 59.6%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot e^{-\log 2} + 0.5 \cdot \frac{e^{-\log 2} \cdot \left(x - 0.5 \cdot x\right)}{s}\right) - 0.25 \cdot \frac{x \cdot e^{-\log 2}}{s}}{s}} \]
   10. Step-by-step derivation

   11. Simplified 49.6%

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

   12. Taylor expanded in s around 0 94.1%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + 0.25 \cdot \left(x + -0.5 \cdot x\right)}{s}}}{s} \]

   13. Taylor expanded in x around 0 94.1%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.7% accurate, 620.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 0.0)

C

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

Fortran

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

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(0.0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 99.2%

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

   1. fabs-neg 99.2%

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

   2. distribute-frac-neg 99.2%

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

   3. distribute-frac-neg2 99.2%

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

   4. fabs-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. fabs-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. fabs-neg 99.2%

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

3. Simplified 99.3%

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

6. Applied egg-rr 87.1%

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

   1. associate-*r/ 87.1%

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

   2. *-rgt-identity 87.1%

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

8. Simplified 87.1%

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

9. Taylor expanded in s around inf 66.2%

   \[\leadsto \color{blue}{\frac{\left(0.5 \cdot e^{-\log 2} + 0.5 \cdot \frac{e^{-\log 2} \cdot \left(x - 0.5 \cdot x\right)}{s}\right) - 0.25 \cdot \frac{x \cdot e^{-\log 2}}{s}}{s}} \]
10. Step-by-step derivation

11. Simplified 53.1%

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

12. Taylor expanded in s around 0 75.2%

    \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + 0.25 \cdot \left(x + -0.5 \cdot x\right)}{s}}}{s} \]

13. Taylor expanded in x around 0 75.2%

    \[\leadsto \color{blue}{0} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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))))))